Operator Theory and Indefinite Inner Product Spaces : Presented on the Occasion of the Retirement of Heinz Langer in the Colloquium on Operator Theory,

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Operator Theory and Indefinite Inner Product Spaces Presented on the occasion of the retirement of Heinz Langer in the Colloquium on Operator Theory, Vienna, March 2004

Matthias Langer Annemarie Luger Harald Woracek Editors

Birkhäuser Verlag Basel . Boston . Berlin

Editors: Matthias Langer Department of Mathematics University of Strathclyde 26 Richmond Street Glasgow G1 1XH UK e-mail: [email protected]

Annemarie Luger Harald Woracek Institut für Analysis und Scientific Computing Technische Universität Wien Wiedner Hauptstrasse 8–10 / 101 1040 Wien Austria e-mail: [email protected] [email protected]

2000 Mathematics Subject Classification Primary 46C20, 47B50; Secondary 34L05, 47A57, 47A75

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

ISBN 3-7643-7515-9 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2006 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ' Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN-10: 3-7643-7515-9 e-ISBN: 3-7643-7516-7 ISBN-13: 978-3-7643-7515-7 987654321

www.birkhauser.ch

Heinz Langer

Table of Contents Introduction Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Laudation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Speech of Heinz Langer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Conference Programme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv Bibliography of Heinz Langer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx

Research articles V.M. Adamyan and I.M. Tkachenko General Solution of the Stieltjes Truncated Matrix Moment Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo Q-functions of Quasi-selfadjoint Contractions . . . . . . . . . . . . . . . . . . . . . . . .

23

J. Behrndt A Class of Abstract Boundary Value Problems with Locally Definitizable Functions in the Boundary Condition . . . . . . . . . . .

55

´ P. Binding and B. Curgus Riesz Bases of Root Vectors of Indefinite Sturm-Liouville Problems with Eigenparameter Dependent Boundary Conditions, I . . . . . . . . . . . . .

75

A. Dijksma, A. Luger and Y. Shondin Minimal Models for Nκ∞ -functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A. Fleige, S. Hassi, H.S.V. de Snoo and H. Winkler Generalized Friedrichs Extensions Associated with Interface Conditions for Sturm-Liouville Operators . . . . . . . . . . . . . . . . . . . 135 K.-H. F¨ orster and B. Nagy Spectral Properties of Operator Polynomials with Nonnegative Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 B. Fritzsche, B. Kirstein and A. Lasarow Szeg˝o Pairs of Orthogonal Rational Matrix-valued Functions on the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 M. Kaltenb¨ ack, H. Winkler and H. Woracek Singularities of Generalized Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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L.V. Mikaelyan Orthogonal Polynomials on the Unit Circle with Respect to a Rational Weight Function . . . . . . . . . . . . . . . . . . . . . . . . . 249 D. Popovici Bi-dimensional Moment Problems and Regular Dilations . . . . . . . . . . . . . 257 U. Prells and P. Lancaster Isospectral Vibrating Systems, Part 2: Structure Preserving Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 V. Strauss A Functional Description for the Commutative W J ∗ -algebras of the Dκ+ -class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 F.H. Szafraniec On Normal Extensions of Unbounded Operators: IV. A Matrix Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337

B. Textorius Directing Mappings in Kre˘ın Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351

Survey Article Z. Sasv´ ari The Extension Problem for Positive Definite Functions. A Short Historical Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Preface In this volume we present a collection of research papers which mainly follow lectures given at the “Colloquium on Operator Theory”. This conference was held at the Vienna University of Technology in March 2004 on the occasion of the retirement of Heinz Langer. The present volume of the series “Operator Theory: Advances and Applications” is dedicated to him and his scientific work. The book starts with an introductory part which provides some information about the colloquium itself. We have also included the laudation given by Aad Dijksma and a list of the recent publications of Heinz Langer, which updates his bibliography given in OT 106. The main part of the book consists of fifteen original research papers, which deal with various aspects of operator theory and indefinite inner product spaces. It concludes with a historical survey on the theory of positive definite functions. It would not have been possible to bring together so many colleagues in Vienna without financial support provided by several organizations. We wish to thank the Außeninstitut der TU Wien ¨ ¨ Osterreichische Forschungsgemeinschaft (OFG) Research Training Network HPRN-CT-2000-00116 of the European Union Vienna Convention Bureau ¨ Osterreichisch-Ukrainisches Kooperationsb¨ uro ¨ ¨ Osterreichische Mathematische Gesellschaft (OMG) and everybody who has contributed to make this conference and this book possible. Our special thanks also go to the referees who did a lot of work and in some cases made valuable and essential suggestions, which decisively improved the quality of the papers. Photos were provided by Dr. Mathias Beiglb¨ ock (conference photo) and Peter Geisenberger (photo of Heinz Langer). As organizers of the “Colloquium on Operator Theory” it was a great pleasure for us that so many colleagues participated in the conference; as editors of the present volume we are happy that a large part of them decided to contribute to these proceedings. We greatly acknowledge the interesting experience.

Vienna, July 2005

Matthias Langer Annemarie Luger Harald Woracek (the editors)

Laudation By Aad Dijksma, Vienna, March 4, 2004 Dear Vice-Rector Kaiser, Dean Dorninger, and Professor Inge Troch, dear Heinz, dear colleagues, Ladies and Gentlemen.

Introduction Heinz Langer is a world leading expert in spectral analysis and its applications, in particular in operator theory on spaces with an indefinite inner product. He has moved mathematical boundaries and opened new areas of research. He has the talent to focus on what is fundamental and to give directions to what is possible. By sharing his ideas he has stimulated many to explore unknown mathematical territory. Heinz is co-author of a book with M.G. Krein and I.S. Iokhvidov and he has published more than 170 papers with 45 co-authors from all over the world. He has directed the research of about 25 Ph.D. students. His results are still being applied and quoted in international journals in mathematics and theoretical physics. As one of his collaborators I can attest to the fact that Heinz is still very active. When the organizers of this workshop invited me to give this laudation speech I immediately said yes. I deem it an honor to pay tribute to Heinz. Not only because I admire his work, but also because I have learned a lot from him and still do and because I consider him a close friend. I feel privileged and pleased to speak on this occasion. First I will outline Heinz’s biography and highlight his successful years at the Vienna University of Technology. Then I will try to clarify what I meant when I said that Heinz has moved mathematical boundaries and opened new areas of research by discussing some of the main themes in his work. Finally, I will speak about Heinz’s connection with The Netherlands and especially with Groningen.

Biography Heinz Langer was born in Dresden on August 8, 1935. He lived in Dresden for almost 55 years. He attended the Gymnasium there, studied mathematics at the Technical University of Dresden, where he became an assistant and where he obtained his Ph.D. in 1960 and his Habilitation in 1965. He was appointed professor in mathematics at his Alma Mater in 1966, at the age of 31. At that time there was officially no research group in analysis, let alone functional analysis, so Heinz joined the group in stochastics headed by his Ph.D. advisor Prof. P.H. M¨ uller. He did research in and lectured on semigroups, one-dimensional Markov processes, and the spectral theory of Krein-Feller differential operators, but with Ph.D. students and colleagues from abroad he could work in his favorite topic: operator theory. Heinz declined a position at the prestigious Mathematical Institute of the Academy of Sciences of the GDR, because he liked to teach and to work with Ph.D. and post-doctoral students.

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At the beginning of his career Heinz spent two years abroad, first as a postdoc in Odessa in 1961/62 upon the invitation of M.G. Krein and later in 1966/67, shortly after his appointment as professor, on a fellowship of the National Research Council of Canada which was arranged by Professor I. Halperin in Toronto. The contact with Krein began with Heinz posting a handwritten manuscript in German, in the mailbox at the central station of Dresden on December 31, 1959. I will say more about the contents of the manuscript when I speak about the main themes in Heinz’s work. The subsequent stay in Odessa and the many shorter visits after that had a tremendous influence on Heinz, his career in mathematics, and, I think, on his style of doing mathematics. When Heinz mentions Krein it is with great affection and respect, and I know that Krein thought of Heinz as one his most brilliant students and collaborators. In between the two trips abroad Heinz had married Elke and in May 1967 their daughter Henriette was born. In the 1970’s and 1980’s Heinz spent several extended periods away from home in, for example, Jyv¨ askyl¨ a, Stockholm, Uppsala, Link¨ oping, Antwerp, Groningen, Amsterdam, and Regensburg. In all these places Heinz left his mark. He established some form of research co-operation, set up Ph.D. projects and made it possible for students and colleagues to come to Dresden. My account on Heinz’s connection with Groningen later on serves as an example of his stimulating influence on other mathematicians. Heinz remained in Dresden until 1989. In October of that year, shortly before the fall of the Berlin wall, he and his family gave up hearth and home, left the GDR and went to Regensburg. Turbulent times followed. Thanks to Albert Schneider Heinz obtained a professorship for one year at the University of Dortmund. After that year, with the support of Reinhard Mennicken, Heinz became professor at the University of Regensburg. Finally, in August 1991, Heinz moved to Vienna where he began a new and successful period in his life and in his mathematical career. Heinz was offered the prestigious chair “Anwendungsorientierte Analysis” at the Vienna University of Technology, previously held by Professor Edmund Hlawka, but under a different name. The University could hardly have found a more worthy successor. For several years Heinz was chairman of the Institute. He was pragmatic and apparently did his job well because he was re-elected. In no time the Vienna University of Technology became an internationally renowned research center of operator theory with young people doing challenging research. In Vienna Heinz guided 7 Ph.D. and 3 post-doctoral students. The center attracted many visitors from all over the world for visits to do research and for workshops. During the past 12 years Heinz has organized 4 workshops, one in co-operation with the Schr¨ odinger Institute and one in 2001 when the Vienna University of Technology awarded Professor Israel Gohberg from Israel an honorary doctorate. He obtained various long term research grants including funds for Ph.D. and post-doctoral projects. Heinz has clear and practical ideas and a keen sense on what to apply for and on how to formulate it. Two of these grants were from the Austrian “Fonds zur F¨ orderung der Wissenschaftlichen Forschung (FWF),” the last of which was

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awarded very recently for a project on canonical systems. As to a third grant: Heinz is the leader of the Austrian-German team in a Research Training Network of the European Union, in which 10 universities from 9 different countries participate. In recognition of his work Heinz was elected corresponding member of the Austrian Academy of Sciences.

Main themes in his work Heinz’s lifelong mathematical interest has been in the theory of operators, in particular operators on indefinite inner product spaces and its applications. This subject was suggested by Professor P.H. M¨ uller, his thesis advisor. Heinz has initiated many new projects and made fundamental contributions to their development. His work draws the attention of several mathematicians and physicists. To illustrate this I elaborate on four of the main themes in his work: 1. From the Ph.D. period: the invariant subspace theorem. 2. From the Habilitation period: definitizable operators. 3. From the period 1975–1985: extension theory. 4. From the Vienna period: block operator matrices. These main themes are interconnected and overlap in time. The division in periods is made only to facilitate the exposition. The invariant subspace theorem Now I come back to the handwritten German manuscript. For his Ph.D. thesis, so in the late fifties and early sixties, Heinz read the papers by L.S. Pontryagin, M.G. Krein, and I.S. Iokhvidov. Pontryagin in 1944 published his famous theorem that a self-adjoint operator A in a Πκ space with a κ-dimensional negative subspace has a maximal invariant non-positive subspace and that the spectrum of A restricted to this subspace lies in the closed upper half-plane. In 1956 Krein gave a different proof of Pontryagin’s theorem using a fixed point theorem but then for unitary operators. Iokhvidov used the Cayley transform to show that the theorems of Pontryagin and Krein are equivalent. Heinz generalized Pontryagin’s theorem to self-adjoint operators on a Krein space. The sole assumption was a simple compact corner condition to have control over the operator when restricted to the possibly infinite-dimensional negative subspace of the Krein space. It was a truly remarkable achievement. Heinz wrote it up by hand and in German and sent the manuscript to Krein on New Year’s eve in 1959. It would later be the main result in Heinz’s Ph.D. thesis. Iokhvidov in Odessa had tried to prove the same theorem before but he had not succeeded. So naturally Krein was very interested in Heinz’s proof and he must have been impressed because in one of his Crimean lecture notes he refers to the generalized invariant subspace theorem as the Pontryagin-Langer theorem. In any case, Krein reacted by inviting Heinz to come to Odessa. The visit marked the beginning of a fruitful co-operation that lasted for more than 25 years, until the death of Krein in 1989, and resulted in 13 joint papers. But the story goes on: One of the first things they discovered was a beautiful and unexpected application of the invariant subspace theorem. With the

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help of this theorem they proved that a self-adjoint quadratic operator pencil has a root with some specified spectral properties. The two papers “On some mathematical principles in the linear theory of damped oscillations”, which deal with this factorization, led to new publications in spectral theory and applications to mechanics and physics. They are probably their most frequently cited joint papers. Definitizable operators In 1962 Krein and Heinz proved that a self-adjoint operator in a Pontryagin space has a generalized spectral function. Krein showed that an integral operator with a positive kernel gives rise to a positive operator on a Krein space and that this operator also has a generalized spectral function. Heinz discovered that these are two examples of a special class of self-adjoint operators on a Krein space, namely the class of definitizable operators. The concept and the name are due to Heinz. A self-adjoint operator A is definitizable if it has a nonempty resolvent set and for some polynomial p, p(A) is nonnegative. Like the compact corner condition in the Pontryagin-Langer theorem, definitizability is a way to keep control over the nonpositivity of the inner product of the Krein space. In his Habilitation submitted in 1965 Heinz shows that a definitizable operator has a generalized spectral function and he applies his theory to quadratic operator pencils. It is no exaggeration when I say that this work is a genuine corner stone in the spectral theory of operators in spaces with an indefinite metric. Like its Hilbert space counterpart, the spectral function in a Krein space has a wide range of applications such as to quadratic pencils, just mentioned, Sturm-Liouville problems with indefinite weight, elliptic problems, and variational principles. Extension theory The research of Heinz and Krein in the extension theory of symmetric operators in Pontryagin and Krein spaces began with generalizing Krein’s theory of generalized resolvents, resolvent matrices, and entire operators to an indefinite setting with applications involving new classes of meromorphic functions with finitely many ¨ poles and canonical systems. It culminated in the 4 seminal “Uber einige Fortsetzungsprobleme . . . .” papers published jointly with Krein in the period 1977–85. These papers are still quoted and applied today. In fact, they could be called trendsetters, because there is a growing interest in generalizing, where possible, positive definite results to an indefinite setting. Of the many examples I only mention the indefinite version of the de Branges theory of entire functions and canonical systems. The research in this area is carried out by the group around Heinz here at the Vienna University of Technology. The results are important and of a high quality. Heinz can be proud of the research team he leaves behind. Block operator matrices This topic was taken up by Heinz jointly with Reinhard Mennicken in Regensburg and came to bloom in Vienna. 2 × 2 block operator matrices are common in the theory of operators on Krein spaces. But now the emphasis is different. The problem is to describe the spectral properties of an operator defined on a product of two

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Hilbert spaces and given as a 2×2 block operator matrix in terms of the properties of the operator entries of the matrix. Typical examples come from mathematical physics or system theory, where the entries are differential operators of different order and hence unbounded. One of the first problems is to define the domain of definition of the block operator matrix. Many papers have appeared since 1991. They concern, for example, the location of the essential spectrum, the solution of a Riccati equation, and block diagonalization. A new concept initiated and further developed jointly with others in the last 5 years is that of the quadratic numerical range. It is a new tool for localizing the spectrum of a block operator matrix.

The Netherland connection Let me begin with how I came to meet Heinz. I wrote a letter to him with some results on differential operators with eigenvalue depending boundary conditions on February 25, 1980. Heinz responded on March 19, 1980 with detailed answers to my questions. Some years before, Rien Kaashoek had visited Heinz in Dresden and organized a return visit for Heinz to Amsterdam. This took place in October 1981 and Heinz used the opportunity to come to Groningen as well. This visit was an immediate success, we were on first name basis within a few minutes, and his visit to my house and family went as smooth as pie. With Henk de Snoo we started to work on classes of meromorphic functions which arose in extension theory of symmetric operators in spaces with an indefinite metric and applied the theory to self-adjoint boundary eigenvalue problems with eigenvalue depending boundary conditions. At one time Heinz predicted that what we were about to start would lead to many publications. Before I realized it, I asked “Oh, how many?” (I dislike vague remarks.) Heinz actually paused to think and answered “About 10.” I was impressed. How could he predict that many? But how right he was: Many papers have appeared since then. First jointly with Henk de Snoo, even more than 10, and later also with others. Since that first visit in 1981, we have met at numerous working visits, conferences, and workshops. When near the Mediterranean we would go out to swim. Out of all these contacts grew a friendship which I treasure very much. During his many visits to Groningen, Heinz generously shared many ideas with us and stimulated us to work on them by ourselves. They resulted in a master thesis and a Ph.D. thesis about extension theory and interpolation, some papers with postdoctoral students about extension theory and commutant lifting, and even a book about Schur functions, operator colligations, and reproducing kernel Pontryagin spaces which my co-authors D. Alpay, J. Rovnyak, and H.S.V. de Snoo and I dedicated to him in appreciation, admiration and amity. Right now Heinz and I together with Daniel Alpay (from Israel), Tomas Azizov, and Yuri Shondin (both from Russia) and others are working on two projects. One is about an indefinite version of the Schur algorithm. The other concerns singular perturbations of self-adjoint operators with applications to quantum physics and to, for example, the Bessel and Laguerre differential operators. I very much enjoy working with Heinz. His enthusiasm for the problem at hand is stimulating

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and I am impressed by the ease with which Heinz finds the right wording for a paper. Our style of working together depends on where we are. When in Vienna, we sit on a couch in Heinz’s office, we have scratch paper on our knees, and our hands are blue from writing, so to speak. When in Groningen, we stand before a big green blackboard and our hands are white from the chalk. Upon the invitation of Rien Kaashoek, Heinz regularly visited the Free University in Amsterdam. Heinz took part in the reading and examination committee for some of Rien’s Ph.D. students. He also gave a series of lectures at the Thomas Stieltjes Institute. Heinz was an important participant in the INTAS-projects which Rien arranged. As to mathematics, Heinz’s visits to the Free University resulted in joint works about solutions of the Riccati equation with André Ran and his students. Heinz was co-promotor for one of them. Heinz’s relation with the Netherlands is not restricted to the Free University and the State University of Groningen only. At present Heinz is a member of the international evaluation committee to evaluate the output over the period 1996–2001 of the Mathematics Departments of all Dutch Universities. It is yet another indication that Heinz is recognized as an authority in mathematics with an international reputation.

Concluding remark I have said a lot and inadvertently I may have omitted things I should have said, but I should finish. Allow me to make one last remark: some years ago at some place Heinz and I had to fill in some forms and we discovered that at the space where we had to write down our profession we had entered different things. I had written “Professor” and Heinz had filled in “Mathematician.” Heinz may have retired as a professor but I hope and wish that he will not retire as a mathematician for many years to come, for mathematics and for the sake of all of us, I am sure.

Speech of Heinz Langer By Heinz Langer, Vienna, March 4, 2004

Magnifizenz, Spectabilis, Inge and Aad, Ladies and Gentlemen! I feel very honoured by all these nice words, thank you very much indeed. And I am deeply touched by the fact that so many friends and colleagues have come to this conference, although at the beginning it was intended to be just a small meeting. My thanks go to the organizers, Institutsvorstand Inge Troch, Harald Woracek, Annemarie Luger, Matthias Langer, and Fritz Vogl, who have put a lot of efforts in bringing us together for these days. Many things have been said about the 45 years of my life as a mathematician. All this may look very smooth from outside, but in fact I am not somebody who likes to make long-term plans for the future. Looking back, I see two special features which have determined my life. The first one is that I often made decisions which at the time were rather unpopular and not understandable for others. Let me give three examples. At the time of my diploma at the end of the 1950’s, under the strong influence of Bourbaki, abstract linear topological spaces and, in particular, locally convex spaces were very much in fashion. However, I chose a more classical topic, namely operator theory in Hilbert spaces which at that time by some colleagues was considered a bit outdated, but from today’s viewpoint it turned out to be a very good decision. And I am very grateful to my teacher P.H. M¨ uller that he proposed to me to study indefinite inner product spaces. Another example is my stay in Odessa at the beginning of the sixties. Krein had invited me to spend a year with him within an existing exchange programme between the GDR and the Soviet Union. Officially the German-Soviet friendship was a big issue, but many people were feeling differently, and not all of my friends could understand why I wanted to follow this invitation. But I insisted: even after my first application in 1960 had been rejected for political reasons, I tried again and succeeded one year later in 1961. In fact, this year in Odessa was one of the most important turning points of my life. I was deeply impressed by Krein’s personality, and his school of Functional Analysis in Odessa did not only influence me strongly as a mathematician, but was also the origin of true friendships which last until today. And I am very happy that some of these friends are here. The third example is my decision not to return to the GDR in October 1989 and to give up a secure position for a rather uncertain future. But, of course, the price for this security was very high, and this brings me to the second special feature, which I mentioned before: during a great part of my life I had to find ways to cope with the boundary conditions imposed by the political system of the GDR. So it was not always possible to do what you wanted and when you wanted it.

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For example, – I was not allowed to study physics (which was maybe not the worst thing in the end), – I was allowed to go to Odessa only one year later, – contacts to colleagues in the West were very restricted, and permissions to travel abroad were often denied or delayed, like my first trip to Canada: I got the invitation from Israel Halperin to Toronto for 1965. This was not so long after the Berlin wall had been built, and so this first invitation was not even considered seriously by the authorities. However, Israel Halperin was very insistent and renewed the invitation for the next year, then even by a letter to the minister. And he succeeded. This was somehow a typical situation, and I am very grateful also to other colleagues who made it possible for me to visit them in the West by not giving up after a first or even a second failure: Ilppo Simo Louhivaara who at the time was rector in Jyv¨ askyl¨ a, Rudi Hirschfeld in Antwerp who visited the GDR with an official delegation from Belgium and on this occasion established contacts with me, G¨ oran Borg, who was the rector of the Royal Institute of Technology in Stockholm, and Bj¨ orn Textorius, ˚ Ake Pleijel in Uppsala, Hrvoje Kraljevic and the late Branko Najman in Zagreb, Rien Kaashoek and Israel Gohberg in Amsterdam, and finally Reinhard Mennicken in Regensburg. All these contacts to colleagues from abroad were very important for me, also because they allowed me to continue to work in operator theory while in Dresden I held a professorship in probability theory. And even in the most critical situation, when I arrived in West Germany in October 1989 and did not have a job, colleagues from there, in particular, Reinhard Mennicken and Albert Schneider from Dortmund helped me to keep my period of unemployment to 3 weeks. Now you may wonder how it came to this happy end in Austria. Well, I don’t know myself, and this is one of the big miracles in my life. Certainly, it had nothing to do with the fact that Austria had been present for me already since my childhood like a fairy tale. I am still keeping an Edelweiss in Meyers Ostalpen from my father which he found in 1926 at the Karlingerboden near Kaprun, and in our living room we had a painting called “Im Stubaital”. After the Second world war it was basically impossible to visit Austria from East Germany, and my first visit in 1955 via West Germany was illegal from the point of view of GDR and kind of adventurous. Hitchhiking with a friend from Munich to Salzburg, but in separate cars, we had agreed to meet at a bridge over the Salzach mentioned in my Baedeker from the 1920’s. I arrived first, but the bridge had disappeared . . . . Only 35 years later, in 1990, I could return to Austria, when I applied for the Chair previously held by Edmund Hlawka. That I finally got this position was for me like a dream came through. In the past almost 13 years here at the University of Technology in Vienna I had the freedom to follow my own scientific interests, to build up a research group in operator theory, and to cooperate with colleagues from all over the world. I am

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very grateful to the University and the former Institute of Analysis and Technical Mathematics for giving me this opportunity and supporting me in many ways. It has always been a pleasure to work in the friendly atmosphere here. And I also enjoyed teaching several generations of students of electrical engineering and mathematics. You may now have got the impression that I am just another example of a ¨ Saxonian as described by Franz Grillparzer in his “Loblied auf Osterreich” where it reads S’ist m¨ oglich, daß in Sachsen und beim Rhein Es Leute gibt, die mehr in B¨ uchern lasen than people in Austria, I should add. However, I really enjoy living in Vienna with all its theaters and museums, opera houses and concert halls, cafés and vineyards, and nearby mountains and valleys. And, being attentive, you can even find a number of close cultural ties between Vienna and Dresden, like the Albertina or Gottfried Semper’s architecture. I was happy when I received the Austrian citizenship 13 years ago. Indeed, I do now share the passionate love of the Austrians for this beautiful country and its rich culture. And I have at least tried to adopt the Austrian mentality which in his “Loblied” Grillparzer describes as follows: Allein, was not tut und was Gott gef¨ allt, Der klare Blick, der off’ne, richt’ge Sinn, ¨ Da tritt der Osterreicher hin vor jeden, denkt sich sein Teil und l¨ aßt die andern reden!

Programme of the “Colloquium on Operator Theory” Thursday, 4 March

9’15

Opening: H. Kaiser (Vice-Rector of the Vienna University of Technology) D. Dorninger (Dean of the Faculty for Mathematics and Geoinformation) I. Troch (Head of the Institute for Analysis and Scientific Computing) Laudation: A. Dijksma Heinz Langer

10’30 I. Gohberg: Continuous analogue of orthogonal polynomials Coffee break Chair: A. Luger 11’30 C. Tretter: Spectral theory of block operator matrices and applications in mathematical physics 12’15 V. Adamyan: Matrix continuous analogues of orthogonal trigonometric polynomials 12’45 Organizational remarks, conference photo Lunch break Chair: R. Mennicken 14’30 P. Lancaster: An inverse quadratic eigenvalue problem 15’00 H. de Snoo: Singular Sturm–Liouville problems nonlinear in the eigenvalue parameter ¨ck: Indefinite canonical systems and 15’30 M. Kaltenba the inverse spectral theorem Coffee break Chair: A. Fleige 16’30 F. Szafraniec: q-disease in operator theory: some cases 17’00 B. Textorius: Directing mappings in Krein spaces 17’30 H. Winkler: Isometric isomorphisms between strings and canonical systems

Programme

xxi

Friday, 5 March Chair: B. Kirstein 9’00

A. Dijksma: The algorithm of Issai Schur in an indefinite setting

9’45

¨ rster: On matrix and operator polynomials with K.-H. Fo nonnegative coefficients Coffee break

Chair: O. Staffans 11’00 A. Ran: Some remarks on LQ-optimal control: asymptotics and the inverse problem 11’30 A. Gheondea: The indefinite Caratheodory problem 12’00 D. Alpay: Rational functions and backward shift operators in the hyperholomorphic case Coffee break Chair: B. Textorius 13’00 M. Brown: Inverse resonance problems for the Sturm–Liouville problem and for the Jacobi matrix 13’30 P. Binding: Oscillation of indefinite Sturm–Liouville eigenfunctions ¨ ller: The spectrum of the multiplication operator 14’00 M. Mo associated with a family of operators in a Banach space Afternoon in Vienna Conference dinner

xxii

Programme

Saturday, 6 March, morning Chair: M. Langer 9’00

P. Kurasov: Pontryagin type models for soliton potentials: inverse scattering method for operator extensions Chair: H. de Snoo

9’30

10’00 10’30

D. Volok: De Branges– Rovnyak spaces and Schur functions: the hyperholomorphic case X. Mary: Subdualities and associated (reproducing) kernels

S. Hassi: Boundary relations and Weyl families of symmetric operators

Y. Shondin: Pontryagin space boundary value problems for a singular differential expression

V. Strauss: On J-symmetric operators with square similar to a bounded symmetric operator

A. Batkai: Polynomial stability of operator semigroups

Coffee break Chair: A. Ran 11’30

´ B. Curgus: Indefinite Sturm–Liouville problems with eigenparameter dependent boundary conditions Chair: P. Binding

12’00

12’30

C. Trunk: Spectral points of type π+ for closed operators in Krein spaces J. Behrndt: Finite-dimensional perturbations of locally definitizable selfadjoint operators in Krein spaces

D. Popovici: Moment theorems for commuting multi-operators H. Lundmark: Direct and inverse spectral problem for a non-selfadjoint third order generalization of the discrete string equation

Lunch break

Programme

Saturday, 6 March, afternoon Chair: C. Tretter 14’30

V. Pivovarchik: Shifted Hermite–Biehler functions Chair: C. Trunk

15’00

G. Wanjala: The Schur transform at the boundary point z = 1

15’30

L. K´ erchy: Canonical factorization of vectors with respect to an operator

A. Lasarow: Some basic facts on orthogonal rational matrix-valued functions on the unit circle L. Mikayelyan: Orthogonal polynomials on the unit circle with respect to a rational weight function

Coffee break Chair: H. Woracek 16’30 17’15

´ri: The extension problem for positive definite functions. Z. Sasva A historical survey M. Kaashoek: Metric constrained interpolation problems and control theory Closing

xxiii

Participants of the “Colloquium on Operator Theory”

28

36

59

34

6

53

26

46

19

4

9

37

54

50

24

33

55

8

42

14

3

15

57

16

25

35

10

39

41

43

49

23

1

40

44

22

27

45

18

51

5

13

7 29

21

30

32 2

12 47 20

38

17

58

31

56

Participants xxv

xxvi

Participants

List of Participants 1. Vadim Adamyan, I.I. Mechnikov Odessa National University, Ukraine, [email protected], [email protected] 2. Daniel Alpay, Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel, [email protected] 3. Andras Batkai, Department of Applied Analysis, ELTE TTK, Budapest, Hungary, [email protected] 4. Jussi Behrndt, Institut f¨ ur Mathematik, TU-Berlin, Germany, [email protected] 5. Christa Binder, Institut f¨ ur Analysis und Scientific Computing, TU Wien, Austria, [email protected] 6. Paul Binding, Department of Mathematics and Statistics, University of Calgary, Canada, [email protected] 7. Bernhard Bodenstorfer, Frankfurt/Main, Germany, [email protected] 8. Malcolm Brown, Department of Computer Science, Cardiff University, United Kingdom, [email protected] ´ 9. Branko Curgus, Department of Mathematics, Western Washington University, United States, [email protected] 10. Aad Dijksma, Department of Mathematics, University of Groningen, The Netherlands, [email protected] 11. David Eschwé, Wien, Austria, [email protected] 12. Andreas Fleige, Dortmund, Germany, [email protected] 13. Karl-Heinz Förster, Institut f¨ ur Mathematik, TU-Berlin, Germany, [email protected] 14. Aurelian Gheondea, Department of Mathematics, Bilkent University, Ankara, Turkey, [email protected] 15. Israel Gohberg, School of Mathematical Sciences, Tel-Aviv University, Israel, [email protected]

Participants

xxvii

16. Seppo Hassi, Department of Mathematics and Statistics, University of Vaasa, Finland, [email protected] 17. Rien Kaashoek, Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands, [email protected] 18. Michael Kaltenbäck, Institut f¨ ur Analysis und Scientific Computing, TU Wien, Austria, [email protected] 19. Victor Katsnelson, Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot, Israel, [email protected] 20. László Kérchy, University of Szeged, Hungary, [email protected] 21. Bernd Kirstein, Mathematisches Institut, Universität Leipzig, Germany, [email protected] 22. Hrvoje Kraljevic, Zagreb, Croatia, [email protected] 23. Pavel Kurasov, Department of Mathematics, Lund Institute of Technology, Sweden, [email protected] 24. Peter Lancaster, University of Calgary, Canada, [email protected] 25. Heinz Langer, Institut f¨ ur Analysis und Scientific Computing, TU Wien, Austria, [email protected] 26. Matthias Langer, Institut f¨ ur Analysis und Scientific Computing, TU Wien, Austria, [email protected] 27. Andreas Lasarow, Departement Computerwetenschappen, K. U. Leuven, Heverlee, Belgium, [email protected] 28. Annemarie Luger, Institut f¨ ur Analysis und Scientific Computing, TU Wien, Austria, [email protected] 29. Hans Lundmark, Department of Mathematics, Link¨ oping University, Sweden, [email protected] 30. Xavier Mary, ENSAE – CREST LS, 3, avenue Pierre Larousse, 92245 Malakoff Cedex, Paris, France, [email protected]

xxviii

Participants

31. Vladimir Matsaev, School of Mathematics, Tel Aviv University, Israel, [email protected] 32. Reinhard Mennicken, Naturwissenschaftliche Fakult¨ at I, Universit¨ at Regensburg, Germany, [email protected] 33. Levon Mikaelyan, Department of Informatics and Applied Mathematics, Yerevan State University, Armenia, [email protected] 34. Manfred M¨ oller, School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa, [email protected] 35. Maria Magdalena Nafalska, Institut f¨ ur Mathematik, TU-Berlin, Germany, m [email protected] 36. Vjacheslav Pivovarchik, Odessa State Academy of Structure and Architecture, Ukraine, [email protected], [email protected] 37. Dan Popovici, Department of Mathematics and Computer Science, University of the West Timi¸soara, Romania, [email protected] 38. Andre Ran, Department of Mathematics/FEW, Vrije Universiteit, Amsterdam, The Netherlands, [email protected] 39. Adrian Sandovici, Department of Mathematics, University of Groningen, The Netherlands, [email protected] 40. Zoltán Sasvári, Institut f¨ ur Mathematische Stochastik, TU Dresden, Germany, [email protected] 41. Wilfried Schenk, Institut f¨ ur Mathematische Stochastik, TU Dresden, Germany, [email protected] 42. Yuri G. Shondin, Faculty of Physics, Nizhny Novgorod State Pedagogical University, Russia, [email protected] 43. Henk de Snoo, Department of Mathematics, University of Groningen, The Netherlands, [email protected] 44. Olof Staffans, Abo Akademi, Finland, [email protected]

Participants

xxix

45. Vladimir Strauss, Depto de Matemáticas Puras y Aplicadas, Universidad Simón, Caracas, Venezuela, [email protected] 46. Franciszek Hugon Szafraniec, Instytut Matematyki, Uniwersytet Jagiello´ nski, Krak´ ow, Poland, [email protected], [email protected] 47. Bj¨ orn Textorius, Department of Mathematics, Link¨ oping University, Sweden, [email protected] 48. Igor Tkachenko, Polytechnic University of Valencia, Spain, [email protected] 49. Christiane Tretter, FB 3 – Mathematik, Universit¨ at Bremen, Germany, [email protected] 50. Inge Troch, Institut f¨ ur Analysis und Scientific Computing, TU Wien, Austria, [email protected] 51. Carsten Trunk, Institut f¨ ur Mathematik, TU-Berlin, Germany, [email protected] 52. Fritz Vogl, Institut f¨ ur Analysis und Scientific Computing, TU Wien, Austria, [email protected] 53. Dan Volok, Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel, [email protected] 54. Markus Wagenhofer, FB 3 – Mathematik, Universit¨ at Bremen, Germany, [email protected] 55. Gerald Wanjala, Department of Mathematics, University of Groningen, The Netherlands, [email protected] 56. Henrik Winkler, Department of Mathematics, University of Groningen, The Netherlands, [email protected] 57. Monika Winklmeier, FB 3 – Mathematik, Universit¨ at Bremen, Germany, [email protected] 58. Harald Woracek, Institut f¨ ur Analysis und Scientific Computing, TU Wien, Austria, [email protected] 59. Rosalinde Pohl (Secretary)

Bibliography of Heinz Langer In the volume OT 106 of the series “Operator Theory: Advances and Applications” which appeared in 1998, a bibliography of Heinz Langer up to this year was compiled. The following list updates this bibliography; it collects his recent papers. [131] Direct and inverse spectral problems for generalized strings, Integral Equations Operator Theory 30 (1998), 409–431 (with H. Winkler) [132] A factorization result for generalized Nevanlinna functions, Integral Equations Operator Theory 36 (2000), 121–125 (with A. Dijksma, A. Luger, and Yu. Shondin) [133] Classical Nevanlinna–Pick interpolation with real interpolation points, Oper. Theory Adv. Appl. 115 (2000), 1–50 (with D. Alpay and A. Dijksma) [134] The spectral shift function for certain block operator matrices, Math. Nachr. 211 (2000), 5–24 (with V. Adamjan) [135] A class of 2 × 2–matrix functions, Glas. Matem. Ser. III 35(55) (2000), 149–160 (with A. Luger) [136] Linearization and compact perturbation of self-adjoint analytic operator functions, Oper. Theory Adv. Appl. 118 (2000), 255–285 (with A. Markus and V. Matsaev) [137] Self-adjoint differential operators with inner singularities and Pontryagin spaces, Oper. Theory Adv. Appl. 118 (2000), 105–175 (with A. Dijksma, Yu. Shondin, and C. Zeinstra) [138] On singular critical points of positive operators in a Krein space, Proc. Amer. ´ Math. Soc. 128 (2000), 2621–2626 (with B. Curgus and A. Gheondea) [139] Variational principles for real eigenvalues of selfadjoint operator pencils, Integral Equations Operator Theory 38 (2000), 190–206 (with D. Eschwé and P. Binding) [140] Dissipative eigenvalue problems for a Sturm–Liouville operator with a singular potential, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 1237–1257 (with B. Bodenstorfer and A. Dijksma) [141] Diagonalization of certain block operator matrices and applications to Dirac operators, Oper. Theory Adv. Appl. 122 (2001), 331–358 (with C. Tretter) [142] Existence and uniqueness of contractive solutions of some Riccati equations, J. Funct. Anal. 179 (2001), 448–473 (with V. Adamyan and C. Tretter) [143] A spectral theory for a λ-rational Sturm–Liouville problem, J. Differential Equations 171 (2001), 315–345 (with V. Adamyan and M. Langer) [144] Compact perturbation of definite type spectra of self–adjoint quadratic operator pencils, Integral Equations Operator Theory 39 (2001), 127–152 (with V. Adamyan and M. M¨ oller)

Bibliography

xxxi

[145] A new concept for block operator matrices: The quadratic numerical range, Linear Algebra Appl. 330 (2001), 89–112 (with A. Markus, V. Matsaev, and C. Tretter) [146] Corners of numerical ranges, Oper. Theory Adv. Appl. 124 (2001), 385–400 (with A. Markus and C. Tretter) [147] Elliptic eigenvalue problems with eigenparameter dependent boundary conditions, J. Differential Equations 174 (2001), 30–54 (with P. Binding, R. Hryniv, and B. Najman) [148] A relation for the spectral shift function of two self–adjoint extensions, Oper. Theory Adv. Appl. 127 (2001), 437–445 (with V. A. Yavrian and H. de Snoo) [149] The Schur algorithm for generalized Schur functions I: coisometric realizations, Oper. Theory Adv. Appl. 129 (2001), 1–36 (with D. Alpay, T. Ya. Azizov, and A. Dijksma) [150] Invariant Subspaces of infinite-dimensional Hamiltonians and solutions of the corresponding Riccati equations, Oper. Theory Adv. Appl. 130 (2001), 235–254 (with A. C. M. Ran and B. A. van de Rotten) [151] On the Loewner problem in the class Nκ , Proc. Amer. Math. Soc. 130 (2002), 2057–2066 (with D. Alpay and A. Dijksma) [152] Triple variational principles for eigenvalues of self-adjoint operators and operator functions, SIAM J. Math. Anal. 34 (2002), 228–238 (with D. Eschwé) [153] Variational principles for eigenvalues of block operator matrices, Indiana Univ. Math. J. 51 (2002), 1427–1459 (with M. Langer and C. Tretter) [154] The Schur algorithm for generalized Schur functions II: Jordan chains and transformations of characteristic functions, Monatsh. Math. 138 (2003), 1–29 (with D. Alpay, T. Ya. Azizov, and A. Dijksma) [155] Self-adjoint block operator matrices with non-separated diagonal entries and their Schur complements, J. Funct. Anal. 199 (2003), 427–451 (with A. Markus, V. Matsaev, and C. Tretter) [156] The Schur algorithm for generalized Schur functions III: J-unitary matrix polynomials on the circle, Linear Algebra Appl. 369 (2003), 113–144 (with D. Alpay, T. Ya. Azizov, and A. Dijksma) [157] Continuous embeddings, completions and complementation in Krein spaces, ´ Rad. Mat. 12 (2003), 37–79 (with B. Curgus) [158] A basic interpolation problem for generalized Schur functions and coisometric realizations, Oper. Theory Adv. Appl. 143 (2003), 39–76 (with D. Alpay, T. Ya. Azizov, A. Dijksma, and G. Wanjala) [159] Rank one perturbations at infinite coupling in Pontryagin spaces, J. Funct. Anal. 209 (2004), 206–246 (with A. Dijksma and Yu. Shondin) [160] Continuations of Hermitian indefinite functions and canonical systems: an example, Methods Funct. Anal. Topology 10 (2004), 39–53 (with M. Langer and Z. Sasv´ ari)

xxxii

Bibliography

[161] Solution of a multiple Nevanlinna–Pick problem via orthogonal rational functions, J. Math. Anal. Appl. 293 (2004), 605–632 (with A. Lasarow) [162] Oscillation results for Sturm–Liouville problems with an indefinite weight function, J. Comput. Appl. Math. 171 (2004), 93–101 (with P. Binding and M. M¨ oller) [163] Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions, Linear Algebra Appl. 387 (2004), 313–342 (with D. Alpay and A. Dijksma) [164] The Schur algorithm for generalized Schur functions IV: unitary realizations, Oper. Theory Adv. Appl. 149 (2004), 23–45 (with D. Alpay, T. Ya. Azizov A. Dijksma, and G. Wanjala) [165] A Krein space approach to PT-symmetry, Czechoslovak J. Phys 54 (2004), 1113–1120 (with C. Tretter) [166] Minimal realizations of a scalar generalized Nevanlinna function related to their basic factorization, Oper. Theory Adv. Appl. 154 (2004), 69–90 (with A. Dijksma, A. Luger, and Yu. Shondin) [167] Partial non-stationary perturbation determinants, Oper. Theory Adv. Appl. 154 (2004), 1–18 (with V. Adamyan) [168] Spectrum of definite type of self-adjoint operators in Krein spaces, Linear and Multilinear Algebra 53 (2005), 115–136 (with M. Langer, A. Markus, and C. Tretter) [169] Spectral problems for operator matrices, Math. Nachr. 278 (2005), 1408– 1429 (with A. Batkai, P. Binding, A. Dijksma, and R. Hryniv) [170] Bounded normal operators in a Pontryagin space, Oper. Theory Adv. Appl. 162 (2006), 231–251 (with F. Szafraniec) [171] Partial non-stationary perturbation determinants for a class of J-symmetric operators, Oper. Theory Adv. Appl. 162 (2006) 1–18 (with V. Adamyan)

Operator Theory: Advances and Applications, Vol. 163, 1–22 c 2005 Birkh¨ auser Verlag Basel/Switzerland

General Solution of the Stieltjes Truncated Matrix Moment Problem Vadim M. Adamyan and Igor M. Tkachenko To Heinz Langer with admiration and gratitude for fruitful co-operation

Abstract. The description of all solutions of the truncated Stieltjes matrix moment problem consisting in finding all s × s matrix measures dσ (t) on [0, ∞) with given first 2n + 1 power s × s matrix moments (Cj )n j=0 is obtained in a general case, when the block Hankel matrix Γn := (Cj+k )n j,k=0 may be non-invertible. Special attention is paid to the description of canonical solutions for which dσ (t) is a sum of at most sn + s point matrix “masses” with the minimal sum of their ranks. Mathematics Subject Classification (2000). Primary 30E05, 30E10; Secondary 82C70, 82D10. Keywords. Stieltjes moments problem, matrix functions, Nevanlinna’s formula.

1. Introduction In this paper we consider the following problem: Given a set of Hermitian s × s matrices {C0 , C1 , C2 , . . . , C2n },

n = 0, 1, 2, . . . .

Find all non-negative matrix measures dσ (t) such that ∞ tk dσ (t) = Ck , k = 0, 1, 2, . . . , 2n.

(1.1)

(1.2)

0

This is a matrix version of the well-known Stieltjes truncated moment problem. The Stieltjes problem was considered in different settings in many monographs and papers starting from the classical memoirs by Stieltjes himself [14, 15]. Classical The authors appreciate the fair work and helpful suggestions and corrections of the referees of this paper.

2

V.M. Adamyan and I.M. Tkachenko

results on the topic are contained in the books [3, 10, 13] and the papers [12, 11]; more recent developments on the truncated moment problems can be found in [5, 7, 6, 1, 8, 9]. The above matrix problem was considered recently in [2]. The main results of that paper can be summarized in the following two theorems. Theorem 1.1. A system of Hermitian matrices {C0 , C1 , C2 , . . . , C2n }, 0, 1, 2, . . . , admits the representation ∞ tk dσ (t) = Ck , k = 0, 1, 2, . . . , 2n,

n =

(1.3)

0

if and only if a) the block Hankel matrix Γn := (Ck+j )nk,j=0 is non-negative; b) for any set ξ 0 , . . . , ξr ∈ Cs , 0 ≤ r ≤ n − 1, and with (ξ, η) being the standard scalar product in Cs , the condition r (Cj+k ξ k , ξ j ) = 0 (1.4) j,k=0

implies

r j,k=0

(Cj+k+2 ξ k , ξ j ) = 0;

(1.5)

(1)

c) the block Hankel matrix Γn−1 := (Ck+j+1 )n−1 k,j=0 is non-negative and for any set ξ 0 , . . . , ξ r ∈ Cs , 0 ≤ r ≤ n − 1, the condition r (Cj+k+1 ξ k , ξ j ) = 0 (1.6) j,k=0

implies (1.5). In what follows we will denote by R the Nevanlinna class of holomorphic dissipative s × s matrix functions on the upper half-plane, i.e., matrix functions with non-negative imaginary parts and by S the subset of R consisting of all Nevanlinna s × s matrix functions t (z), Imz > 0, which admit the integral representation ∞ t (z) = 0

1 dρ (t) t−z

with a non-decreasing s × s matrix function (Cs operator) ρ (t) such that the condition ∞ d (ρ (t) h, h) < ∞ 0

holds for any h ∈ Cs . Theorem 1.2. Let the conditions of Theorem 1.1 hold and det Γn > 0, and let ∆Ξ (z) be the upper-left block of the matrix function −1 (1) Γn , Γn ΓΞ;n (z) − zΓn

Stieltjes Truncated Matrix Moment Problem where

⎛ ⎜ ⎜ (1) ΓΞ;n (z) := ⎜ ⎝

Cn+1 .. .

(1)

Γn−1 Cn+1

. . . C2n

C2n Ξ (z)−1 + z

3 ⎞ ⎟ ⎟ ⎟ ⎠

and Ξ ∈ S. Then the relation ∞ 1 dσ Ξ (t) = ∆Ξ (z) , Imz > 0, t−z 0

by means of the Stieltjes inversion formula (σ Ξ (β)h, h) − (σ Ξ (α)h, h) = − π1 lim η↓0

β α

Im (∆Ξ (t + iη)h, h) dt,

σ Ξ (α − 0) = σ Ξ (α + 0), σ Ξ (β − 0) = σ Ξ (β + 0), h ∈ Cs , −∞ < α < β < ∞, establishes a one-to-one correspondence between a set of all solutions σ Ξ (t) of the truncated Stieltjes matrix moment problem with the given moments {C0 , . . . , C2n } and the subset of all Nevanlinna matrix functions Ξ from S which satisfy the condition n−1 −1 Ξ (λ) − Cn+j+1 bjk (λ)Cn+k+1 > 0, λ < 0 , (1.7) j,k=0

with s × s matrices bjk (z) defined by the equality −1 (1) Γn−1 − z = (bjk (z))n−1 / [0, ∞). j,k=0 , z ∈ The aim of the present work is to describe all solutions of the Stieltjes truncated matrix moment problem in the degenerate case: det Γn = 0. In Section 2 following ideas of M.G. Krein [11] we show how the truncated matrix moment problems can be reduced to problems of the extension theory for symmetric operators. In Section 3 we consider the so-called completely degenerate truncated moment problem, we find here its unique solution in an explicit form. A special class of the canonical solutions of the truncated Stieltjes problem is described in Section 4. For this class of solutions the sought non-decreasing matrix function σ (t) has no more than ns + s points of growth. An algorithm for the construction of such solutions is given here as well. In the last section we give, omitting the detailed proof, the description of all solutions of the problem under consideration.

2. Reduction to the extension theory problem From now on we will assume that the conditions of Theorem 1.1 hold. Notice that due to the conditions a) and c) of this theorem, the quadratic forms of the matrices Cj are non-negative, or briefly Cj ≥ 0, j = 0, . . . , 2n. Without loss of

4


generality we can assume that all matrix moments Cj are invertible, i.e., that Cj > 0, j = 0, . . . , 2n. Indeed, let Nj ⊂ Cs be the null-spaces of Cj . Due to the condition a) of Theorem 1.1 for any vector η ∈ Cs the equality C0 η = 0 implies Cj η = 0, j = 1, . . . , 2n. Besides, since for any vector η ∈ Cs and any solution σ (t) of the moment problem all integrals ∞ tk d(σ (t) η, η), 1 ≤ k ≤ 2n 0

vanish simultaneously, any equality Cj η = 0 implies Ck η = 0, 1 ≤ k ≤ 2n. Hence, N0 ⊂ N1 = Nk , k = 1, . . . , 2n and for a suitable basis in Cs the matrices Cj can be reduced to the form

0 0

j 0 C C , j = 1, . . . , 2n, (2.1) ; C = C0 = j

0 0 0 0 C

j > 0, j = 0, 1, . . . , 2n. where by construction det C

j is solvIf the truncated Stieltjes problem for invertible matrix moments C able, then the initial problem is solvable as well, and its general solution σ (t) can be presented as

(t) σ 0 σ (t) = U U∗ ,

0 ϑ (t) 0 C where U is a fixed s × s-unitary matrix, 0, t ≤ 0, ϑ (t) = 1, t > 0,

(t) runs through the set of solutions of the truncated Stieltjes matrix moment and σ

j. problem for the moments C Let a non-decreasing matrix function σ (t), 0 ≤ t < ∞, be a solution of the truncated Stieltjes matrix moment problem, i.e., ∞ tk dσ (t) = Ck , k = 0, 1, 2, . . . , 2n. (2.2) 0

Consider the set of continuous vector functions f (t) , 0 ≤ t < ∞, with values in Cs , which satisfy the condition ∞ (dσ (t) f (t) , f (t)) < ∞. (2.3) 0

Construct a pre-Hilbert space L of such vector functions taking the bilinear functional ∞ f , g = (dσ (t) f (t) , g (t)) (2.4) 0

as the scalar product. Notice that by (1.2) the vector polynomials f (t) = ξ 0 + t · ξ 1 + · · · + tn · ξn ,

ξ 0 , . . . , ξ n ∈ Cs ,

(2.5)

of degree n belong to L. We will denote the linear subset of these polynomials by Pn .

Stieltjes Truncated Matrix Moment Problem

5

Let L0 be the subspace of L consisting of all vector functions f such that f := f , f = 0. If g = f + f0 , where f ∈ L, f0 ∈ L0 , than, due to the Schwartz inequality f , f0 = 0 and, hence, g = f . Let us denote by L1 the factor space L/L0 . For the class of = f +L0 of this factor space we set elements g gL1 = f . Taking the completion of L1 with respect to this norm, we obtain the Hilbert space L2σ (Cs ). We keep the same symbol ., . for the scalar product in L2σ (Cs ). Let Ln be the subspace of L2σ (Cs ) generated by the subset of vector polynomials Pn . By (2.2) and (2.4) for f , g ∈Pn , f (t) =

n

tr · ξ r ,

g (t) =

l=0

n

tr · η r ,

ξ 0 , . . . , η n ∈ Cs ,

l=0

we have: f , g =

n

(Cj+k ξ k , η j ).

(2.6)

j,k=0

Therefore the scalar products on Ln in the spaces L2σ (Cs ) for all non-decreasing ˜ (t) satisfying matrix functions σ (t) which satisfy (1.2) must coincide. Take any σ

0 of multiplication by the (1.2) and consider the non-negative symmetric operator A independent variable t on the subset of all continuous compact vector functions in

of A

0 is a self-adjoint operator [4]. Let us the related space L2σ (Cs ). The closure A denote by Ln−1 the subspace of Ln generated by vector polynomials of the degree

to Ln−1 is a non-negative symmetric ≤ n − 1. The restriction A0 of the operator A

operator which by definition of A actually does not depend on the choice of a ˜ (t) solution of the truncated Stieltjes moment problem. Therefore each solution σ

of A0 . of this problem generates some non-negative self-adjoint extension A

t , 0 ≤ t < ∞, be now the spectral function of some simple non-negative Let E

1 of A0 . For the canonical orthonormal basis {e1 , . . . , es } in Cs we extension A introduce the set of classes { e10 , . . . , es0 } ⊂ Ln which contain zero degree vector monomials e10 (t) ≡ e1 , . . . , es0 (t) ≡ es , ˜ (t) = respectively. Let us consider the non-decreasing s × s matrix function σ ( σµν (t))sµ,ν=1 , 0 ≤ t < ∞,

t σ

µν (t) := E eν0 , eµ0 , 1 ≤ µ, ν ≤ s. (2.7) Ln

for the classes { e1k , . . . , esk } ⊂ Ln which contain the By definition of A0 and A, vector monomials e1k (t) = tk e1 , . . . , esk (t) = tk es , 0 ≤ k ≤ n, 1 A self-adjoint extension A of A acting in a Hilbert space H ⊇ L is called simple if A has no n 0 invariant subspaces in H Ln .

6


it holds:

k eµk = Ak0 eµ0 = A eµ0 , 1 ≤ µ ≤ s, Hence, for 0 ≤ j, k ≤ n and 1 ≤ µ, ν ≤ s we have (Cj+k )µ,ν

0 ≤ k ≤ n.

= (Cj+k eν , eµ )Cs =

k

j eν0 , A eµ0 eµj Ln = A = eνk , Ln ∞ ∞

t eν0 , = tj+k d E eµ0 = tj+k d σµν (t) . Ln

0

0

We proved the following Proposition 2.1. The formula

t ˜ µν (t) = E eν0 , eµ0 σ

Ln

, 0 ≤ t < ∞,

(2.8)

t establishes a one-to-one correspondence between the set of spectral functions E

of A0 and the set of solutions σ ˜ (t) of the of non-negative simple extensions A truncated Stieltjes matrix moment problem.

3. Solution in the completely degenerate case The non-decreasing matrix functions σ (t) which satisfy (2.2) and for which L2σ (Cs ) =Ln are called canonical. Equivalently, solutions of the truncated matrix Stieltjes prob t is the spectral function of lem given by the expression (2.7) are canonical if E

of A0 , i.e., a non-negative some non-negative canonical self-adjoint extension A self-adjoint extension without leaving Ln . Due to (2.6), a canonical σ (t) is a nondecreasing matrix function which has only a finite number of points of growth and the sum of the ranks of all jumps of σ at such points is ≤ ns. It was proven in [2] that the set of canonical solutions of the truncated matrix Stieltjes moment problem is non-empty whenever this problem is solvable, i.e., whenever the conditions of Theorem 1.1 hold. Formula (2.8) which establishes a one-to-one correspondence between the set of non-negative self-adjoint extensions of A0 without leaving Ln and the set of canonical solutions of the Stieltjes problem, makes it possible to find, under the conditions of Theorem 1.1, an explicit algebraic formula for the description of the sought canonical solutions. To this end we can use as a starting point (2.7) and the relation ∞ ∞ 1 1

t d σµν (t) = d E eν0 , eµ0 = (3.1) t−z t−z Ln 0 0

− z)−1 = (A eν0 , eµ0 . Ln

From now on we will assume that det Γn = 0, i.e., we will consider the degenerate case of the above problems.


7

Let Ln (Cs ) denote the (n + 1)s-dimensional linear space of column vectors ξ=

ξ0

···

ξn

, ξ 0 , . . . , ξ n ∈ Cs ,

(3.2)

( stands for the transposition operation) with the scalar product (ξ, η) =

n

j=0

ξj , η j

Cs

.

We will denote as before by Ln the same linear vector space but with the scalar product n ξ, η = (Γn ξ, η) = Cj+k ξk , η j C . j,k=0

s

Ln was considered above as the space of vector polynomials. Let Ln−1 (Cs ) be the subspace of Ln (Cs ) which consists of vectors (3.2) but with ξ n = 0 and let Nn = Ln (Cs ) Ln−1 (Cs ) . We denote by Pn the orthogonal projector in Ln (Cs ) onto Nn . In the natural basis of subspaces of Ln (Cs ) this projector is evidently given as the following (n + 1) × (n + 1) block operator matrix ⎞ ⎛ 0 ... 0 ⎟ ⎜ Pn = ⎝ ... . . . ... ⎠ , 0 ...

I

where I is the s × s unit matrix. Let us denote by Qn the orthogonal projector in Ln (Cs ) onto the null-space Mn of the operator Γn and let Q⊥ n = I − Qn . We call the truncated matrix Stieltjes moment problem completely degenerate if Pn Mn = Nn . We will also use the same definition for non-negative Hankel matrices Γn having the property: Pn Mn = Nn . In the completely degenerate case for each vector ⎛ ⎞ 0 ⎜ .. ⎟ ⎜ ⎟ eni = ⎜ . ⎟ ⎝ 0 ⎠ ei of the canonical orthonormal basis {en1 , . . . , ens } in Nn there is a non-zero vector ⎛ ⎞ ξ ni,0 ⎜ .. ⎟ ⎜ ⎟ ξni = ⎜ . ⎟ ⎝ ξ ni,n−1 ⎠ ei

8


such that

⎛

⎜ ⎜ Γn ξ ni = ⎜ ⎝

Cn .. .

Γn−1 Cn

. . . C2n−1

C2n−1 C2n

⎞⎛

ξ ni,0 ⎟ ⎜ .. ⎟⎜ . ⎟⎜ ⎠ ⎝ ξ ni,n−1 ei

⎞ ⎟ ⎟ ⎟ = 0, i = 1, . . . , s. ⎠

(3.3) Note that at least some of ξni,k = 0 in (3.3), k = 0, . . . , n − 1. Indeed, otherwise we would have that Cn ei = · · · = C2n ei = 0. But by our assumption all matrices C0 , . . . , C2n are invertible, a contradiction. As follows, the class from Ln containing the monomial tn ei contains also the vector polynomial dni (t) = −ξ ni,0 − · · · − tn−1 ξ ni,n−1 . Hence, in the completely degenerate case Ln = Ln−1 . This equality implies that the symmetric operator A0 from Ln−1 into Ln defined above by relations ek = ek+1 , k = 0, . . . , n − 1, A0 actually is self-adjoint. As follows, in the completely degenerate case the solution of the truncated Stieltjes problem is unique. For vectors ξ n0 , . . . , ξ ns appearing in (3.3) let us introduce vector polynomials φi (z) := ξ ni,0 + · · · + z n−1 ξ ni,n−1 + z n ei , i = 1, . . . , s, and the s × s matrix function D(z), the corresponding columns of which are φi (z). Since n D(z) = z n I + o (|z| ) , (3.4) z→∞

the matrix function is invertible everywhere except of at most sn points. Let σ(D) be the spectrum of the matrix function D(z), i.e., the set of those z ∈ C for which there exists a non-zero h ∈ Cs such that D(z)h = 0. Proposition 3.1. In the completely degenerate case the spectrum σ(A0 ) of the nonnegative self-adjoint operator A0 coincides with σ(D) and for z ∈ / σ(D) the resol−1 vent (A0 − z) of A0 acts on a vector polynomial g(t) of degree ≤ n − 1 representing a class of polynomials from Ln−1 by the formula 1 (A0 − z)−1 g (t) = g(t) − D(t)D(z)−1 g(z) . (3.5) t−z Proof. In the completely degenerate case each vector of Ln can be represented as a vector polynomial of degree n − 1. Correspondingly A0 acts on such vector polynomials as the multiplication operator by t. Let us consider the equation (t − z) f (t) = ((A0 − z)f )(t) = g(t), z ∈ / σ(D), (3.6)


9

and assume that the degree of the representing polynomial g(t) is ≤ n and that the degree of the sought polynomial f (t) is ≤ n − 1. If g(z) = 0 then we put −1

f (t) = (t − z)

g(t).

Otherwise we can substitute on the right-hand side of the equation (3.6) instead of g(t) the equivalent polynomial

(t) = g(t) − D(t)D(z)−1 g(z), g and get as a solution of (3.6) the vector polynomial 1 g(t) − D(t)D(z)−1 g(z) , f (t) = (t − z) the degree of which is, evidently, ≤ n − 1. Hence for z ∈ / σ(D) and any vector polynomial g(·) ∈ Ln the equation (3.6) has a solution in the form of a vector polynomial from Ln−1 . Therefore C \ σ(D) belongs to the resolvent set of A0 and −1 (A0 − z) for z ∈ C \ σ(D) is given by (3.5). On the other hand, if z ∈ σ(D), then there is a non-zero vector h ∈ Cs such that D(z)h = 0. Note that by (3.4) D(t)h = tn h + o(tn ) t→∞

and thus D(t)h = 0 identically. If (t − z)−l+1 D(t)h, 1 ≤ l ≤ n, belongs to the zero class in Ln , but f0 (t) := (t − z)−l D(t)h is a non-zero element from Ln , then f0 (t) is a non-trivial solution of the homogeneous equation (A0 − z) f = 0. Remark 3.2. Since A0 is a non-negative operator, we have σ(D) ⊂ [0, ∞). Let σ 0 (t) be the (unique) solution of the truncated Stieltjes matrix moment problem and let ∞ 1 dσ 0 (t) (D(t) − D(z)) . E(z) = (3.7) t−z 0

Note that if D(z) = z n I + z n−1 An−1 + · · · + zA1 + A0 , then E(z) =

n−1

z k Cn−1−k +

0

n−2

z k Cn−2−k An−1 + · · · + C0 A1 .

(3.8)

0

It stems from formula (3.5) applied to the classes which contain the zerodegree polynomials and the expression (3.1) that the following assertion is true. Proposition 3.3. In the completely degenerate case the unique solution σ0 (t) of the truncated Stieltjes matrix moment problem satisfies the relation ∞ 1 dσ 0 (t) = −E(z)D(z)−1 . (3.9) t−z 0

10


The point of Proposition 3.3 is that it allows one to calculate “explicitly” the unique solution σ 0 (t) of the truncated Stieltjes moment problem in the completely degenerate case. To this end it is enough to find, following the above prescription, the matrix polynomial D(z), take E(z) from (3.8) and apply the Stieltjes inversion formula to (3.9). By this formula for each h ∈ Cs and β > α such that det D(α) = 0, det D(β) = 0, we have β 1 (σ 0 (β)h, h) − (σ 0 (α)h, h) = − lim Im E(t + iη)D(t + iη)−1 h, h dt. (3.10) π η↓0 α

4. Canonical solutions Now we will omit the assumption that the projection Pn Mn of the null-space Mn onto the subspace Nn necessarily covers all this subspace. Theorem 4.1. Let C0 , C1 , C2 , . . . , C2n be any set of Hermitian s×s matrices which satisfy the conditions of Theorem 1.1 and let σ(t) be any solution of the truncated Stieltjes matrix moment problem for this set. Then the extended set of non-negative Hermitian matrices ∞ ∞ C0 , . . . , C2n , C2n+1 = t2n+1 dσ (t) , C2n+2 = t2n+2 dσ (t) (4.1) 0

0

satisfies all conditions of Theorem 1.1 with n replaced by n+1 and for the extended set (4.1) the truncated Stieltjes moment problem is completely degenerate if and only if σ(t) is a canonical solution. Proof. Since the Stieltjes problem for the set (4.1) is evidently solvable, this set satisfies the conditions of Theorem 1.1. Suppose that σ(t) is a canonical solution of the Stieltjes problem for the set C0 , . . . , C2n . Recall that the solvability of the Stieltjes problem for this set guarantees the existence of canonical σ(t) [2]. By the definition of canonical solutions for σ(t) we have Ln+1 =Ln =L2σ (Cs ). Therefore for any non-zero vector h ∈ Cs there is a vector polynomial Ph (t) = tn ξhn + · · · + ξh0 such that the vector polynomials tn+1 h and Ph (t) belong to the same class or, more precisely, the polynomial tn+1 h − Ph (t) belongs to the zero-class in L2σ (Cs ). n+1 Since the block Hankel matrix Γn+1 = (Cj+k )j,k=0 is non-negative definite and (C2n+2 h, h) − (C2n+1 h, ξhn ) − (C2n+1 ξhn , h) − · · · + (C2n+1 ξh0 , ξh0 ) = ∞ dσ (t) tn+1 h − Ph (t) , tn+1 h − Ph (t) = 0, 0

we see that the vector

⎞ −ξh0 ⎜ .. ⎟ ⎜ . ⎟ ⎟ ⎜ ⎝ −ξhn ⎠ h ⎛

(4.2)


11

belongs to the null-space of Γn+1 . As this is true for any h ∈ Cs we conclude that for the set (4.1) the Stieltjes problem is completely degenerate. Let us assume now that the solution σ(t) is such that for the extended set (4.1) the Stieltjes problem is completely degenerate. Then for each vector h ∈ Cs there is a vector of the form (4.2) from the null-space of Γn+1 or, equivalently, there is a vector polynomial tn+1 h − tn ξhn − · · · − ξh0 from the zero-class in L2σ (Cs ). Therefore, for the given solution σ(t) we have Ln+1 = Ln = L2σ (Cs ) This implies that σ(t) is a canonical solution. Theorem 4.1 gives us a hint for the description of all canonical solutions of the Stieltjes moment problem. To this end it is enough first to find all those extensions C0 , C1 , . . . , C2n , C2n+1 , C2n+2 of the given set of moments C0 , C1 , . . . , C2n , for which corresponding Stieltjes problems turn out to be completely degenerate, and then to find unique solutions of the arising completely degenerate problems as it has been done in the previous section. We start the description of the demanded extensions of a given moment set with the following general proposition. Theorem 4.2. Let A be a bounded non-negative self-adjoint operator in a Hilbert space H1 and B be a bounded operator from H1 into a Hilbert space H2 . Put Q(s) = B(A + s)−1 B ∗ , 0 < s < ∞. The following conditions are equivalent: • there are bounded non-negative self-adjoint extensions of the operator A + B : H1 → H1 ⊕ H2 to the Hilbert space H = H1 ⊕ H2 ; • the operator Q0 in H2 defined in a way that Q0 f = lim Q(s)f, s→0+

f ∈ H2 ,

is bounded.

of If these conditions hold, then bounded non-negative self-adjoint extensions A (A + B)|H1 have the form of block operator matrices A B∗

= , (4.3) A B D where D runs through the set of non-negative self-adjoint operators in H2 and satisfies the condition D − Q0 ≥ 0.

(4.4)

Proof. Since the operators A and B are bounded, we have that each bounded self-adjoint extension of (A + B)|H1 to H can be represented in the form (4.3)

is non-negative with a bounded self-adjoint D. Note further that an extension A

− λ is strictly positive. By the if and only if for every λ < 0 the operator A

12


Schur-Frobenius factorization formula we have I 0

A−λ= B(A − λ)−1 I A−λ 0 I × 0 0 D − λ − B (A − λ)−1 B ∗

(A − λ)−1 B ∗ I

(4.5) , λ < 0.

is non-negative if and only if Therefore the extension A W (λ) := D − λ − B (A − λ)

−1

B ∗ > 0, λ < 0.

Let the operator Q0 exist and be bounded. Since the operator function W (λ) is

is non-negative if and only non-increasing on the half-axis (−∞, 0), the operator A if the inequality (4.4) holds. Suppose now that for some f ∈ H2 we have lims→0+ (Q(s)f, f ) = +∞. As D is a bounded operator we conclude that for s small enough and the same f ∈ H2 the inequality (W (−s)f, f ) < 0 holds. Therefore there are no non-negative operators among bounded self-adjoint extensions of (A + B)|H1 . Remark 4.3. Let the operators A and B be as in Theorem 4.2 and the space H1 be finite-dimensional. Then among bounded self-adjoint extensions of (A + B)|H1 to the Hilbert space H = H1 ⊕ H2 there are non-negative operators if and only if the null-space of A is contained in the null-space of B. Remark 4.4. Let us assume that the operator A in Theorem 4.2 has a bounded inverse. Then the condition D ≥ BA−1 B ∗ for a bounded self-adjoint D in (4.3)

given by the block operator is necessary and sufficient for the non-negativity of A matrix (4.3). Proposition 4.5. Let operators A : H1 → H1 and B : H1 → H2 be as in Theorem

0 in the Hilbert 4.2 and assume that A has a bounded inverse. Then the operator A space H = H1 ⊕ H2 given as the block operator matrix A B∗

0 = (4.6) A B B ∗ A−1 B is the unique non-negative self-adjoint extension of (A+B)|H1 to H which satisfies the condition: P2 MA 0 = {0} ⊕ H2 , (4.7) where P2 is the orthogonal projector in H from H to {0} ⊕ H2 and MA 0 is the

0 . null-space of A

Proof. By assumptions of the proposition and Remark 4.4 the extension (4.6) is bounded, self-adjoint and non-negative with D = BA−1 B ∗ . Let us take an arbitrary vector h ∈ H2 and consider the vector −A−1 B ∗ h

∈ H. h= h

0 It is evident that A h = 0. Hence, (4.7) holds for the extension (4.6).


13

be any bounded non-negative self-adjoint extension which satisfies Now, let A the condition (4.7). Then for any h ∈ H2 there is a vector h1 ∈ H1 such that

that is h = h1 + h ∈ H is a null-vector of A, Ah1 + B ∗ h = 0, Bh1 + Dh = 0.

(4.8)

As A is invertible we deduce from (4.8) that the equality −BA−1 B ∗ + D h = 0 is true for any h ∈ H2 . Hence D = BA−1 B ∗ . Having disposed of these preliminary steps, we can now return to the description of all canonical extensions of the truncated Stieltjes moment problem. Let C0 , C1 , C2 , . . . , C2n be any sequence of Hermitian s×s matrices, which satisfy all the conditions of Theorem 1.1. We begin with the non-degenerate case and assume that the non-negative n block Hankel matrix Γn = (Cj+k )j,k=0 is invertible. This implies that the non(1)

n−1

negative block Hankel matrix Γn−1 = (Cj+k+1 )j,k=0 is invertible as well. Indeed, (1)

(2)

if Γn−1 is non-invertible, then by the conditions of Theorem 1.1 the matrix Γn−1 = (2) (Cj+k+2 )n−1 j,k=0 is also non-invertible. But Γn−1 is a diagonal block of the positive definite matrix Γn , a contradiction. Put ⎛ ⎞ Cn+1 ⎜ ⎟ .. S1 = ⎝ ⎠. . C2n By Theorem 4.2 and Remark 4.4 for any non-negative definite s × s matrix W the n × n block Hankel matrix ⎛ ⎞ (1) Γ S 1 n−1 −1 ⎝ ⎠ Γ(1) (4.9) n = (1) S∗1 W + S∗1 Γn−1 S1 (1)

is a non-negative definite n × n block Hankel extension of the block matrix Γn−1 . Let ⎛ ⎞ Cn+1 .. ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟. S=⎜ ⎟ C ⎝ ⎠ 2n −1 (1) W + S∗1 Γn−1 S1 −1 (1) By taking W + S∗1 Γn−1 S1 as C2n+1 and the matrix S∗ Γ−1 n S as C2n+2 , we construct the (n + 2) × (n + 2) block Hankel matrix S Γn n+1 . Γn+1 = (Cj+k )j,k=0 = S∗ S∗ Γ−1 n S

(4.10)

14


From Proposition 4.5 we can conclude that the matrix (4.10) is the unique nonnegative definite extension of Γn for which the condition Pn+1 Mn+1 = Nn+1 holds; as before Mn+1 is the null-space of Γn+1 and Pn+1 is the orthogonal projector in the space Ln+1 of column vectors ⎛ ⎞ ξ0 ⎜ .. ⎟ ⎜ ⎟ ξ = ⎜ . ⎟ , ξ 0 , . . . , ξn+1 ∈ Cs , ⎝ ξn ⎠ ξn+1 onto the subspace Nn+1 = Ln+1 Ln of column vectors with ξ 0 = · · · = ξ n = 0. Proposition 4.6. For the set of Hermitian s × s matrices C0 , C1 , . . . , C2n which satisfy the conditions of Theorem 1.1 and such that the block Hankel matrix Γn = n (Cj+k )j,k=0 is invertible, all extensions C2n+1 , C2n+2 that generate completely degenerate truncated Stieltjes matrix moment problems are described by formulas ⎛ ⎞ Cn+1 −1 ⎜ ⎟ (1) .. (4.11) C2n+1 = W + (Cn+1 , . . . , C2n ) Γn−1 ⎝ ⎠, . ⎛

C2n ⎞

Cn+1 ⎟ .. C2n+2 = (4.12) ⎠, . C2n+1 where the “parameter” W runs through the set of all non-negative definite s × s matrices. (Cn+1 , . . . , C2n+1 ) Γ−1 n

⎜ ⎝

Proof. It remains to prove that (4.11) with (4.12) provides all demanded extensions. Let now C2n+1 , C2n+2 be an extension generating a completely degenerate Stieltjes problem. By Theorem 1.1 this means, in particular, that the extended (1) Hankel matrix Γn should be non-negative definite. According to Remark 4.4 this implies the expression (4.11) for C2n+1 with some non-negative definite matrix parameter W. By Proposition 4.5, C2n+2 for a certain appropriate C2n+1 cannot differ from that given by expression (4.11). From now on we assume that the block Hankel matrix Γn is not invertible. The problem of describing all extensions C2n+1 , C2n+2 which transform the given Stieltjes (2n + 1) moment problem into completely degenerate (2n + 3) ones, can be handled in the following way. We can take any sequence of Hermitian s×s matrices G0 , G1 , G2 , . . . , G2n such that the block Hankel matrices ∆n := (Gk+j )nk,j=0 (1)

and ∆n−1 := (Gk+j+1 )n−1 k,j=0 are invertible and positive definite. Then for any ε > 0 the perturbed Hankel matrices (1)

(1)

(1)

Γn (ε) := Γn + ε∆n , Γn−1 (ε) := Γn−1 + ε∆n−1

(4.13)


15

are invertible and positive definite. Taking for the set of moments C0 (ε) := C0 + εG0 , . . . , C2n (ε) := C2n + εG2n the solutions (4.11), (4.12) of the extension problem under consideration, we can further obtain the demanded solutions by passing to the limit ε ↓ 0. To this end let us first show that under the conditions of Theorem 1.1 there exists a finite limit of non-negative matrices ⎛

⎛

⎞⎞ Cn+1 (ε) ⎜ ⎜ ⎟⎟ .. ⎝S1 (ε) = ⎝ ⎠⎠ . C2n (ε)

−1 (1) S∗1 (ε) Γn−1 (ε) S1 (ε), as ε ↓ 0. (1)

We denote by P1 (Γn−1 ) the orthogonal projector onto the null-space MΓ(1) ⊂ n−1

(1)

Ln−1 (Cs ) of the non-negative block Hankel matrix Γn−1 := (Ck+j+1 )n−1 k,j=0 and (1)

(1)

set P2 (Γn−1 ) = I − P1 (Γn−1 ). With respect to the representation Ln−1 (Cs ) = M⊥ ⊕ MΓ(1) (1) Γ

n−1

n−1

and by the definition of MΓ(1) and (4.13) we can write n−1

(1) Γn−1 (ε)

(1)

=

(1)

(1)

(1)

ε∆n−1,12 (1) (1) Γn−1,22 + ε∆n−1,22

ε∆n−1,11 (1) ε∆n−1,21 (1)

(1)

Γn−1,ij + ε∆n−1,ij = Pi (Γn−1 )Γn−1 (ε) |Pj (Γ(1)

n−1 )Ln−1 (Cs )

Put

−1 (1) (1) (1) Υn−1 = lim P2 Γn−1 Γn−1 − λ |M⊥

(1) Γ n−1

λ↑0 (1)

,

, i, j = 1, 2.

.

(1)

Note that the matrices Υn−1 and ∆n−1,11 are invertible. The Schur-Frobenius factorization formula gives: 1 (1) Γn−1 (ε)−1 =

(1)

∆n−1,11 0

−1

0 0

+ ε ⎞ ⎛ −1 −1 −1 (1) (1) (1) (1) (1) (1) (1) (1) − ∆n−1,11 ∆n−1,12 Υn−1 ⎟ ⎜ ∆n−1,11 ∆n−1,12 Υn−1 ∆n−1,21 ∆n−1,11 −1 ⎠ ⎝ (1) (1) (1) (1) Υn−1 ∆n−1,21 ∆n−1,11 Υn−1 +O(ε).

(4.14)

16


Note further that for any h ∈ Cs we have −1 −1 (1) (1) ∗ S1 (ε)h, h = Γn−1 (ε) k(ε), k(ε) , S1 (ε) Γn−1 (ε) ⎞ ⎛ 0 ⎜ .. ⎟ ⎟ ⎜ (2) (2) k(ε) = Γn−1 (ε) ⎜ . ⎟ , Γn−1 (ε) = (Ck+j+2 + εGk+j+2 )n−1 k,j=0 . ⎝ 0 ⎠ h (1)

Recall that by the assumptions of Theorem 1.1, the null space of Γn−1 belongs to the null-space of (2) (2) Γn−1 = Γn−1 (0) . Therefore taking into account (4.14), we get −1 (1) (1) (1) (1) S1 (ε) = S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0). lim S∗1 (ε) Γn−1 (ε) ε↓0

(1)

Note that by Proposition 4.6 for ε > 0 a self-adjoint Hankel extension Γn (ε) of (1) Γn−1 (ε) is non-negative definite if and only if −1 (1) C2n+1 (ε) = W + S∗1 (ε) Γn−1 (ε) S1 (ε), where W is any non-negative definite s × s matrix. As follows, for each fixed (1)(0) with the limit non-negative definite W the limit Hankel matrix Γn (1) (1) (1) (4.15) C2n+1 (0) = W + S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0) is non-negative definite. On the other hand, as −1 (1) (1) (1) (1) S1 (0), S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0) = lim S∗1 (0) Γn−1 − λ λ↑0

it stems from Proposition 4.6 that only extensions C2n+1 given by (4.15) with (1) W ≥ 0 provide the non-negativity of Γn . To find now the set of all appropriate extensions C2n+2 it remains to select among the non-negative definite s× s matrices W those for which there exist finite limits of the matrices S∗ (ε)Γn (ε)−1 S(ε) as ε ↓ 0. To this end note that taking any h ∈ Cs and setting ⎛ ⎞ ⎞ ⎛ Cn+1 (ε)h 0 ⎜ ⎟ .. ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ k=⎜ ⎟, ⎜ ⎟ = Γ(1) n (ε)k, (ε)h C 2n ⎟ ⎝ 0 ⎠ ⎜ −1 ⎝ ⎠ (1) h S1 (ε) h W + S∗1 (ε) Γn−1 (ε)

Stieltjes Truncated Matrix Moment Problem we can write

(1) S∗ (ε)Γn (ε)−1 S(ε)h, h = Γn (ε)−1 Γ(1) (ε)k, Γ (ε)k . n n

17

(4.16)

Let P1 (Γn ) be the orthogonal projector onto the null-space MΓn ⊂ Ln (Cs ) of the non-negative block Hankel matrix Γn := (Ck+j )nk,j=0 , P2 (Γn ) = I − P1 (Γn ) and let −1 Υn = lim P2 (Γn ) (Γn − λ) |M⊥ . Γ λ↑0

n

Note that for any ξ ∈ MΓn we have

Γ(1) n (0)ξ

= j

n−1

Cj+1+k ξk = (Γn ξ)j+1 = 0, 0 ≤ j ≤ n − 1.

(4.17)

k=0

Let Pn be the orthogonal projector onto the subspace Pn Mn . Taking into account (1) that the matrices Γn (0), Γn are Hermitian, we deduce from (4.17) that (1) (1) (1) Γ(1) n (0)P1 (Γn )Γn (0) = Γn (0)P1 (Γn )Pn P1 (Γn ) Γn (0).

(4.18)

Applying the same arguments as above to (4.16) and by virtue of (4.18), we obtain that ∗ (1) (1) S (ε)Γn (ε)−1 S(ε)h, h = 1ε Pn P1 (Γn )Γn (0)h, Pn Γn (0)h ε↓0 (4.19) (1) (1) + Υn P2 (Γn )Γn (0)h, P2 (Γn )Γn (0)h + O(ε). Hence, the condition Pn P1 (Γn )Γ(1) n (0)Pn = 0,

(4.20)

or, equivalently, Pn Γ(1) n (0)P1 (Γn )Pn = 0,

is necessary and sufficient for the existence of a finite limit of the matrix function S∗ (ε)Γn (ε)−1 S(ε) as ε ↓ 0. Remark 4.7. The condition (4.20) as well as the condition (2) (0)P (Γ )P = 0 P P (Γ )Γ (0)P = 0 Pn Γ(2) 1 n n 1 n n n n n

(4.21)

are necessary for an extended set of matrices (1) (1) (1) C0 , . . . , C2n , W + S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0), C2n+2 to be, in the degenerate case, the set of the first 2n + 3 moments of some nondecreasing measure on the half-axis (0, ∞). Indeed, let the condition of the remark hold. Then by Theorem 1.1, both Hankel matrices Γn and Γn+1 are non-negative definite. Let ξ ∈ Ln (Cs ) be a non-zero null-vector of Γn . Then for the vector ξ ∈ Ln+1 (Cs ) ξ= 0

18 we have:

V.M. Adamyan and I.M. Tkachenko Γn+1 ξ, ξ = (Γn ξ, ξ) = 0.

ξ is a null-vector of Γn+1 . Therefore, ξ is also Since Γn+1 ≥ 0, this means that (1) a null-vector of the Hankel matrix Γn (0), which implies (4.20). The condition (4.21) can be verified in the same manner. Let us introduce the linear subspace ⎧ ⎫ ⎞ ⎛ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎜ .. ⎟ ⎜ . ⎟ (4.22) Nn := h ∈ Cs : ⎜ ⎟ ∈ Pn Mn . ⎪ ⎪ ⎝ 0 ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ h By definition (4.22) for any h ∈ Nn there is n−1

ξ(h) ∈ Ln−1 (Cs ) , ξ(h) = (ξj (h))j=0 such that

ξ(h) h

∈ Mn .

The application of (4.20) and (4.21) to such a null-vector yields (1) (1) (1) W + S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0) h = −Cn+1 h−. . .−C2n h, (4.23) (1) (1) (1) C2n+2 h = −Cn+2 h − . . . − W + S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0) h. (4.24) Thus all admissible extensions (1) (1) (1) C2n+1 = W + S∗1 (0)P2 Γn−1 Υn−1 P2 Γn−1 S1 (0), C2n+2 of the moment set must coincide on the subspace Mn . Since admissible matrices C2n+1 , C2n+2 are Hermitian, they can differ only on the subspace N⊥ n = Cs N n . We proved that for any set of Hermitian s × s matrices C0 , C1 , C2 , . . . , C2n which satisfy the conditions of Theorem 1.1, the set of extensions C2n+1 , C2n+2 which generates a completely degenerate Stieltjes problem, is always non-empty and the variety of such extensions is exhausted by the expressions ⎛ ⎞ Cn+1 ⎜ ⎟ (1) (1) (1) .. C2n+1 = S∗1 P2 Γn−1 Υn−1 P2 Γn−1 S1 + W, S1 = ⎝ (4.25) ⎠, . C2n n (1) , Γ(1) (4.26) C2n+2 = Γ(1) n P2 (Γn )Υn P2 (Γn )Γn n = (Cj+k+1 )j,k=0 , nn

where W runs through the set of all non-negative s × s-matrices which satisfy the condition WNn = {0} and (. . .)nn stands for the nnth block of the corresponding block-matrix.


19

Proceeding from the assertion of Theorem 4.1 and the results of this and previous sections, we arrive at the following algorithm for finding all canonical solutions of the truncated Stieltjes matrix moment problem: • for the set of matrix moments C0 , C1 , C2 , . . . , C2n which satisfy the condi(1) tions of Theorem 1.1, find the orthogonal projectors P1 Γn−1 and P1 (Γn ) (1)

onto the null-spaces of Hankel matrices Γn−1 = (Cj+k+1 )n−1 j,k=0 and Γn = n (Cj+k )j,k=0 , respectively; (1)

• find Hermitian matrices Υn−1 , Υn which satisfy the equations (1) (1) (1) (1) Γn−1 Υn−1 = P2 Γn−1 = I − P1 Γn−1 , Γn Υn = P2 (Γn ) = I − P1 (Γn ) and the conditions (1) (1) Υn−1 P1 Γn−1 = Υn P1 (Γn ) = 0; • taking any matrix W≥0 which satisfies the condition WPn P1 (Γn ) = 0, define the extension C2n+1 by (4.25) and then the extension C2n+2 by formula (4.26); n+1 • for the Hankel matrix Γn+1 = (Cj+k )j,k=0 determine the (n + 2) × s matrix = X

X I

⎛

⎞ X0 ⎜ ⎟ , X = ⎝ ... ⎠ , Xn

(4.27)

= 0; then with s × s matrices Xj , I, which satisfies the equation Γn+1 X determine the (n + 2) × s matrix ⎛

Y0 .. .

⎜ ⎜ Y=⎜ ⎝ Yn−1 Yn

⎞ j ⎟ ⎟ Ck Xn+1−j+k , Xn+1 = I; ⎟ , Yj = ⎠ k=0

• for the matrix polynomials D(z) = z n+1 I +

n k=0

z k Xk , E(z) =

n k=0

z k Yk

20

V.M. Adamyan and I.M. Tkachenko find all poles tmin = t1 < t2 < · · · < tν = tmax , ν ≤ ns, of the rational matrix function E(z)D(z)−1 and its residues M1 , . . . , Mν at these poles, and put ⎧ ⎪ 0, t ≤ t1 , ⎪ ⎪ ⎪ ⎪ ⎪ M , t 1 1 < t ≤ t2 , ⎨ σ(t) = M1 + M2 , t2 < t ≤ t3 , ⎪ ⎪ ⎪. . . ⎪ ⎪ ⎪ ⎩M + · · · + M , t > t . 1 ν ν

5. Non-canonical solutions By (4.25), (4.26) the set of all canonical solutions of the truncated matrix Stieltjes moment problem is parametrized in the degenerate but not the completely degenerate case by the set of s × s matrices W ≥ 0 which satisfy the condition WNn = {0}. To find the description of all canonical solutions σ W (t), notice that for the extended Hankel matrix Γn+1 the (n + 1) × s matrix X (4.27) is of the form X = X⎛0 + XW⎞ , ⎛ ⎞ X00 Cn+1 ⎜ ⎟ ⎜ ⎟ .. X0 = ⎝ ... ⎠ := Υn ⎝ ⎠, . X0n C2n+1 (1) (1) (1) ∗ C2n+1 = S1 P2 Γn−1 Υn−1 P2 Γn−1 S1 ; ⎛ ⎛ ⎛ ⎞ ⎞ X10 W X10 ⎜ ⎜ ⎜ .. ⎟ ⎜ ⎟ . . XW = ⎝ ⎠ , X1 = ⎝ . ⎠ := Υn ⎜ . ⎝ X1n X1n W Put D0n (z) = z n+1 I + E0n (z) = E1n (z) =

n # k=0 n−1 # k=0

n #

z k X0k , D1n (z) =

k=0

z k Y0k, Y0j = z k Y1k, Y1j =

j # k=0 j #

n #

0 .. . 0 I − Pn

⎞

(5.1)

⎟ ⎟ ⎟. ⎠

z k X1k ;

k=0

Ck X0n+1−j+k , X0n+1 = I,

(5.2)

Ck X0n−j+k .

k=0

It stems from the above results that the Nevanlinna type formula ∞ −1 1 dσ W (t) = − E0n (z) + E1n (z)W D0n (z) + D1n (z)W t−z

(5.3)

0

establishes a one-to-one correspondence between the set of all canonical solutions σ W of the truncated matrix Stieltjes moment problem and the set of s × s matrix W ≥ 0 which satisfy the condition WNn = {0}.


21

Notice also that the matrix functions D0n (z), D1n (z), E0n (z), E1n (z) are independent of the choice of W ≥ 0, WNn = {0}. Let us denote by P n the orthoprojector onto the subspace Nn in Cs . By definition of these matrix functions or, irrespectively of this, from (5.3) it follows that D0n (z)∗ E0n (z) = E0n (z)∗ D0n (z), D1n (z)∗ E1n (z) = E1n (z)∗ D1n (z); (5.4) D0n (z)∗ E1n (z) − E0n (z)∗ D1n (z) = I − P n , Im z ≥ 0. Besides, by (5.3), for any W ≥ 0, WP n = 0 the matrix function D0n (z) + D1n (z)W is invertible in the upper half-plane and on the half-axis (−∞, 0). By applying these facts and the arguments used in [1], [2] and not adduced here, we come to the following description of all solution of the truncated matrix Stieltjes moment problem in the degenerate case. Theorem 5.1. Let the conditions of Theorem 1.1 hold. Then the formula ∞ 1 dσ Ξ (t) t−z 0 −1 − E0 (z) + E1 (z) (Ξ (z) + zI)(I − P n ) × D0 (z) + D1 (z) Ξ (z)−1 + zI (I − P n ) ,

(5.5)

establishes a one-to-one correspondence between the set of all solutions σ Ξ (t) of the truncated matrix Stieltjes problem and the subset S of the Nevanlinna matrix functions Ξ(z) with values in Cs Nn . Remark 5.2. The subset of canonical solutions is generated by substitution into −1 (5.5) of matrix functions Ξ(z) = (W − zI) with W ≥ 0 such that Nn ⊂ ker W. Acknowledgement The financial support of the Polytechnic University of Valencia is gratefully acknowledged. V. Adamyan was also supported by the USA Civil Research and Development Foundation and the Government of Ukraine (CRDF grant UM12567-OD-03).

References [1] V.M. Adamyan, I.M. Tkachenko, Solution of the truncated matrix Hamburger moment problem according to M.G. Krein, Operator Theory: Advances and Applications, 118 (2000), 33–52 (Proceedings of the Mark Krein International Conference on Operator Theory and Applications, Vol.II, Operator Theory and Related Topics), Birkh¨ auser, Basel. [2] V.M. Adamyan, I.M. Tkachenko, Solution of the Stieltjes Truncated Matrix Moment Problem, Opuscula Mathematica, 25 (2005), no.1, 5–24. [3] N.I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner, N.Y., 1965.

22


[4] N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Spaces, Frederick Ungar Publishing Co., N.Y., 1966. [5] V. Bolotnikov, Degenerate Stieltjes moment problem and J-inner polynomials, Z. Anal. Anwendungen 14 (1995), no. 3, 441–468. [6] R.E. Curto and L.A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math., 17 (1991), 603–635. [7] H. Dym, On Hermitian Block Hankel Matrices, Matrix Polynomials, the Hamburger Moment Problem, Interpolation and Maximum Entropy, Integral Equations Operator Theory, 12 (1989), 757–812. [8] G.-N. Chen, Y.-J. Hu, A unified treatment for the matrix Stieltjes moment problem in both nondegenerate and degenerate cases, J. Math. Anal. Appl. 254 (2001), no. 1, 23–34. [9] Y.-J. Hu, G.-N. Chen, A unified treatment for the matrix Stieltjes moment problem, Linear Algebra Appl. 380 (2004), 227–239. [10] S. Karlin, W.S. Studden, Tschebyscheff systems: with applications in analysis and statistics, Interscience, 1966. [11] M.G. Krein, The theory of extensions of semi-bounded Hermitian operators and its applications, Mat. Sbornik I, 20 (1947), 431–495; II, 21 (1947), 365–404 (in Russian). [12] M.G. Krein, M.A. Krasnoselskii, Fundamental theorem on the extension of Hermitian operators and certain of their applications to the theory of orthogonal polynomials and the problem of moments, Uspekhi. Matem. Nauk (N.S.) 2 (1947), no. 3(19), 60–106 (in Russian). [13] M.G. Krein, A.A. Nudel’man, The Markov moment problem and extremal problems, “Nauka”, Moscow, 1973 (in Russian). English translation: Translation of Mathematical Monographs AMS, 50, 1977. [14] T.J. Stieltjes, Recherches sur les fractions continues, Annales de la Faculté des Sciencies de Toulouse, 8 (1894), 1–122; 9 (1895), 1–47. [15] T.J. Stieltjes, Collected Papers, G. van Dijk (ed.), Springer, Berlin, 1993. Vadim M. Adamyan Department of Theoretical Physics I.I. Mechnikov Odessa National University 65026 Odessa Ukraine e-mail: [email protected] Igor M. Tkachenko Department of Applied Mathematics Polytechnic University of Valencia 46022 Valencia Spain e-mail: [email protected]


Q-functions of Quasi-selfadjoint Contractions Yury Arlinski˘ı, Seppo Hassi and Henk de Snoo To Heinz Langer on the occasion of his retirement

Abstract. A bounded everywhere defined operator T in a Hilbert space H is said to be a quasi-selfadjoint contraction or (for short) a qsc-operator, if T is a contraction and ker (T − T ∗ ) = {0}. For a closed linear subspace N of H containing ran (T − T ∗ ) the operator-valued function QT (z) = PN (T − zI)−1 N, |z| > 1, where PN is the orthogonal projector from H onto N, is said to be a Q-function of T acting on the subspace N. The main properties of such Q-functions are studied, in particular the underlying operator-theoretical aspects are considered by using some block representations of the contraction T and analytical characterizations for such functions QT (z) are established. Also a reproducing kernel space model for QT (z) is constructed. In the special case where T is selfadjoint QT (z) coincides with the Q-function of the symmetric operator A := T (H N) and its selfadjoint extension T = T ∗ in the usual sense. Mathematics Subject Classification (2000). Primary: 47A10, 47A56, 47A64; Secondary 47A05, 47A06, 47B15. Keywords. Symmetric contraction, contractive extension, quasi-selfadjoint operator, Q-function, operator model, resolvent.

1. Introduction The concept of a Q-function was introduced by M.G. Kre˘ın for the case of a densely symmetric operator S in a Hilbert space H with equal defect numbers by means of a selfadjoint extension A of S, cf. [26], [28], [33], and also [29], [30], [31]. Such a function belongs to the class N of Nevanlinna (or Herglotz-Nevanlinna) functions, i.e., Q(z) ∈ N if it is holomorphic in the open upper and lower half-planes and satisfies the conditions Q(¯ z ) = Q(z)∗ and (Im z) (Im Q(z)) ≥ 0, z ∈ C+ ∪ C− . The This work was supported by the Research Institute for Technology at the University of Vaasa. The first author was also supported by the Academy of Finland (projects 203227, 208057) and the Dutch Organization for Scientific Research NWO (B 61–553).

24

Yu.M. Arlinski˘ı, S. Hassi and H.S.V. de Snoo

Q-function plays an essential role in Kre˘ın’s resolvent formula, which describes all (generalized resolvents of) selfadjoint extensions of S. In fact, all generalized resolvents (canonical as well as exit space) were first described independently by M.A. Naimark [41] and M.G. Kre˘ın [26]; see also [29] for further historical remarks. A characteristic property of a Q-function Q(z) in the class of Nevanlinna functions is that Im Q(z) is invertible (at some or equivalently at every point z ∈ C+ ∪ C− ): every Nevanlinna function with this property is a Q-function of some simple symmetric operator S and a selfadjoint extension A of S in a Hilbert space H. Moreover, the simple (completely non-selfadjoint) symmetric operator S and its selfadjoint extension A are essentially unique in the sense that the Q-function of S determines S and A uniquely up to unitary equivalence. Another approach for describing selfadjoint as well as non-selfadjoint intermediate extensions of a symmetric operator is via a boundary value space and the corresponding Weyl function, see [21], [19], [18], and the references therein. Two special subclasses of Q-functions, consisting of the so-called Qµ - and QM -functions, which belong to the class N of Nevanlinna functions were defined and investigated by M.G. Kre˘ın and I.E. Ovcharenko in [32]. Here the underlying symmetric operator is a non-densely defined contraction. In a recent paper [7] by the authors some extensions of Qµ - and QM -functions were introduced; in fact, this paper contains also some corrections to the result stated in [32]. Some other type of Q-function associated to a non-densely defined symmetric contraction has been considered in [47], including the resolvent formulas for the selfadjoint (canonical and exit space) extensions. In this paper a class of operator-valued Q-functions associated with a nondensely defined symmetric contraction A and its, in general, non-selfadjoint contractive extensions T is introduced. By definition a bounded operator T in the Hilbert space H is a quasi-selfadjoint contraction or, for short, a qsc-operator if dom T = H, T ≤ 1, and ker (T − T ∗ ) = {0}. Let T be a qsc-operator and let N be a proper subspace of H which contains ran (T − T ∗ ). Define the operator-valued function Q(z) as follows Q(z) = PN (T − zI)−1 N,

|z| < 1.

(1.1)

In what follows the function Q(z) in (1.1) will be called a Q-function of T with respect to the subspace N ⊂ H. Observe, that if T is selfadjoint then the function Q defined by (1.1) is an ordinary Q-function associated with T and the symmetric restriction A := T H0 of T , where H0 = H N. However, if T is not selfadjoint this function in general is not a Nevanlinna function. A qsc-operator T may be considered as a contractive, in general, non-selfadjoint extension of the symmetric contraction A = T H0 which is also called a quasi-selfadjoint contractive extension of A; here A is symmetric due to H0 ⊂ ker (T − T ∗ ). Such kind of extensions were parametrized and investigated in [10] and [12]. The special case of selfadjoint contractive extensions was investigated by M.G. Kre˘ın [27] and by M.G. Kre˘ın and I.E. Ovcharenko [32]. In particular, in [32] two special Q-functions of the Nevanlinna class for the symmetric contraction were defined and studied

Q-functions of Quasi-selfadjoint Contractions

25

and the resolvent formulas for selfadjoint contractive extensions (sc-extensions) were established. These formulas were extended in [10] and [12] for qsc-extensions. A boundary value space approach for describing extensions of dual pairs of densely defined operators appears in [37] and for dual pairs of linear relations and their canonical and generalized resolvents in [39], [40], see also [34], [35]. In the present paper the approach can be seen as a non-selfadjoint counterpart of the Q-function approach developed and systematically used in the papers of M.G. Kre˘ın and H. Langer, cf., e.g., [29]–[31]. The contents of this paper will be briefly described. In Section 2 some preliminary notions are introduced. The extension theory for closed symmetric contractions is developed in Section 3. This includes a discussion of minimality of the underlying symmetric operator A and its contractive extensions. The Q-functions for intermediate contractive extensions as in (1.1) are introduced in Section 4, where also a number of associated nonnegative kernels will appear. A resolvent formula for qsc-extensions of a symmetric contraction A is derived in Section 5. It involves a Q-function of the form (1.1) for a given qsc-extension T of A. In Section 6 a model for such Q-functions is constructed by means of a qsc-operator acting in a reproducing kernel Hilbert space and it is proved that two N-minimal qscoperators whose Q-functions in (1.1) coincide are unitarily equivalent. This model is used to establish some characteristic properties of Q-functions of qsc-operators in Section 7. In Section 8 linear fractional transformations of Q-functions are considered. The results in the present paper can be connected with and augmented by the study of a certain class of passive systems. In particular, the Q-functions of quasi-selfadjoint operators investigated in the present paper are in one-to-one correspondence with the transfer functions of so-called passive quasi-selfadjoint systems, which are introduced and investigated in [8]. The work in this paper is a continuation of some investigations done by several authors in the early eighties, including Heinz Langer and Bj¨ orn Textorius, being influenced by ideas developed by M.G. Kre˘ın and I.E. Ovcharenko. It is a pleasure to record the mathematical indebtedness of the authors of the present paper to Heinz Langer over a period of many years.

2. Preliminaries 2.1. Basic notations The class of all continuous linear operators defined on a complex Hilbert space H1 and taking values in a complex Hilbert space H2 is denoted by L(H1 , H2 ) and L(H) := L(H, H). The domain, the range, and the null-space of a linear operator T are denoted by dom T , ran T , and ker T . For T ∈ L(H) the operators TR = (T + T ∗ )/2, TI = (T − T ∗ )/2i are said to be the real and the imaginary part of T . For a contraction T ∈ L(H1 , H2 ) the defect operator DT of T is defined by DT := (I − T ∗ T )1/2 .

(2.1)

26


It is a nonnegative contraction and satisfies the well-known commutation relation T DT = DT ∗ T,

(2.2)

cf. [46]. The closure of the range ran DT is denoted by DT and ρ(T ) stands for the set of all regular points of a closed operator T . If Rl and Rr are two nonnegative operators in L(H) and S0 ∈ L(H), then the symbol B(S0 , Rl , Rr ) denotes the operator ball in L(H) with the center S0 and the left and right radii Rl and Rr , 1/2 1/2 respectively, i.e., the set of all operators in L(H) of the form T = S0 + Rl XRr , where X is a contraction from ran Rr into ran Rl . It is well known, see [43], [44], that a necessary and sufficient condition for T ∈ L(H) to belong to B(S0 , Rl , Rr ) is the following: 2

|((T − S0 )f, g)| ≤ (Rr f, f ) (Rl g, g) for all f, g ∈ H.

(2.3)

If Rl = Rr = R the corresponding operator ball is denoted by B(S0 , R). 2.2. Quasi-selfadjoint contractions Recall that T ∈ L(H) is a quasi-selfadjoint contraction (a qsc-operator) if dom T = H,

T ≤ 1, and ker (T − T ∗ ) = {0}.

A qsc-operator T is said to be a quasi-selfadjoint contractive extension or qscextension of a closed symmetric contraction A if dom A ⊂ ker (T − T ∗ ) or equivalently ran (T − T ∗ ) ⊂ (dom A)⊥ , cf. [10], [12]. Clearly, an operator T ∈ L(H) is a qsc-extension of A if and only if A ⊂ T and A ⊂ T ∗ , or, equivalently, if T is an intermediate extension of A. A qsc-operator T has always symmetric restrictions A for which T is a qsc-extension. Namely, with a subspace N ⊃ ran (T − T ∗ ) define dom A = H N,

A = T dom A.

Then dom A ⊂ ker (T − T ∗). A qsc-operator T is called completely non-selfadjoint if there is no non-zero invariant subspace on which the restriction of T is selfadjoint. Lemma 2.1. [15] A qsc-operator T is completely non-selfadjoint if and only if span { ran T n (T − T ∗ ) : n = 0, 1, . . . } = H. 2.3. The classes C(α) Let α ∈ [0, π/2) and denote by S(α) the following sector of the complex plane: S(α) = { z ∈ C : | arg z| ≤ α } . A linear operator S, in general unbounded, in a Hilbert space H is said to be sectorial with vertex at the origin and semiangle α, if its numerical range W (S) = { (Sf, f ) : f = 1, f ∈ dom S }


27

is contained in the sector S(α), cf. [25]. This condition is equivalent to |Im (Sf, f )| ≤ (tan α) Re (Sf, f ) for all f ∈ dom S. If the resolvent set of S is not empty then S is called maximal sectorial. A maximal sectorial operator S is densely defined and its adjoint S ∗ is also a maximal sectorial operator. A bounded operator T on a Hilbert space H is said to belong to the class C(α), α ∈ (0, π/2), if T sin α ± i cos α I ≤ 1, (2.4) cf. [3]. Clearly, T belongs to C(α) if and only if T ∗ belongs to C(α). Moreover, it follows from (2.4) that the operators belonging to C(α) are contractive. The condition (2.4) is equivalent to each of the following two conditions: |(TI f, f )| ≤ or

tan α DT f 2 2

for all f ∈ H;

(2.5)

the operator (I − T ∗ )(I + T ) is sectorial with vertex at the origin and semiangle α,

(2.6)

cf. [4]. Note that the linear fractional transformation T = (I − S)(I + S)−1 of a maximal sectorial operator S with vertex at the origin and semiangle α is an operator of the class C(α). Let $

= C {C(α) : α ∈ [0, π/2)}.

were studied in [3], [4]. In particular, Some properties of the operators in the class C

in [3] it was proved that T ∈ C implies that ran DT n = ran DT ∗n = ran DTR ,

n = 1, 2, . . . ,

where TR is the real part of T . Furthermore it was proved in [3] that the subspace DT reduces the operator T , that the operator T ker DT is selfadjoint and unitary, and that T DT is a completely non-unitary contraction of the class C00 , i.e., lim T n f = lim T ∗n f = 0

n→∞

n→∞

for all f ∈ DT ,

cf. [46]. 2.4. The Schur-Frobenius formula for the resolvent of a block operator Let the Hilbert space H be decomposed as H = H1 ⊕ H2 and decompose T ∈ L(H) accordingly: T11 T12 T = , Tij ∈ L(Hi , Hj ). (2.7) T21 T22 Define the operator-valued functions VT (z) = T21 (T11 − zI)−1 T12 − T22 ,

WT (z) = −zI − VT (z),

z ∈ ρ(T11 ). (2.8)

28


By the Schur-Frobenius formula the resolvent (T − z)−1 of T can be rewritten in the block form (T11 − zI)−1 I + T12 WT (z)−1 T21 (T11 − zI)−1 −(T11 − zI)−1 T12 WT−1 (z) WT−1 (z) −WT−1 (z)T21 (T11 − zI)−1 (2.9) for z ∈ ρ(T ) ∩ ρ(T11 ). In particular, −1

PH2 (T − zI)−1 H2 = − (VT (z) + zI)

,

z ∈ ρ(T ) ∩ ρ(T11 ).

(2.10)

2.5. Nevanlinna functions Let N be a Hilbert space. An operator-valued function V (z), z ∈ C \ R, with values in L(N) is said to be a Nevanlinna function or an R-function, cf. [24], if V (z) is holomorphic on C \ R, V ∗ (z) = V (z), and Im z Im V (z) ≥ 0 for all z ∈ C \ R. The subclass of Nevanlinna functions V (z) which are holomorphic on the domain Ext [−1, 1] := C \ [−1, 1] is denoted by NN [−1, 1]. By the general theory of Nevanlinna functions, cf. [24], [15], every function V (z) in NN [−1, 1] has an integral representation of the form 1 V (z) = Γ + −1

dG(t) , t−z

where Γ is a bounded selfadjoint operator on N and the L(N)-valued function G(t) is nondecreasing, nonnegative, normalized by G(−1 − 0) = 0, and has finite total variation concentrated on [−1, 1]. Clearly, V (∞) := s − lim V (z) = Γ. The next z→∞

result is also well known, cf. [15]. Theorem 2.2. Let N be a Hilbert space and let V (z) ∈ NN [−1, 1]. Then there exist a Hilbert space H, a selfadjoint contraction B on H, and F ∈ L(N, H), such that V (z) = V (∞) + F ∗ (B − zI)−1 F,

z ∈ Ext [−1, 1].

(2.11)

In what follows the subclass of functions V (z) in NN [−1, 1] which have the limit values V (±1) in L(N) plays a central role. In this case Theorem 2.2 can be completed as follows. Theorem 2.3. Let N be a Hilbert space and let V (z) ∈ NN [−1, 1]. If for all f ∈ N the limit values lim (V (x)f, f ) , lim (V (x)f, f ) (2.12) x↑−1

x↓1

are finite, then there exist a Hilbert space H, a selfadjoint contraction B in H, and an operator G ∈ L(N, DB ), such that 2 (B − zI)−1 G, V (z) = V (∞) + G∗ DB

z ∈ Ext [−1, 1].

(2.13)

Conversely, for every function V (z) of the form (2.13) the limit values (2.12) exist for all f ∈ N and are finite.


29

Proof. By Theorem 2.2 V (z) has the representation (2.11), where B is a selfadjoint contraction in a Hilbert space H and F ∈ L(N, H). Since the limits in (2.12) exist for all f ∈ N, one concludes that ran F ⊂ ran (I − B)1/2 ∩ ran (I + B)1/2 . Consequently, ran F ⊂ ran DB and this implies that F = DB G for some operator G ∈ L(N, DB ), cf. [23]. Conversely, if V (z) is of the form (2.13) then ran DB ⊂ ran (B ± I)1/2 and this implies the existence of the limit values (2.12) for all f ∈ N, cf. [32]. It follows from Theorem 2.3 that V (−1) := s − lim V (x) = V (∞) + G∗ (I − B)G ∈ L(N), x↑−1

V (1) := s − lim V (x) = V (∞) − G∗ (I + B)G ∈ L(N),

(2.14)

x↓1

so that V (−1) + V (1) = 2V (∞) − 2G∗ BG,

V (−1) − V (1) = 2G∗ G.

(2.15)

2.6. Nonnegative kernels, reproducing kernel Hilbert spaces, and sectorial kernels An operator-valued function K(z, ξ) : Ω × Ω → L(N), Ω ⊂ C, is said to be a nonnegative kernel [1], [13], [42], if n

(K(wj , wi )fi , fj )N ≥ 0

i,j=1

for every choice of points {wi }ni=1 ⊂ Ω and vectors {fi }ni=1 ⊂ N. With the kernel K(z, ξ) is associated a reproducing kernel Hilbert space HK . It is the completion of the linear space of vectors of the form n

K(·, wi )fi ,

{wi }ni=1 ⊂ Ω,

{fi }ni=1 ⊂ N,

n ∈ N,

i=1

with respect to the inner product ⎛ ⎞ n m ⎝ K(·, wi )fi , K(·, µj )gj ⎠ i=1

j=1

=

HK

m n

(K(µj , wi )fi , gj )N .

i=1 j=1

Then the Hilbert space HK consists of the N-valued functions f (·) such that for every h ∈ N the reproducing property holds: (f (·), K(·, w)h)HK = (f (w), h)N ,

w ∈ Ω.

Observe that an L(N)-valued function V (z) belongs to the Nevanlinna class N(N) if and only if the function K(z, ξ) =

V (z) − V (ξ)∗ , z−ξ

z, ξ ∈ C \ R,

30


is a nonnegative kernel. Also note that the kernel associated with generalized resolvents (of selfadjoint exit space extensions) in a Hilbert space is given by K(z, ξ) =

V (z) − V (ξ)∗ − V (z)V (ξ)∗ , z−ξ

z, ξ ∈ C \ R.

An operator-valued function K(z, ξ) : Ω × Ω → L(N), Ω ⊂ C, is said to be an α-sectorial kernel, if n (K(wj , wi )fi , fj )N ∈ S(α) i,j=1

for every choice of points {wi }ni=1 ⊂ Ω and vectors {fi }ni=1 ⊂ H, i.e., % % ⎛ ⎞ % % n n % % %Im (K(wj , wi )fi , fj )N %% ≤ (tan α) Re ⎝ (K(wj , wi )fi , fj )N ⎠ , % % % i,j=1 i,j=1 cf. [5]. For α = 0 the corresponding kernel is nonnegative.

3. qsc-extensions of closed symmetric contractions 3.1. A decomposition of closed symmetric contractions Let A be a non-densely defined closed symmetric contraction in the Hilbert space H with the domain dom A =: H0 and let N := H dom A. Let P0 and PN be the orthogonal projections in H onto H0 and N, respectively. Then the operator A0 = P0 A is contractive and selfadjoint in the subspace H0 . Let DA0 = (I −A20 )1/2 be the defect operator determined by A0 . The operator A21 = PN A is also contractive. 2 Moreover, it follows from A∗ A ≤ I that A∗21 A21 ≤ DA . Therefore, the identity 0 K0 DA0 f = PN Af,

f ∈ dom A,

defines a contractive operator K0 from DA0 := ran DA0 into N, cf. [20], [23]. This gives the following decomposition for the symmetric contraction A A0 A = A0 + K0 DA0 = . (3.1) K 0 D A0 3.2. A matrix representation for qsc-extensions Let the closed symmetric contraction A be defined on the subspace H0 = dom A and decompose A according to H = H0 ⊕ N as in (3.1). Let T be a qsc-extension of A, so that A ⊂ T and A ⊂ T ∗ , and decompose T = (Tij ) also with respect ∗ to H = H0 ⊕ N, cf. (2.7). Then clearly T11 = A0 , T12 = T21 = K0 DA0 . The next result gives a parametrization of all qsc-extensions of A and some of its subclasses by means of block formulas, cf. [14], [17], [45], and [10], [12]. For completeness a short, simple proof is presented. Theorem 3.1. Let A be a closed symmetric contraction A in H = H0 ⊕ N with dom A = H0 and decompose A as in (3.1). Then:

Q-functions of Quasi-selfadjoint Contractions (i) the formula A0 T = K 0 D A0

DA0 K0∗ −K0 A0 K0∗ + DK0∗ XDK0∗

31

H0 H0 → : N N

(3.2)

gives a one-to-one correspondence between all qsc-extensions T of the symmetric contraction A = A0 + K0 DA0 and all contractions X in the subspace DK0∗ := ran DK0∗ ⊂ N; (ii) T in (3.2) belongs to the class C(α) if and only if X belongs to the class C(α), α ∈ (0, π/2); (iii) T is a selfadjoint contractive extension of A if and only if X in (3.2) is a selfadjoint contraction in DK0∗ . Proof. (i) Every operator T ∈ L(H) satisfying the conditions A ⊂ T and A ⊂ T ∗ admits the block-matrix representation of the form H0 H0 A0 DA0 K0∗ : → , T = K 0 D A0 D N N where D ∈ L(N). Then I − T ∗ T is given in the block form 2 − DA0 K0∗ K0 DA0 −A0 DA0 K0∗ − DA0 K0∗ D DA ∗ 0 . I −T T = 2 2 ∗ ∗ −K0 DA0 A0 − D∗ K0 DA0 DK ∗ − K0 A0 K0 − D D 0 Contractivity of T means that 0 ≤ DA0 f − A0 K0∗ h2 + DK0∗ h2 − K0 DA0 f + Dh2 ,

(3.3)

ran K0∗

for all f ∈ H0 and h ∈ N. Since ⊂ DA0 and A0 DA0 ⊂ DA0 , there exists a sequence {fn }∞ n=1 ⊂ DA0 such that for a given h ∈ N the equality lim DA0 fn = A0 K0∗ h

n→∞

holds. Hence, it follows from (3.3) that E := K0 A0 K0∗ + D satisfies Eh2 ≤ DK0∗ h2 ,

E ∗ h2 ≤ DK0∗ h2 ,

h ∈ N,

(3.4)

where the second inequality follows from the first one by taking into account that T ∗ is a contraction, too. By the second inequality in (3.4) there exists a contraction Z ∈ L(N, DK0∗ ) such that E = DK0∗ Z, i.e., D = −K0 A0 K0∗ + DK0∗ Z. By substituting this into (3.3) one obtains 0 ≤ DK0 (DA0 f − A0 K0∗ h) − K0∗ Zh2 + DK0∗ h2 − Zh2 ,

f ∈ H0 ,

h ∈ N, (3.5)

since by means of (2.2) one has −K0(DA0 f − A0 K0∗ h) + DK0∗ Zh2 = −K0 (DA0 f − A0 K0∗ h)2 − Zh2 + K0∗ Zh2 − 2Re (DK0 (DA0 f − A0 K0∗ h), K0∗ Zh) . Due to the inclusion ran Z ⊂ DK0∗ , one can choose a sequence {fn }∞ n=1 ⊂ DA0 such that for a given h ∈ N the equality lim DK0 DA0 fn = DK0 A0 K0∗ h + K0∗ Zh

n→∞

(3.6)

32


holds. Now (3.5) shows that Zh2 ≤ DK0∗ h2 for all h ∈ N, so that Z = XDK0∗ for some contraction X in DK0∗ . Therefore, E = DK0∗ Z = DK0∗ XDK0∗ and D = −K0 A0 K0∗ + DK0∗ XDK0∗ .

(3.7)

Conversely, let D be of the form (3.7), where X is a contraction in DK0∗ . 2 ≥ 0 implies that T given by (3.2) satisfies Then DX f f , (I − T ∗ T ) h h (3.8) ∗ ∗ 2 2 ∗ ∗ = DK0 (DA0 f − A0 K0 h) − K0 XDK0 h + DX DK0 h ≥ 0, cf. (3.5). Thus, every contraction X in DK0∗ defines a qsc-extension T of A via (3.2). (ii) It follows from (3.2) and (3.8) that T satisfies (2.5) if and only if |(XI DK0∗ h, DK0∗ h)| (3.9) tan α DK0 (DA0 f − A0 K0∗ h) − K0∗ XDK0∗ h2 + DX DK0∗ h2 ≤ 2 holds for all f ∈ H0 , h ∈ N. In view of (3.6) the condition (3.9) is equivalent to tan α DX h2 (3.10) |(XI h, h)| ≤ 2 for all h ∈ DK0∗ . (iii) The statement is clear since T in (3.2) is selfadjoint if and only if X is selfadjoint in DK0∗ . The class of all selfadjoint contractive (sc-) extensions of A in part (iii) of Theorem 3.1 forms an operator interval [Aµ , AM ]. Using the block representation (3.2) the endpoints of [Aµ , AM ] are given by DA0 K0∗ A0 , (3.11) Aµ = 2 K0 DA0 −K0 A0 K0∗ − DK ∗ 0 and

AM =

A0 K 0 D A0

DA0 K0∗ 2 −K0 A0 K0∗ + DK ∗ 0

,

(3.12)

with X = −I DK0∗ and X = I DK0∗ , respectively. From the formulas (3.11) and (3.12) it is seen that Aµ + AM 0 0 AM − Aµ A0 DA0 K0∗ = = , . 2 0 DK ∗ K0 DA0 −K0 A0 K0∗ 2 2 0 This means that all qsc-extensions in (3.2) of the symmetric contraction A form an operator ball Aµ + AM AM − Aµ , B 2 2 with center (Aµ + AM )/2


33

and equal left and right radii

√ Rl = Rr = (AM − Aµ )1/2 / 2.

The one-to-one correspondence between all qsc-extensions T of A and all contractions X in Theorem 3.1 can be reformulated also as follows 1/2 1/2 Aµ + AM AM − Aµ AM − Aµ T = + X , (3.13) 2 2 2 where the parameters X are contractions in the subspace ran (AM − Aµ ), cf. [10], [11], [12]. It is easy to see from (3.2), (3.11), and (3.12), that if T is a qsc-extension of A such that TR = (T + T ∗ )/2 = Aµ (AM ), then in fact T = Aµ (AM ). Namely, X = XR + iXI satisfies 2 0 ≤ X ∗ X = XR + i(XR XI − XI XR ) + XI2 ≤ I, (3.14) ∗ 2 0 ≤ XX = XR − i(XR XI − XI XR ) + XI2 ≤ I, 2 2 so that 0 ≤ XR + XI2 ≤ I and here clearly XR = I implies XI = 0.

The description of all contractive selfadjoint extensions of a symmetric contraction A as the operator interval [Aµ , AM ] is due to M.G. Kre˘ın [27]. In that paper the notion of shorted operators was also introduced and used for instance to establish the following characterization for Aµ and AM : ran (I + Aµ )1/2 ∩ N = {0},

ran (I − AM )1/2 ∩ N = {0},

(3.15)

cf. [7], [22]. Block formulas for describing all contractive extensions of a dual pair appear in [14], [17], [45], a description in Crandall’s form [16] in [2]. The one-toone correspondence between all qsc-extensions T of A of the class C(α) and all operators X in ran (AM − Aµ ) belonging to the class C(α) by means of (3.13) was proved in a different way in [12], another proof based on (3.2) was given in [38]. 3.3. Simplicity of the symmetric operator According to [32] a closed symmetric contraction A is said to be simple if there is no non-zero subspace in dom A which is invariant under A. Since A is symmetric, simplicity of A is equivalent to A being completely non-selfadjoint, i.e., to A having no selfadjoint parts. Lemma 3.2. Let the closed symmetric contraction A = A0 +K0 DA0 in H = H0 ⊕N, H0 = dom A, be decomposed as in (3.1) with K0 : DA0 → N. Then A is simple if and only if the subspace & ' Hs0 := span (A0 − zI)−1 K0∗ N : z ∈ ρ(A0 ) (3.16) = span { An0 K0∗ N : n = 0, 1, . . . } coincides with H0 . In this case, DA0 = H0 , K0 : H0 → N, and A0 f < f for all f ∈ H0 \{0}.

34


Proof. Suppose that A is simple. Then clearly ker DA0 = {0} or equivalently A0 f < f for all f ∈ H0 \ {0}, so that DA0 = H0 and K0 : H0 → N. Observe, that the subspace Hs0 in (3.16) and therefore also H0 Hs0 is invariant under A0 = A∗0 . Then the subspace H0 Hs0 is also invariant under DA0 . Moreover, H0 Hs0 = { f ∈ H0 : K0 An0 f = 0, n = 0, 1, . . . } .

(3.17)

It follows that K0 DA0 f = 0 for all f ∈ H0 Hs0 . Hence, in view of (3.1) Af = A0 f for all f ∈ H0 Hs0 . This means that the subspace H0 Hs0 is invariant under A. Since A is a simple, one concludes that Hs0 = H0 . Conversely, assume that Hs0 = H0 . Since ran K0∗ ⊂ DA0 and DA0 is invariant under A0 , the definition of Hs0 in (3.16) shows that Hs0 ⊂ DA0 . Hence, the assumption implies that H0 = DA0 = ran DA0 , so that ker DA0 = {0}. Now,

0 ⊂ H0 is a subspace which is invariant under A. Then for every suppose that H

0 , so that K0 DA0 f = 0 for all f ∈ H

0

f ∈ H0 one has Af = A0 f + K0 DA0 f ∈ H

0 = A0 H

0 . Hence, H

0 is invariant under A0 and DA . Moreover, since and A H 0

0 . This implies that K0 H

0 is dense in H

0 = {0} ker DA0 = {0} the image DA0 H n n

and since A0 H0 ⊂ H0 one has K0 A0 H0 = {0} for all n = 0, 1, . . ., i.e.,

0 ⊂ { f ∈ H0 : K0 An f = 0, n = 0, 1, . . . } = H0 Hs = {0}, H 0 0

cf. (3.17). Therefore, A is simple.

Let T be a qsc-extension of A in the Hilbert space H = H0 ⊕ N with H0 = dom A. It is evident that the subspace & ' HT := span (T − zI)−1 N : |z| > 1 = span { T n N : n = 0, 1, 2, . . . } , (3.18) is invariant under T , and that the subspace HT := H HT , ∗

is invariant under T . Since N ⊂

HT ,

(3.19)

one obtains

HT ⊂ N⊥ = dom A ⊂ ker (T − T ∗ ). Therefore the restriction of T ∗ to HT is a selfadjoint operator in HT . The restriction T HT (= PHT T HT ) is called the N-minimal part of T . Moreover, T is said to be N-minimal if the equality H = HT holds. If T be a qsc-extension of A then its adjoint T ∗ is also a qsc-extension of A and one can associate with it the subspace HT ∗ and the corresponding N-minimal part of T ∗ . The next result shows the Nminimal parts of T and T ∗ are qsc-extensions of the simple part A Hs0 of A in the same subspace HT = HT ∗ . Proposition 3.3. Let A be a symmetric contraction in H = H0 ⊕ N with H0 = dom A, let T be a qsc-extension of A in H, and let T ∗ be its adjoint. Then the subspaces HT , HT ∗ , and Hs0 of H = H0 ⊕ N as defined in (3.18) and (3.16) are connected by (H :=) HT = HT ∗ = Hs0 ⊕ N. (3.20)


35

In particular, the symmetric contraction A is simple if and only if the qsc-extension T , or equivalently T ∗ , of A is N-minimal. Proof. It follows from the Schur-Frobenius formula (2.9) that −(A0 − z)−1 DA0 K0∗ N −1 , |z| > 1, (T − z) N = N which implies that & ' span (T − zI)−1 N : |z| > 1 & ' = span (A0 − zI)−1 DA0 K0∗ N : z ∈ ρ(A0 ) ⊕ N & ' = (clos DA0 span (A0 − zI)−1 K0∗ N : z ∈ ρ(A0 ) ) ⊕ N. This shows that

HT = (clos DA0 Hs0 ) ⊕ N. (3.21) ∗ s Since ran K0 ⊂ DA0 and DA0 is invariant under A0 one has H0 ⊂ DA0 . In particular, Hs0 ∩ ker DA0 = {0}, which together with DA0 Hs0 ⊂ Hs0 implies that clos DA0 Hs0 = Hs0 . Hence, (3.21) implies the equality HT = Hs0 ⊕ N. It follows from (T ∗ − zI)−1 − (T − zI)−1 = (T − zI)−1 [T − T ∗ ](T ∗ − zI)−1 ,

|z| > 1,

∗

and the inclusion ran (T − T ) ⊂ N that (T ∗ − zI)−1 N ⊂ (T − zI)−1 N ⊂ HT , HT ∗

|z| > 1.

HT

Therefore, ⊂ and the reverse inclusion follows by symmetry. This completes the proof of (3.20). The last statement is clear from (3.20). For selfadjoint extensions of A the result in Proposition 3.3 has been given in [32]. In the case of closed densely defined symmetric operators A there is an equivalent criterion for the simplicity of A due to M.G. Kre˘ın based on the defect elements: span { ker (A∗ − λ) : λ ∈ C \ R } = H, cf. Lemma 3.2. This characterization has been extended to non-densely defined symmetric operators in [36].

4. Q-functions of qsc-operators Let T be a qsc-operator in a separable Hilbert space H and let N be a subspace of H such that N ⊃ ran (T − T ∗ ). The operator-valued function QT (z) = PN (T − zI)−1 N,

|z| > 1,

(4.1)

where PN is the orthogonal projection in H onto N, is said to be a Q-function associated with T and the subspace N. Clearly, it has the limit value QT (∞) = 0 and the Q-functions of T and T ∗ in N are connected by QT ∗ (z) = QT (¯ z )∗ ,

|z| > 1.

(4.2)

36


If T is a selfadjoint contraction then the Q-function QT (z) in (4.1) is a Nevanlinna function of the class NN [−1, 1]. The next result contains some basic properties for the Q-function QT (z) of a qsc-operator T as defined in (4.1). Proposition 4.1. Let QT (z) be a Q-function of a qsc-operator T as defined in (4.1). Then: (i) QT (z) has the following asymptotic expansion: 1 1 1 , z → ∞, (4.3) QT (z) = − I + 2 F + o z z z2 where F = −PN T N; (ii) Q−1 T (z) ∈ L(N) for all |z| > 1; −1 (iii) Q−1 T (z) has strong limit values QT (±1): −1 Q−1 T (−1) = lim QT (x), x↑−1

−1 Q−1 T (1) = lim QT (x); x↓1

(iv) for all f, g ∈ N the following inequality holds: % −1 % % Q (−1) + Q−1 (1) f, g %2 T T −1 −1 ≤ Q−1 QT (−1) − Q−1 T (−1) − QT (1) f, f T (1) g, g ; (v) the function −Q−1 T (z) − F − zI is an operator-valued Nevanlinna function; (vi) QT (z) ∈ NN [−1, 1] if and only if F = F ∗ . Moreover, if T is decomposed as in (3.2) with H0 = H N and A = T H0 , then F = K0 A0 K0∗ − DK0∗ XDK0∗ , ∗ ∗ Q−1 T (−1) = DK0 (X + I)DK0 ,

−Q−1 T (z)

(4.4)

∗ ∗ Q−1 T (1) = DK0 (X − I)DK0 ,

− F − zI = K0 (I −

A20 )(A0

−1

− zI)

K0∗ .

−1

Proof. (i) Clearly, limz→∞ zQT (z)h = limz→∞ zPN (T − zI) h ∈ N. Moreover, for all h ∈ N

(4.6)

h = −PN h for all

lim z(I + zQT (z))h = lim zPN T (T − zI)−1 h = −PN T h.

z→∞

(4.5)

z→∞

(4.7)

Hence, QT (z) admits the asymptotic expansion (4.3). (ii) Let |z| > 1, let f ∈ N, and let ϕ = (T − zI)−1 f . Then f ≤ (1 + |z|)ϕ and % % |(QT (z)f, f )| = % (T − zI)−1 f, f % = |(ϕ, (T − zI)ϕ)| % % |z| − 1 = %(ϕ, T ϕ) − z¯ϕ2 % ≥ f 2 . (|z| + 1)2 Since |(QT (z)f, f )| = |(QT (z)∗ f, f )|, this implies that QT (z)f ≥

|z| − 1 f , (|z| + 1)2

Therefore, Q−1 T (z) ∈ L(N) for all |z| > 1.

QT (z)∗ f ≥

|z| − 1 f . (|z| + 1)2


37

(iii) Decompose H = H0 ⊕N and write T in block form as in (3.2), where H0 = 1/2 , HN, A = T H0 , A0 = P0 A is a selfadjoint contraction in H0 , DA0 = I − A20 K0 ∈ L (DA0 , N) is a contraction, and X is a contraction in the subspace DK0∗ ⊂ N. The formula (4.4) for F is immediate from (3.2). Write Q−1 T (z) as in (2.10), Q−1 T (z) = −VT (z) − zI, where

|z| > 1,

VT (z) = K0 A0 + (A0 − zI)−1 (I − A20 ) K0∗ − DK0∗ XDK0∗ .

(4.8)

This shows that the limit values Q−1 T (±1) exist and that they are given by (4.5). (iv) It follows from (4.5) that −1 Q−1 T (−1) + QT (1) = DK0∗ XDK0∗ , 2 −1 Q−1 2 ∗ T (−1) − QT (1) = DK ∗ = I − K0 K0 . 0 2

(4.9)

2 It remains to apply the criterion (2.3) with S0 = 0 and Rl = Rr = DK ∗. 0 (v) It follows from (4.4) and (4.8) that (4.6) holds. Clearly, the function in (4.6) is a Nevanlinna function. (vi) If QT (z) ∈ NN [−1, 1], then −QT (z)−1 is a Nevanlinna function and now part (v) implies that F = F ∗ . Conversely, if F = F ∗ then the function VT (z) in (4.8) and −QT (z)−1 = VT (z) + zI are Nevanlinna functions. Therefore, QT (z) ∈ NN [−1, 1].

Let T be a qsc-operator, let QT (z) be defined by (4.1), and let F be defined by F = −PN T N. Associate with QT (z) the following kernels: GT (z, ξ) :=

QT (z) − QT (ξ)∗ − QT (z)(F − F ∗ )QT (ξ)∗ , z − ξ¯

(4.10)

¯ T (z, ξ), MT (z, ξ) := I + zQT (z) + ξQT (ξ)∗ + z ξG

(4.11)

LT (z, ξ) := GT (z, ξ) − MT (z, ξ),

(4.12)

KT (z, ξ) := LT (z, ξ) − QT (z)(F − F ∗ )QT (ξ)∗ ,

(4.13)

and ¯ |z|, |ξ| < 1. The insertion of the definition of GT (z, ξ) in LT (z, ξ) and with z = ξ, KT (z, ξ) leads to the identities ¯ T (z, ξ) = (1 − z 2 )QT (z) − (1 − ξ¯2 )QT (ξ)∗ (z − ξ)L ¯ T (z)(F − F ∗ )QT (ξ)∗ − (z − ξ)I, ¯ − (1 − z ξ)Q and ¯ T (z, ξ) = (1 − z 2 )QT (z) − (1 − ξ¯2 )QT (ξ)∗ (z − ξ)K ¯ T (z)(F − F ∗ )QT (ξ)∗ − (z − ξ)I. ¯ − (1 + z)(1 − ξ)Q

38


Proposition 4.2. Let T be a qsc-operator, let QT (z) be defined by (4.1), and let F be defined by F = −PN T N. Let the kernels associated with QT (z) be given by (4.10), (4.11), (4.12), and (4.13). Then the following equalities hold for every ¯ |z|, |ξ| > 1: z = ξ, GT (z, ξ) = PN (T − zI)−1 (T ∗ − ξI)−1 N, −1

MT (z, ξ) = PN (T − zI)

∗

∗

−1

T T (T − ξI)

N,

(4.14) (4.15)

and

LT (z, ξ) = PN (T − zI)−1 (I − T T ∗)(T ∗ − ξI)−1 N. (4.16) The operator-valued functions GT (z, ξ), MT (z, ξ), and LT (z, ξ) are nonnegative kernels. If, in addition, the operator T belongs to the class C(α) then the function KT (z, ξ) = PN (T − zI)−1 (I + T )(I − T ∗ )(T ∗ − ξI)−1 N

(4.17)

with |z|, |ξ| > 1 is an α-sectorial kernel. Proof. Note that ran (T − T ∗ ) ⊂ N implies that N⊥ ⊂ ker (T − T ∗ ), and hence T − T ∗ = PN (T − T ∗ )PN . Therefore, for every f, g ∈ N, ¯ −1 f, g ((QT (z) − Q∗T (ξ))f, g) = PN (T − zI)−1 f − PN (T ∗ − ξI) ¯ −1 f, g = PN (T − zI)−1 (T ∗ − T )(T ∗ − ξI) ¯ PN (T − zI)−1 (T ∗ − ξI) ¯ −1 f, g + (z − ξ) = (QT (z)(F − F ∗ )QT (ξ)∗ f, g) ¯ PN (T − zI)−1 (T ∗ − ξI) ¯ −1 f, g . + (z − ξ) Hence, it follows that ¯ N (T − zI)−1 (T ∗ − ξI) ¯ −1 N, QT (z) − Q∗T (ξ) = QT (z)(F − F ∗ )Q∗T (ξ) + (z − ξ)P and this proves (4.14). The identity (4.15) follows now from ∗ ∗ ¯ −1 f, T ∗ (T ∗ − z¯I)−1 g = f + ξ(T ¯ ∗ − ξI) ¯ −1 f, g + z¯(T ∗ − z¯I)−1 g T (T − ξI) ¯ ∗ (ξ)f, g) + z ξ(G ¯ T (z, ξ)f, g), f, g ∈ N. = (f, g) + z(QT (z)f, g) + ξ(Q T

Subtracting (4.15) from (4.14) gives immediately the identity (4.16). It is clear from the given formulas (4.14), (4.15), and (4.16), that the functions GT (z, ξ), MT (z, ξ), and LT (z, ξ) are nonnegative kernels. Since T − T ∗ = PN (T − T ∗ )PN , the definitions of QT (z) and F in (4.1), (4.7) show that ¯ −1 . −QT (z)(F − F ∗ )Q∗ (ξ) = PN (T − zI)−1 (T − T ∗ )(T ∗ − ξI) T

Combining this identity with (4.16) leads to (4.17). It is a consequence of (2.6) that KT (z, ξ) is an α-sectorial kernel.

Proposition 4.3. Let T be a qsc-operator in a Hilbert space H, N ⊃ ran (T − T ∗ ). Suppose that T is N-minimal, i.e., H = span { (T − z)−1 N : |z| > 1 }. Then the following conditions are equivalent: (i) N = H;


39

(ii) GT (z, z) = QT (z)QT (z)∗ for at least one (and equivalently for every) z with |z| > 1, where QT (z) is Q-function of T defined by (4.1) and GT (z, ξ) is defined by (4.10); (iii) the operator-valued function Q−1 T (z) + zI is constant. Proof. (i) ⇒ (ii) & (iii) If N = H then QT (z) = (T − zI)−1 and the equality GT (z, z) = QT (z)QT (z)∗ for all z, |z| > 1, follows immediately from (4.14). Besides, Q−1 T (z) + zI = T for all z, |z| > 1. (ii) ⇒ (i) Now suppose that GT (z, z) = QT (z)QT (z)∗ for some z, |z| > 1. Then (4.1) and (4.14) yield (T ∗ − z¯I)−1 f = PN (T ∗ − z¯I)−1 f for every f ∈ N. Therefore, (T ∗ − z¯I)−1 N ⊂ N which implies that the subspace N is invariant under T ∗ , and hence also under T , since ran (T − T ∗ ) ⊂ N. Because T is N-minimal, this leads to N = H. (iii) ⇒ (ii) Suppose that Q−1 T (z) + zI is constant for |z| > 1. According to Proposition 4.1 the function −Q−1 T (z) − zI − F has a holomorphic continuation onto Ext [−1, 1] as a Nevanlinna function. Since −Q−1 T (z) − zI − F is constant for |z| > 1, one has −∗ −Q−1 + z¯I + F ∗ = 0, T (z) − zI − F + QT (z)

|z| > 1.

It follows that −∗ − (F − F ∗ ) −Q−1 T (z) + QT (z) = I, z − z¯

|z| > 1,

and thus

−∗ QT (z) −Q−1 − (F − F ∗ ) QT (z)∗ T (z) + QT (z) = QT (z)QT (z)∗ , z − z¯ Therefore G(z, z) = QT (z)QT (z)∗ for all z, |z| > 1.

|z| > 1.

Observe, that the equality (4.14) can be rewritten in the following two equivalent forms: −QT (z)−1 − F − (−QT (ξ)−1 − F )∗ z − ξ¯ (4.18) ¯ −1 QT (ξ)−∗ , = QT (z)−1 PN (T − zI)−1 (T ∗ − ξI) and −QT (z)−1 − F − zI − (−QT (ξ)−1 − F − ξI)∗ z − ξ¯ ¯ −1 QT (ξ)−∗ . = QT (z)−1 PN (T − zI)−1 (I − PN )(T ∗ − ξI)

(4.19)

These formulas show that −QT (z)−1 − F and −QT (z)−1 − F − zI indeed are Nevanlinna functions. In particular, the conditions (i)–(iii) in Proposition 4.3 are equivalent to the right side of (4.19) to vanish.

40


Remark 4.4. The Q-function QT (z) as defined in (4.1) can be interpreted as the Weyl function for a special kind of boundary value space of a dual pair of operators, cf. [37], [39], [40]. To explain this, let A = A0 + K0 DA0 be a Hermitian contraction and let T be a qsc-extension of A, i.e., T is a contractive extension of a dual pair {A, A}. Let A∗ be the adjoint linear relation of A in the Cartesian product H × H. Then A∗ can be represented as follows: ( ) ( ) A∗ = {f, T f + ϕ} : f ∈ H, ϕ ∈ N = {f, T ∗ f + ψ} : f ∈ H, ψ ∈ N . Define the following bounded linear operators acting from A∗ into N: Γ0 {f, f } = PN f,

Γ1 {f, f } = PN T ∗ f − PN f ,

Γ2 {f, f } = PN T f − PN f ,

where {f, f } ∈ A∗ . Then {N, Γ0 , Γ1 , Γ2 } forms a boundary value space for A∗ . In g = {g, g } ∈ A∗ the following identity holds particular, for all f = {f, f }, (f , g) − (f, g ) = (Γ0 f, Γ1 g ) − (Γ2 f, Γ0 g), and moreover ker Γ1 = T ∗ , ker Γ2 = T , and ( ker Γ0 = {h, A0 h + ϕ} : h ∈ H0 ,

) ϕ∈N .

The corresponding γ-fields are the following operator functions ⎧ −1 ∗ ⎪ ⎪ γ0 (z)ϕ = −(A0 − zI) K0 DA0 ϕ, ⎨ γ1 (z)ϕ = (T ∗ − zI)−1 ϕ, ⎪ ⎪ ⎩ γ2 (z)ϕ = (T − zI)−1 ϕ, where ϕ ∈ N and |z| > 1. It follows that QT (z) = Γ0 γ2 (z) is given by QT (z) = PN (T − zI)−1 N, and that −Q−1 T (z) = Γ2 γ0 (z) is given by −1 (I − A20 ) K0∗ − DK0∗ XDK0∗ + zI N, −Q−1 T (z) = K0 A0 + (A0 − zI) where T is decomposed as in (3.2); see also Proposition 4.1. In particular, this means that QT (z) can be interpreted as the Weyl function corresponding to the boundary value space {N, Γ0 , Γ1 , Γ2 } in the sense of [39], [40].

5. The resolvent formula for qsc-extensions Let A = A0 + K0 DA0 be a closed symmetric contraction in H and let T be a qscextension of A given by the block matrix (3.2). If QT (z) = PN (T − zI)−1 N is the −1 Q-function of T , then by (4.9) the operator (Q−1 T (−1) − QT (1))/2 is nonnegative on N. Let −1 −1 −1 Q−1 T (−1) + QT (1) QT (−1) − QT (1) BQT := B − , (5.1) 2 2


41

be the operator ball in L(N) with center −1 ∗ ∗ −(Q−1 T (−1) + QT (1))/2 = −DK0 XDK0

and equal left and right radii −1 2 (Q−1 T (−1) − QT (1))/2 = DK0∗ .

Recall that it is the set of all operators in N of the form −1 1/2 −1 1/2 −1 Q−1 QT (−1) − Q−1 QT (−1) − Q−1 T (−1) + QT (1) T (1) T (1) + Y , − 2 2 2 where Y ≤ 1. Theorem 5.1. Let A be a closed symmetric operator in a Hilbert space H. Then the formula −1

I + QT (z)B (T − zI)−1 = (T − zI)−1 − (T − zI)−1 B PN (T − zI)−1 (5.2) with |z| > 1 gives a one-to-one correspondence between the resolvents of all qsc belonging to the operator ball BQT in (5.1). extensions T of A and all operators B Proof. By Theorem 3.1 every qsc-extension T of A can be written in the block form A0 DA0 K0∗ T = , (5.3) K0 DA0 −K0 A0 K0∗ + DK0∗ Y DK0∗ where Y ≤ 1. This together with (3.2) gives

:= T − T N = −DK ∗ XDK ∗ + DK ∗ Y DK ∗ (5.4) B 0 0 0 0

∈ BQT . It follows from which in view of (4.9) this means that B

N T − zI = T − zI + BP that

(5.5)

N (T − z)−1 , (T − z)−1 = (T − z)−1 + (T − z)−1 BP

|z| > 1,

and compression to N leads to

Q(z).

Q(z) = Q(z) + Q(z)B

Since Q(z) and Q(z) are invertible by part (ii) of Proposition 4.1, one obtains −1

= Q(z)−1 (I + Q(z)B)

= (I + BQ(z))Q(z)

−1 = Q(z)−1 + B . Q(z)

Therefore, the operators

I + Q(z)B

and I + BQ(z),

|z| > 1,

are invertible in N, too. Furthermore, by rewriting (5.5) in the form

N (T − zI)−1 (T − zI), T − zI = I + BP

42


−1

N (T − zI)−1 ∈ L(H) for every |z| > 1 and it is clear that I + BP −1

N (T − zI)−1 (T − zI)−1 = (T − zI)−1 I + BP , |z| > 1.

(5.6)

It also follows from (5.5) that

N (T − zI)−1 . (T − zI)−1 − (T − zI)−1 = −(T − zI)−1 BP

(5.7)

Now using the identities (5.6), (5.7), and −1 −1

N (T − zI)−1

N I + PN (T − zI)−1 BP

N = BP

N I + BP , BP one obtains (T − zI)−1 − (T − zI)−1

−1

N (T − zI)−1

N (T − zI)−1 = −(T − zI)−1 I + BP BP −1

N I + PN (T − zI)−1 BP

N = −(T − zI)−1 BP PN (T − zI)−1 −1

I + QT (z)B = −(T − zI)−1 B PN (T − zI)−1 ,

which gives the required identity (5.2).

∈ BQT , i.e., that B

is given by Conversely, assume that B 1/2 −1 1/2 −1 −1 Q−1 QT (−1) − Q−1 Q−1 T (−1) + QT (1) T (−1) − QT (1) T (1)

+ Y − 2 2 2

= −DK ∗ XDK ∗ + DK ∗ Y DK ∗ . Consider for some Y ≤ 1. By (4.9) one has B 0 0 0 0

of the form (5.3) which the qsc-extension T of A given by the block operator T

= T − T N. As was shown above, the is determined by Y . Then clearly B

is invertible for all |z| > 1 and the resolvent of T takes the operator I + QT (z)B form (5.2). The one-to-one correspondence is clear from the given arguments. Observe, that the Q-function QT (z) of the operator T in (5.2) and the Qfunction QT (z) of T are connected via QT (z) = PN (T − zI)−1 N

−1 QT (z) = QT (z)(I + BQ

T (z))−1 = (I + QT (z)B)

(5.8)

+ Q−1 (z))−1 . = (B T Remark 5.2. The resolvent formula (5.2) established in the theorem for the qscextensions T of A remains true for all quasi-selfadjoint extensions of A, i.e., T and T in Theorem 5.1 can be taken to be bounded, not necessarily contractive, extensions of A. Indeed, by defining the functions QT (z), |z| > T , and QT (z), |z| > T as in (4.1) then they are bounded and boundedly invertible (cf., e.g.,


43

the proof presented for part (ii) of Proposition 4.1). It remains to repeat the proof of Theorem 5.1 to obtain the resolvent formula (5.2) for two arbitrary bounded quasi-selfadjoint extensions T and T of A in the domain |z| > max{T , T }. The

is still given by (5.4), but now X and Y are not in general contractive parameter B

need not belong to the operator and even if T is a qsc-extension of A (X ≤ 1), B

ball BQT in (5.1). Of course, T can be interpreted as a bounded perturbation of

in (5.4). This allows one to apply the resolvent formula T by the parameter B (5.2) to study, for instance, the behavior of the resolvents under arbitrary small

≤ ε with B

not necessarily belonging to the operator ball BQT perturbations B in (5.1). Descriptions of canonical and generalized resolvents of selfadjoint contractive extensions of a non-densely defined symmetric contraction A were given by M.G. Kre˘ın and I.E. Ovcharenko in [32] by means of Q-functions of the selfadjoint extremal extensions Aµ and AM . Later B. Textorius [47] described this set using

of A and the corresponding special an arbitrary selfadjoint contractive extension A form of its Q-function. In [10], [12], [11] all canonical and generalized resolvents of qsc-extensions were parametrized by means of Qµ - and QM -functions in the sense of Kre˘ın-Ovcharenko in [32]. The proof of the parametrization formula (5.2), which is presented here, is kept very elementary, along the lines of a similar presentation in [9]. An abstract formula for canonical and generalized resolvents of extensions of a dual pair of linear relations in terms of a boundary value space and its Weyl function is derived in [39], [40].

6. A reproducing kernel space model for Q-functions of quasi-selfadjoint contractions Let N be a Hilbert space. An operator-valued function Q(z) with values in L(N) and holomorphic outside the unit disk is said to belong to the class Q(N) if: (i) Q(z) has the expansion 1 1 1 , z → ∞; (6.1) Q(z) = − I + 2 F + o z z z2 (ii) the L(N)-valued function G(z, ξ) =

Q(z) − Q(ξ)∗ − Q(z)(F − F ∗ )Q(ξ)∗ , z − ξ¯

¯ z = ξ,

with |z|, |ξ| > 1, is a nonnegative kernel; (iii) the L(N)-valued function ¯ ¯ − F ∗ )Q(ξ)∗ − (z − ξ)I (1 − z 2 )Q(z) − (1 − ξ¯2 )Q(ξ)∗ − (1 − z ξ)Q(z)(F L(z, ξ) = , z − ξ¯ ¯ |z|, |ξ| > 1, is a nonnegative kernel; with z = ξ,

44


(iv) there exist a complex number z0 , |z0 | > 1, and a vector f ∈ N, such that G(z0 , z0 )f = Q(z0 )Q(z0 )∗ f. If T is a qsc-operator in the Hilbert space H, N is a subspace of H such that N = H and ran (T − T ∗ ) ⊂ N, and QT (z) is its Q-function defined by (4.1), then according to Propositions 4.1, 4.2, and 4.3 the function QT (z) belongs to the class Q(N). The converse statement is also true. Theorem 6.1. Let Q(z) be a function of the class Q(N). Then there exist a Hilbert space H ⊃ N, N = H, and an N-minimal qsc-operator T in H, such that N ⊃ ran (T − T ∗ ) and Q(z) = PN (T − zI)−1 N,

for all |z| > 1.

(6.2)

If, in addition, the L(N)-valued function K(z, ξ) := L(z, ξ) − Q(z)(F − F ∗ )Q(ξ)∗ ¯ ¯ − F ∗ )Q(ξ)∗ − (z − ξ)I (1 − z 2 )Q(z) − (1 − ξ¯2 )Q(ξ)∗ − (1 + z)(1 − ξ)Q(z)(F = z − ξ¯ ¯ |z|, |ξ| > 1, where F is given by (6.1), is an α-sectorial kernel with with z = ξ, α ∈ [0, π/2), then the corresponding operator T belongs to the class C(α).

be the reproducing kernel Hilbert space associated with the Proof. Step 1. Let H

is the completion of nonnegative kernel G(z, ξ), i.e., H span { G(·, w)f : f ∈ N, |w| > 1 } with respect to the norm determined by the inner product (G(·, w)f, G(·, µ)g) H

= (G(µ, w)f, g)N . For all f ∈ N and |w|, |µ| > 1, 2 2 wG(·, w)f − µ ¯ G(·, µ)f 2H

= |w| (G(w, w)f, f ) N + |µ| (G(µ, µ)f, f )N

− µw (G(µ, w)f, f )N − µw (G(w, µ)f, f )N .

(6.3)

In view of (6.1) one has Q(z) = (−1/z)I + o(1/z) as z → ∞, which implies that lim wG(z, w)f = −Q(z)f,

|z| > 1,

w→∞

(6.4)

and moreover that lim µwG(µ, w)f = f,

µ,w* →∞

f ∈ N.

(6.5)

(Here * → stands for the nontangential limit in a sector | arg(z) − π/2| ≤ α < π/2.)

Hence (6.3) and (6.5) imply that the following limit exists in H Kf := − lim wG(z, w)f w* →∞

(6.6)

for which and defines a linear operator K : N → H 2 2 2 Kf 2H

= lim ||wG(·, w)f ||H

= lim |w| (G(w, w)f, f ) N = ||f ||N . w* →∞

w* →∞

(6.7)


45

Thus, K is isometric. It follows from (6.4) that (Kf, G(·, µ)g)H

= − lim w (G(·, w)f, G(·, µ)g) H

w* →∞

= − lim w (G(µ, w)f, g)N = (Q(µ)f, g)N , w* →∞

which shows that K ∗ G(·, µ)g = Q(µ)∗ g,

g ∈ N.

(6.8)

by Step 2. Define the linear relation S in H ++ n , , n n S= G(·, wi )fi , Kfi + w i G(·, wi )fi : fi ∈ N, |wi | > 1 . i=1

i=1

(6.9)

i=1

In fact, S is a contractive linear By definition the domain of S is dense in H.

operator in H, since -2 -2 n n n - - G(·, wi )fi - − Kfi + wi G(·, wi )fi H

i=1

=

n

−

H

i=1

(G(wj , wi )fi , fj )N −

i,j=1 n

=

i=1

n

(fi , fj )N −

i,j=1 n

wj (Q(wj )fi , fj )N −

i,j=1 n

n

w i (Q(wi )∗ fi , fj )N

i,j=1

wj w i (G(wj , wi )fi , fj )N

i,j=1

(L(wj , wi )fi , fj )N ≥ 0,

i,j=1

where (6.7) and (6.8) have been used. Therefore, the operator S has a unique

and for which the same contractive continuation which is defined everywhere on H notation S is preserved. # Step 3. To calculate the imaginary part of S note that for h = ni=1 G(·, wi )fi the following identities holds ⎛ ⎞ n n n Kfi + w i G(·, wi )fi , G(·, wj )fj ⎠ (Sh, h) = ⎝ i=1

=

n

i=1

j=1

(Q(wj )fi + wi G(wj , wi )fi , fj )N .

i,j=1

Similarly one obtains (h, Sh) =

n i,j=1

(Q(wi )∗ fi + wj G(wj , wi )fi , fj )N .

H

46


Since Q(wj ) − Q(wi )∗ + (w i − wj )G(wj , wi ) = Q(wj )(F − F ∗ )Q(wi )∗ , one obtains ⎛ ⎞ n n ⎝(S − S ∗ ) G(·, wi )fi , G(·, wj )fj ⎠ i=1

=

=

n i,j=1 n

j=1

H

(Q(wj )(F − F ∗ )Q(wi )∗ fi , fj )N ((F − F ∗ )K ∗ G(·, wi )fi , K ∗ G(·, wj )fj )N

i,j=1

⎛

= ⎝K(F − F ∗ )K ∗

n

G(·, wi )fi

,

i=1

n

⎞ G(·, wj )fj ⎠ .

j=1

H

This implies that S − S ∗ = K(F − F ∗ )K ∗ . By the definition of S in (6.9) one has (S − wI)G(·, w)f = Kf , so that (S − wI)−1 Kf = G(·, w)f,

f ∈ N,

|w| > 1.

(6.10) (6.11)

∗

Step 4. Since K is isometric, ran K is closed. # Let H0 := ker K and define n H := H0 ⊕ N. Observe, that according to (6.8) h = i=1 G(·, wi )fi belongs to the # n ∗

if and only if

subspace H0 of H i=1 Q(wi ) fi = 0. Now decompose H = H0 ⊕ran K

and define the operator U : H → H by U(x + y) = x + K ∗ y,

x ∈ H0 ,

y ∈ ran K.

Then U maps H onto H and it is unitary. Hence, the operator T defined by T := US ∗ U−1 is contractive in H and (6.10) shows that ran (T − T ∗ ) ⊂ U(ran K) = N. Furthermore, for f, g ∈ N and |z| > 1 the identities (6.8) and (6.11) yield (T − z)−1 f, g H = (S ∗ − zI)−1 U−1 f, U−1 g H

∗ −1 = (S − zI) Kf, Kg H

= Kf, (S − z¯I)−1 Kg H

= (Kf, G(·, z)g)H

= (Q(z)f, g)N . Thus, Q(z) = PN (T − zI)−1 N, |z| > 1. Moreover, it follows from (6.11) that the operator T is N-minimal. Step 5. Finally it is shown that H0 = {0}. If H0 = {0} then N = H and by Proposition 4.3 the equality G(z, z) = Q(z)Q(z)∗ holds for all |z| > 1. But this is impossible due to the condition (iv) of the definition of the class Q(N). Therefore H0 = 0, N = H, and T is a qsc-operator whose Q-function QT (z) coincides with Q(z).


47

As to the last statement observe, that since Q(z) is of the form (6.2), the kernel K(z, ξ) admits the operator representation (4.17) in Proposition 4.2. Since T is N-minimal, it follows from (4.17) and (2.6) that T ∈ C(α). The qsc-operator T constructed in Theorem 6.1 is N-minimal. The next result shows that this model for functions Q(z) belonging to the class Q(N) is essentially unique. Namely, the N-minimal part of a qsc-operator T (and hence also of T ∗ ) is up to unitary equivalence uniquely determined by its Q-function; a fact which is well known in the selfadjoint case. Theorem 6.2. Let H1 = H01 ⊕ N and H2 = H02 ⊕ N be two Hilbert spaces, and let T1 and T2 be qsc-operators in H1 and H2 , respectively, such that ran (T1 − T1∗ ) ⊂ N and ran (T2 − T2∗ ) ⊂ N. If QT1 (z) = QT2 (z) in some neighborhood of infinity then the N-minimal parts of T1 and T2 are unitarily equivalent. Proof. Assume that QT1 (z) = QT2 (z) holds in some neighborhood of infinity, say, for |z| > R > 1. Then these functions coincide everywhere outside the unit disk. It follows from (4.3) and (4.7) that F1 = F2 , while (4.14) implies that PN (T1∗ − ξI)−1 (T1 − zI)−1 N = PN (T2∗ − ξI)−1 (T2 − zI)−1 N, for all |z|, |ξ| > 1; cf. (4.2). Hence, for all f, g ∈ N (T1 − zI)−1 f, (T1 − ξI)−1 g = (T2 − zI)−1 f, (T2 − ξI)−1 g . (6.12) & ' −1 Now define&the linear relation U from ' H1 = span (T1 − zI) N : |z| > 1 into −1 H2 = span (T2 − zI) N : |z| > 1 by the formula , + n n −1 −1 U= (T1 − zk I) fk , (T2 − zk I) fk . k=1

k=1

Then the identity (6.12) implies that U is a unitary operator from H1 onto H2 . In addition, U f = f for all f ∈ N, and n n n −1 −1 U T1 (T1 − zk I) fk = fk + U zk (T1 − zk I) fk k=1

=

n k=1

fk +

k=1 n k=1

−1

zk (T2 − zk I)

fk = T 2 U

k=1 n

−1

(T1 − zk I)

fk

.

k=1

Therefore, the simple parts of T1 and T2 are unitarily equivalent.

7. Characteristic properties of Q-functions of quasi-selfadjoint contractions The definition of the class Q(N) given in Section 6 can be seen as an analytical characterization for Q-functions of qsc-operators T as defined in (4.1). Another characterization is established in the next theorem.

48


Theorem 7.1. Let N be a Hilbert space. The following conditions are equivalent: (i) the function Q(z) belongs to the class Q(N); (ii) (a) Q(z) ∈ L(N) is holomorphic in the domain |z| > 1 and with F ∈ L(N) it has the asymptotic expansion 1 1 1 Q(z) = − I + 2 F + o , z → ∞; z z z2 (b) the function −Q−1 (z) − zI − F is not constant, it has a holomorphic continuation onto Ext [−1, 1] as a bounded Nevanlinna function, and the strong limits Q−1 (±1) exist; (c) Q−1 (−1) − Q−1 (1) ≥ 0 and for all f, g ∈ N the following inequality holds: % −1 % % Q (−1) + Q−1 (1) f, g %2 −1 ≤ Q−1 (−1) − Q−1 (1) f, f Q (−1) − Q−1 (1) g, g . Proof. (i) ⇒ (ii) Let the function Q(z) belong to the class Q(N). Then (a) holds by definition, see (6.1). By Theorem 6.1 the function Q(z) has the operator representation Q(z) = PN (T − zI)−1 N, where T is a qsc-operator in a Hilbert space H ⊃ N, such that ran (T − T ∗ ) ⊂ N. Now (b) follows from parts (ii) and (v) of Proposition 4.1 and Proposition 4.3, see also the identity (4.19). The inequality in (c) is obtained from part (iv) of Proposition 4.1. (ii) ⇒ (i) Now assume that the function Q(z) has the properties (a)–(c). It follows from (a) and (b) that Q−1 (z) = −zI − F − G(z),

G(z) = o(1),

z → ∞.

Here G(z) ∈ NN [−1, 1] and G(∞) = 0. Now it follows from Theorem 2.3 that G(z) has the representation G(z) = K0 (A0 − zI)−1 (I − A20 )K0∗ , where A0 is a selfadjoint contraction in some Hilbert space H0 and K0 ∈ L(H0 , N). Moreover, according to (2.14) G(−1) = −Q−1 (−1) + I − F = K0 (I − A0 )K0∗ , G(1) = −Q−1 (1) − I − F = −K0 (I + A0 )K0∗ . This gives

⎧ −1 −1 ⎪ ⎨ Q (−1) − Q (1) = I − K0 K0∗ , 2 −1 −1 ⎪ ⎩ Q (−1) + Q (1) = K0 A0 K ∗ − F. 0 2 ∗ Now the assumption (c) implies that I − K0 K0 ≥ 0 and |((K0 A0 K0∗ − F )f, g)| ≤ DK0∗ f DK0∗ g,

f, g ∈ N.

(7.1)


49

By (2.3) there exists a contraction X in DK0∗ such that −F = −K0 A0 K0∗ + DK0∗ XDK0∗ .

(7.2)

Consider the Hilbert space H = H0 ⊕ N and let the operator T in H be given by the block form (3.2). Then T is a contraction and, in fact, a qsc-extension of the closed symmetric contraction A = A0 + K0 DA0 defined on H0 . According to Schur-Frobenius formula (see (2.9), (2.10)) PN (T − zI)−1 N = −(G(z) + zI + F )−1 = Q(z),

|z| > 1,

i.e., Q(z) is the Q-function of T . Therefore, Q(z) belongs to the class Q(N).

The model established in Theorem 6.1 yields the following simple characterizations of Q-functions corresponding to the extreme selfadjoint contractive extensions Aµ and AM of A within the class Q(N). Recall, that all sc-extensions T = T ∗ of A can be described as the operator interval [Aµ , AM ], see [27]. Proposition 7.2. Let Q(z) belong to the class Q(N) and suppose that lim inf |(Q(x)f, f )| = ∞,

for all f ∈ N \ {0},

(7.3)

lim inf |(Q(x)f, f )| = ∞,

for all f ∈ N \ {0}.

(7.4)

x↑−1

or x↓1

Then Q(z) is a Nevanlinna function in NN [−1, 1] and it can be represented in the form Q(z) = PN (Aµ − zI)−1 N or Q(z) = PN (AM − zI)−1 N, z ∈ Ext [−1, 1], respectively, where Aµ and AM are the left and right extreme sc-extension of some symmetric contraction A. Proof. According to Theorem 6.1 the function Q(z) has the operator representation Q(z) = PN (T − zI)−1 N, where T is a qsc-operator in a Hilbert space H ⊃ N, such that ran (T − T ∗ ) ⊂ N. Moreover, T is a qsc-extension of the closed symmetric contraction A defined by A = T dom A with dom A = H N. Let TR = (T + T ∗ )/2 and TI = (T − T ∗ )/2 be the real and the imaginary part of T , respectively, so that T = TR + iTI . Then for |x| > 1, −1

(T − xI)−1 = (TR − xI)−1/2 (I + iB)

(TR − xI)−1/2 ,

where

B = (TR − xI)−1/2 TI (TR − xI)−1/2 is a bounded selfadjoint operator. This shows that for all f ∈ N (Q(x)f, f ) = (I + iB)−1 (TR − xI)−1/2 f, (TR − xI)−1/2 f . Since (I + iB)−1 ≤ 1, one obtains

-2 |(Q(x)f, f )| ≤ -(TR − xI)−1/2 f - .

Now the assumption (7.3) implies that -2 lim inf -(TR − xI)−1/2 f - = ∞, x↑−1

for all f ∈ N \ {0}.

50


This means that ran (I + TR )1/2 ∩ N = {0}, cf., e.g., [7]. Since TR is a sc-extension of A, one concludes from the characterizations in (3.15) that TR = Aµ , cf. [27], [7], [22]. Now, in view of (3.14) TI = 0 and T = Aµ . The proof of the other statement is similar. Some further characteristic properties of Q-functions in the selfadjoint case, in particular, of Qµ - and QM -functions corresponding to the sc-extensions Aµ and AM have been established in [7], including some corrections to the results stated in [32].

8. Linear fractional transformations of the class Q(N) The Kre˘ın formula (5.2) and the discussion following it concerning the formulas in (5.8) give rise to a linear fractional transformation of Q-functions. Theorem 8.1. Let Q(z) belong to the class Q(N). Then the function −1

Q(z) (I + BQ(z))

,

|z| > 1,

belongs to the class Q(N) if and only if Q−1 (−1) + Q−1 (1) Q−1 (−1) − Q−1 (1) , . B∈B − 2 2

(8.1)

−1

Moreover, Q(z) (I + BQ(z)) is a Nevanlinna function of the class NN [−1, 1] if and only if B satisfies the conditions B + Q−1 (1) ≤ 0,

B + Q−1 (−1) ≥ 0.

(8.2)

Proof. First observe that, if B ∈ L(N) and (I + BQ(z))−1 ∈ L(N) for all |z| > 1, then it follows from (6.1) that 1 1 1 −1

, z → ∞, Q(z) := Q(z) (I + BQ(z)) = − I + 2 (F − B) + o z z z2

−1 (z) = Q−1 (z) + B. and clearly Q

Now assume that Q(z) ∈ Q(N). Then Q(z) ∈ L(N), |z| > 1, and since by −1 −1 Theorem 7.1 Q(z) , Q(z) ∈ L(N), |z| > 1, one has B, (I + BQ(z))−1 ∈ L(N)

−1 (±1) exist and satisfy for all |z| > 1. Moreover, the limit values Q

−1 (−1) = Q−1 (−1) + B, Q

−1 (1) = Q−1 (1) + B. Q

Now part (c) of Theorem 7.1 implies that % %2 −1 −1 % % % B + Q (−1) + Q (1) f, g % % % 2 −1 −1 Q (−1) − Q−1 (1) Q (−1) − Q−1 (1) f, f g, g ≤ 2 2 holds for all f, g ∈ N. Therefore, the condition (8.1) is satisfied.

(8.3)


51

Conversely, let the operator B ∈ L(N) satisfy the condition (8.1). By assumption Q(z) belongs to Q(N) and Theorem 6.1 shows that Q(z) = PN (T − z)−1 N, where T is qsc-operator in some Hilbert space H ⊃ N. Moreover, T is a qscextension of the symmetric contraction A = T H0 , H0 = H N. Now by Theorem 5.1 the assumption (8.1) means that B defines a qsc-extension T of A whose

= B. According to (5.8) the Q-function Q (z) resolvent is given by (5.2) with B T is of the form QT (z) = Q(z)(I + BQ(z))−1 , |z| > 1, and as a Q-function belongs to the class Q(N); see the discussion preceding Theorem 6.1. To prove the second part of the theorem, observe that in view of (4.5) −1 Q−1 (−1) = DK0∗ (Y + I)DK0∗ ,

(−1) = B + Q T

and

−1 (1) = DK0∗ (Y − I)DK0∗ , Q−1

(1) = B + Q T

where Y is a contraction in the subspace DK0∗ = ran DK0∗ . By Theorem 3.1 T is a selfadjoint contraction if and only if Y is a selfadjoint contraction in DK0∗ , or equivalently, B satisfies the conditions (8.2). Now, if (8.2) holds then T is selfadjoint and Q(z)(I + BQ(z))−1 = QT (z) ∈ NN [−1, 1]. Conversely, if QT (z) ∈ NN [−1, 1] then by part (vi) of Proposition 4.1 one

has F = F ∗ and consequently T = T ∗ , i.e., the conditions (8.2) are satisfied.

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[10] Yu.M. Arlinski˘ı and E.R. Tsekanovski˘ı, Nonselfadjoint contractive extensions of Hermitian contractions and M.G. Kre˘ın’s theorems, Usp. Mat. Nauk, 37, (1982) 131–132. [11] Yu.M. Arlinski˘ı and E.R. Tsekanovski˘ı, Generalized resolvents of quasi-selfadjoint contracting extensions of a Hermitian contraction, Ukrain. Mat. Zh., 35 (1983), 601– 603. [12] Yu.M. Arlinski˘ı and E.R. Tsekanovski˘ı, Quasi-selfadjoint contractive extensions of Hermitian contractions, Teor. Funkts., Funkts. Anal. Prilozhen, 50 (1988), 9–16. [13] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337–404. [14] Gr. Arsene and A. Gheondea, Completing matrix contractions, J. Operator Theory, 7 (1982), 179–189. [15] M.S. Brodski˘ı, Triangular and Jordan representations of linear operators, Nauka, Moscow 1968 (Russian). [16] M.G. Crandall, Norm preserving extensions of linear transformations on Hilbert spaces, Proc. Amer. Math. Soc., 21 (1969), 335–340. [17] Ch. Davis, W.M. Kahan, and H.F. Weinberger, Norm preserving dilations and their applications to optimal error bounds, SIAM J. Numer. Anal., 19 (1982), 445–469. [18] V.A. Derkach, S. Hassi, M.M. Malamud, and H.S.V. de Snoo, Boundary relations and their Weyl families, Trans. Amer. Math. Soc. (to appear). [19] V.A. Derkach and M.M. Malamud, The extension theory of hermitian operators and the moment problem, J. Math. Sciences, 73 (1995), 141–242. [20] R.G. Douglas, On majorization, factorization and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc., 17 (1966), 413–416. [21] V.I. Gorbachuk and M.L. Gorbachuk, Boundary value problems for operator differential equations, Kluwer Acad. Publ., Dordrecht–Boston–London, 1991 (Russian edition: Naukova Dumka, Kiev, 1984). [22] S. Hassi, M.M. Malamud, and H.S.V. de Snoo, On Kre˘ın’s extension theory of nonnegative operators, Math. Nachr., 274/275 (2004), 40–73. [23] P.A. Fillmore and J.P. Williams, On operator ranges, Advances in Math., 7 (1971), 254–281. [24] I.S. Kac and M.G. Kre˘ın, R-functions – analytic functions mapping the upper halfplane into itself, Supplement to the Russian edition of F.V. Atkinson, Discrete and continuous boundary problems, Mir, Moscow 1968 (Russian) (English translation: Amer. Math. Soc. Transl. Ser. 2, 103 (1974), 1–18). [25] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, Heidelberg, 1966. [26] M.G. Kre˘ın, On Hermitian operators whose deficiency indices are 1, Doklady Acad. Sci. SSSR, 43 (1944), 323–326. [27] M.G. Kre˘ın, The theory of selfadjoint extensions of semibounded Hermitian operators and its applications, Mat. Sb., 20 (1947) 431–495; 21 (1947), 365–404. [28] M.G. Kre˘ın, On the resolvents of an Hermitian operator with deficiency index (m, m), Doklady Acad. Sci. SSSR, 52 (1946), 657–660.


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[29] M.G. Kre˘ın and H. Langer, The defect subspaces and generalized resolvents of a Hermitian operator in the space Πκ , Funkcional. Anal. i Priloˇzen, 5 (1971), No. 2, 59–71 (English translation: Funct. Anal. Appl., 5 (1971), 136–146). [30] M.G. Kre˘ın and H. Langer, The defect subspaces and generalized resolvents of a Hermitian operator in the space Πκ , Funkcional. Anal. i Priloˇzen, 5 (1971), No. 3, 54–69 (English translation: Funct. Anal. Appl., 5 (1971), 217–228). ¨ [31] M.G. Kre˘ın and H. Langer, Uber die Q-function eines π-hermiteschen Operators in Raume Πκ , Acta Sci. Math. (Szeged), 34 (1973), 191–230. [32] M.G. Kre˘ın and I.E. Ovcharenko, On the Q-functions and sc-extensions of a Hermitian contraction with nondense domain, Sibirsk. Mat. J., 18 (1977), 1032–1056. [33] M.G. Kre˘ın and Sh.N. Saakyan, On some new results in the theory of the resolvent of Hermitian operators, Doklady Acad. Sci. SSSR, 169 (1966), 1269–1271. [34] H. Langer and B. Textorius, Generalized resolvents of contractions, Acta Sci. Math. (Szeged), 44 (1982), 125–131. [35] H. Langer and B. Textorius, Generalized resolvents of dual pairs of contractions, Operator Theory: Adv. Appl., 6 (1982), 103–118. [36] H. Langer and B. Textorius, On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space, Pacific J. Math., 72 (1977), 135–165. [37] V.E. Lyantse and O.G. Storozh, Methods of the theory of unbounded operators, Naukova Dumka, Kiev, 1983. [38] M.M. Malamud, On extensions of sectorial operators and dual pairs of contractions, Sov. Math. Dokl., 39 (1989), 252–259. [39] M.M. Malamud and V.I. Mogilevski˘ı, On extensions of dual pairs of operators, Reports of National Academy of Sciences of Ukraine, (1997), 30–37. [40] M.M. Malamud and V.I. Mogilevski˘ı, Kre˘ın type formula for canonical resolvents of dual pairs of linear relations, Methods of Functional Analysis and Topology, 8 (2002), 72–100. [41] M.A. Naimark, On spectral functions of a symmetric operator, Izv. Akad. Nauk SSSR, Ser. Matem., 7 (1943), 285–296. [42] S. Saitoh, Theory of reproducing kernels and its applications, Pitman Research Notes in Mathematics Series, 189. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. [43] Yu.L. Shmul’yan, Two-sided division in the ring of operators, Mat. Zametki, 1 (1967), 605–610. [44] Yu.L. Shmul’yan, Operator balls, Teor. Funkci˘ı Funkcional. Anal. i Priloˇzen, 6 (1968), 68–81 (English translation: Integral Equations Operator Theory, 13 (1990), 864–882). [45] Yu.L. Shmul’yan and R.N. Yanovskaya, Blocks of a contractive operator matrix, Izv. Vuzov, Mat., 7 (1981), 72–75. [46] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, NorthHolland, New York, 1970. [47] B. Textorius, On generalized resolvents of nondensely defined symmetric contractions, Acta Sci. Math. (Szeged), 49 (1985), 329–338.

54


Yury Arlinski˘ı Department of Mathematical Analysis East Ukrainian National University Kvartal Molodyozhny 20-A Lugansk 91034 Ukraine e-mail: [email protected] Seppo Hassi Department of Mathematics and Statistics University of Vaasa P.O. Box 700 65101 Vaasa Finland e-mail: [email protected] Henk de Snoo Department of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen Nederland e-mail: [email protected]


A Class of Abstract Boundary Value Problems with Locally Definitizable Functions in the Boundary Condition Jussi Behrndt Dedicated to Professor Heinz Langer

Abstract. For a class of boundary value problems where the spectral parameter appears in the boundary condition in the form of a locally definitizable function linearizations are constructed and their local spectral properties are investigated. Mathematics Subject Classification (2000). Primary: 47B50, 34B07; Secondary: 46C20, 47A06, 47B40. Keywords. Boundary value problems, locally definitizable operators and relations in Krein spaces, locally definitizable functions, boundary value spaces.

1. Introduction In this paper we study a class of boundary value problems with eigenvalue dependent boundary conditions. Let A be a closed symmetric operator or relation of defect one in a separable Krein space K, let {C, Γ0 , Γ1 } be a boundary value space for the adjoint A+ and let τ be a function locally holomorphic in some open subset of the extended complex plane which is symmetric with respect to the real line such that τ (λ) = τ (λ) holds. We investigate boundary value problems of the f following form: For a given k ∈ K find a vector fˆ = f ∈ A+ such that f − λf = k

and

τ (λ)Γ0 fˆ + Γ1 fˆ = 0

(1.1)

holds. Under additional assumptions on τ and A a solution of this problem can be obtained with the help of the compressed resolvent of a selfadjoint extension

of A which acts in a larger Krein space. Making use of the coupling method A

and we study its local spectral from [6] we construct this so-called linearization A properties, which are closely connected with the solvability of (1.1).

56

J. Behrndt

More precisely, let Ω be some domain in C symmetric with respect to the real line such that Ω ∩ R = ∅ and the intersections of Ω with the upper and lower open half-planes are simply connected. We will assume that the selfadjoint extension A0 := ker Γ0 of A is definitizable over Ω, i.e., for every subdomain Ω of Ω with the same properties as Ω, Ω ⊂ Ω, there exists a selfadjoint projection E which reduces A0 such that A0 ∩ (EK)2 is definitizable in the Krein space EK and Ω belongs to the resolvent set of A0 ∩ ((1 − E)K)2 . With the help of approximative eigensequences or the local spectral function of A0 the spectral points of A0 in Ω ∩ R can be classified in points of positive and negative type and critical points (cf. [13], [16]). Further we assume that τ is a function which is definitizable in Ω, that is, for every domain Ω as Ω, Ω ⊂ Ω, the function τ can be written as the sum of a definitizable function (cf. [14], [15]) and a function holomorphic on Ω . Similarly to selfadjoint operators and relations definitizable over Ω the points in Ω ∩ R can be classified in points of positive and negative type and critical points. It was shown in [17] that τ can be represented with a selfadjoint relation T0 definitizable over Ω in some Krein space H such that the sign types of τ and T0 coincide in Ω ∩ R. If, in addition, the sign types of A0 and τ are “compatible” in Ω ∩ R (see Definition 2.8), then the selfadjoint relation A0 × T0 in the Krein space K × H

of the boundary value problem (1.1) is definitizable over Ω . The linearization A turns out to be a two-dimensional perturbation in resolvent sense of A0 ×T0 . Under some additional minimality assumptions on the selfadjoint relations A0 and T0 and with the help of a recent result of T.Ya. Azizov and P. Jonas which states that the inverse of a matrix-valued locally definitizable function is again locally definitizable

is definitizable over Ω . we prove in Theorem 3.6 that A The paper is organized as follows. In Section 2 we introduce the necessary notations and we recall the definitions of locally definitizable operators and relations and locally definitizable functions which can be found in, e.g., [13], [16] and [17]. Section 3 deals with boundary value problems of the form (1.1). After some preparatory work in Section 3.1 and Section 3.2 we formulate and prove the main

of the eigenvalue dependent boundary result in Section 3.3: The linearization A value problem (1.1) is locally definitizable. I thank P. Jonas for encouragement and critical help in the preparation of the manuscript.

2. Locally definitizable selfadjoint relations and locally definitizable functions 2.1. Notations and definitions Let (K, [·, ·]) be a separable Krein space. We study linear relations in K, that is, linear subspaces of K2 . The set of all closed linear relations in K is denoted by

C(K). Linear operators in K are viewed as linear relations via their graphs. For the usual definitions of the linear operations with relations, the inverse etc., we refer

Boundary Value Problems

57 .

to [9]. We denote the sum (direct sum) of subspaces in K2 by (resp. ). The linear space of bounded linear operators defined on a Krein space K1 with values in a Krein space K2 is denoted by L(K1 , K2 ). In the case K := K1 = K2 we simply write L(K). If (H, [·, ·]H ) is another separable Krein space the elements of K × H will be written in the form {k, h}, k ∈ K, h ∈ H. K × H equipped with the inner product [·, ·]K×H defined by [{k, h}, {k , h }]K×H := [k, k ] + [h, h ]H ,

k, k ∈ K,

h, h ∈ H,

is also a Krein space. If S is a relation in K and T is a relation in H we shall write S × T for the direct product of S and T which is a relation in K × H, % . {s, t} % s t S×T = (2.1) % ∈ S, ∈ T . s {s , t } t {s,t} s, tˆ}, where For the pair {s ,t } on the right-hand side of (2.1) we shall also write {ˆ s t sˆ = s , tˆ = t . Let S be a closed linear relation in K. The resolvent set ρ(S) of S is the set of all λ ∈ C such that (S − λ)−1 ∈ L(K), the spectrum σ(S) of S is the complement of ρ(S) in C. The extended spectrum σ

(S) of S is defined by σ

(S) = σ(S) if S ∈ L(K) and σ

(S) = σ(S) ∪ {∞} otherwise. We say that λ ∈ C belongs to the approximate point spectrum of S, denoted by σap (S), if there exists a sequence xn ∈ S, n = 1, 2, . . . , such that xn = 1 and limn→∞ yn − λxn = 0. The yn extended approximate point spectrum σ

ap (S) of S is defined by + σap (S) ∪ {∞} if 0 ∈ σap (S −1 ) σ

ap (S) := . σap (S) if 0 ∈ σap (S −1 )

ap (S). We remark, that the boundary points of σ

(S) in C belong to σ Next we recall the definitions of the spectra of positive and negative type of a closed linear relation (see [16], [19]). For equivalent descriptions of the spectra of positive and negative type we refer to [16, Theorem 3.18]. Definition 2.1. Let S be a closed linear relation in K. A point λ ∈ σap (S) is said to be of positive type (negative type) with respect to S, if for every sequence xynn ∈ S, n = 1, 2 . . . , with xn = 1, limn→∞ yn − λxn = 0 we have lim inf [xn , xn ] > 0 resp. lim sup [xn , xn ] < 0 . n→∞

n→∞

If ∞ ∈ σ

ap (S), ∞ is said to be of positive type (negative type) with respect to S if 0 is of positive (resp. negative) type with respect to S −1 . An open subset ∆ of R is said to be of positive type (negative type) with respect to S if each point λ ∈ ∆ ∩ σ

(S) is of positive (resp. negative) type with respect to S. ∆ is called of definite type with respect to S if ∆ is either of positive or negative type with respect to S.

58

J. Behrndt

Let A be a linear relation in K. The adjoint relation A+ ∈ C(K) is defined as . % h % g ∈A . A+ := % [g , h] = [g, h ] for all h g

A is said to be symmetric (selfadjoint) if A ⊂ A+ (resp. A = A+ ). For a selfadjoint relation A in K the points of definite type introduced in Definition 2.1 to R. In fact, if, e.g., λ = ∞ is of positive type with respect belong to A, and xynn ∈ A is a sequence with xn = 1 and limn→∞ yn − λxn = 0, then |Im λ| lim inf [xn , xn ] = lim inf |Im [yn − λxn , xn ]| ≤ lim yn − λxn = 0 n→∞

n→∞

n→∞

implies λ ∈ R. 2.2. Locally definitizable selfadjoint relations in Krein spaces Let Ω be a domain in C symmetric with respect to the real axis such that Ω∩R = ∅ and the intersections of Ω with the upper and lower open half-planes are simply connected. Let A0 be a selfadjoint relation in the Krein space K such that σ(A0 ) ∩ (Ω\R) consists of isolated points which are poles of the resolvent of A0 , and no point of Ω ∩ R is an accumulation point of the non-real spectrum of A0 . Let ∆ be an open subset of Ω ∩ R. We say that A0 belongs to the class S ∞ (∆), if for every finite union ∆ of open connected subsets, ∆ ⊂ ∆, there exists m ≥ 1, M > 0 and an open neighborhood U of ∆ in C such that (A0 − λ)−1 ≤ M (1 + |λ|)2m−2 |Im λ|−m

(2.2)

holds for all λ ∈ U\R. We remark, that for an open subset ∆ of Ω ∩ R which is of positive type with respect to A0 the estimate (2.2) holds with m = 1 (see [16, Theorem 3.18] and [19] for the case of a bounded operator A0 ). Definition 2.2. Let Ω be a domain as above and let A0 be a selfadjoint relation in K such that σ(A0 ) ∩ (Ω\R) consists of isolated points which are poles of the resolvent of A0 and no point of Ω ∩ R is an accumulation point of the non-real spectrum of A0 in Ω. The relation A0 is said to be definitizable over Ω, if A0 ∈ S ∞ (Ω ∩ R) and every point µ ∈ Ω ∩ R has an open connected neighborhood Iµ in R such that both components of Iµ \{µ} are of definite type with respect to A0 . The next theorem is a variant of [16, Theorem 4.8]. The simple modification of the proof is left to the reader. Theorem 2.3. Let A0 be a selfadjoint relation in K and let Ω be a domain as above. A0 is definitizable over Ω if and only if for every domain Ω with the same properties as Ω, Ω ⊂ Ω, there exists a selfadjoint projection E in K such that A0 can be decomposed in . A0 ∩ ((1 − E)K)2 A0 = A0 ∩ (EK)2 and the following holds:


59

(i) A0 ∩ (EK)2 is a definitizable relation in the Krein space EK. (ii) σ

A0 ∩ ((1 − E)K)2 ∩ Ω = ∅. Let A0 be a selfadjoint relation in K which is definitizable over Ω, let Ω be a domain with the same properties as Ω, Ω ⊂ Ω, and let E be a selfadjoint projection with the properties as in Theorem 2.3. If E is the spectral function of the definitizable selfadjoint relation A0 ∩ (EK)2 in the Krein space EK (cf. [15, page 71], [10] and [18]), then the mapping δ → E (δ)E =: EA0 (δ)

(2.3)

defined for all finite unions δ of connected subsets of Ω ∩ R the endpoints of which belong to Ω ∩R and are of definite type with respect to A0 ∩(EK)2 , is the spectral function of A0 on Ω ∩ R (see [16, Section 3.4 and Remark 4.9]). 2.3. Locally definitizable functions Let Ω be a domain as in the beginning of Section 2.2 and let τ be an L(Cn )valued piecewise meromorphic function in Ω\R which is symmetric with respect to the real line, that is τ (λ) = τ (λ)∗ for all points λ of holomorphy of τ . If, in addition, no point of Ω∩ R is an accumulation point of non-real poles of τ we write τ ∈ M n×n (Ω). The set of the points of holomorphy of τ in Ω\R and all points µ ∈ Ω ∩ R such that τ can be analytically continued to µ and the continuations from Ω ∩ C+ and Ω ∩ C− coincide, is denoted by h(τ ). In the next definition we introduce the sign type of open subsets in Ω ∩ R with respect to functions from the class M n×n (Ω) (see [17]). Definition 2.4. Let τ ∈ M n×n (Ω). An open subset ∆ ⊂ Ω ∩ R is said to be of positive type with respect to τ if for every x ∈ Cn and every sequence (µk ) of points in Ω ∩ C+ ∩ h(τ ) which converges in C to a point of ∆ we have lim inf Im (τ (µk )x, x) ≥ 0. k→∞

An open subset ∆ ⊂ Ω ∩ R is said to be of negative type with respect to τ if ∆ is of positive type with respect to −τ . ∆ is said to be of definite type with respect to τ if ∆ is of positive or negative type with respect to τ . Definition 2.5. A function τ ∈ M n×n (Ω) is called definitizable in Ω if the following holds. (i) Every point µ ∈ Ω ∩ R has an open connected neighborhood Iµ in R such that both components of Iµ \{µ} are of definite type with respect to τ . (ii) For every open subset ∆ in R, ∆ ⊂ Ω ∩ R, there exists m ≥ 1, M > 0 and an open neighborhood U of ∆ in C such that τ (λ) ≤ M (1 + |λ|)2m |Im λ|−m holds for all λ ∈ U\R.

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In [17] it is shown that a function τ ∈ M n×n (Ω) is definitizable in Ω if and only if for every finite union ∆ of open connected subsets of R such that ∆ ⊂ Ω∩R, τ can be written as the sum of an L(Cn )-valued definitizable function (see [14], [15]) and an L(Cn )-valued function which is locally holomorphic on ∆. The following theorem will be used in Section 3.2 and Section 3.3. It states that a locally definitizable function can be represented with a locally definitizable selfadjoint relation. A proof can be found in [17]. Theorem 2.6. Let τ be an L(Cn )-valued locally definitizable function in Ω and let Ω be a domain with the same properties as Ω, Ω ⊂ Ω. Then there exists a Krein space H, a selfadjoint relation T0 in H definitizable over Ω and a mapping γ ∈ L(Cn , H) with the following properties. (a) ρ(T0 ) ∩ Ω = h(τ ) ∩ Ω . (b) For a fixed λ0 ∈ ρ(T0 ) ∩ Ω and all λ ∈ ρ(T0 ) ∩ Ω τ (λ) = Re τ (λ0 ) + γ + (λ − Re λ0 ) + (λ − λ0 )(λ − λ0 )(T0 − λ)−1 γ holds. (c) For any finite union ∆ of open connected subsets of R, ∆ ⊂ Ω ∩ R, such that the boundary points of ∆ are of definite type with respect to τ the spectral projection ET0 (∆) is defined. If Ω is a domain with the same properties as Ω and Ω , Ω ⊂ Ω , and if we set E := ET0 (∆) + ET0 (Ω \R), then the minimality condition & ' EH = clsp 1 + (λ − λ0 )(T0 − λ)−1 Eγ x | λ ∈ ρ(T0 ) ∩ Ω , x ∈ Cn is fulfilled. (d) Any finite union ∆ of open connected subsets of R, ∆ ⊂ Ω ∩ R, is of positive (negative) type with respect to τ if and only if ∆ is of positive (resp. negative) type with respect to T0 . If τ and T0 are as in Theorem 2.6 we shall say that T0 is an Ω -minimal representing relation for τ . Remark 2.7. Let τ be an L(Cn )-valued locally definitizable function in Ω and let Ω be a domain with the same properties as Ω, Ω ⊂ Ω. If, in addition, τ is the restriction of a definitizable function (see [14], [15]) or if, in addition, the boundary of Ω is contained in h(τ ), then the selfadjoint relation T0 in Theorem 2.6 can be chosen such that the minimality condition & ' H = clsp 1 + (λ − λ0 )(T0 − λ)−1 γ x | λ ∈ ρ(T0 ) ∩ Ω , x ∈ Cn holds. The next definition connects sign types of locally definitizable functions and sign types of spectral points of locally definitizable relations. Definition 2.8. Let τ be an L(Cn )-valued locally definitizable function in Ω and let A0 be a selfadjoint relation in the Krein space K which is definitizable over Ω.


61

We say that the sign types of τ and A0 are d-compatible in Ω if for every point µ ∈ Ω ∩ R there exists an open connected neighborhood Iµ ⊂ Ω ∩ R of µ such that each component of Iµ \{µ} is either of positive type with respect to τ and A0 or of negative type with respect to τ and A0 . If τ is a function which is definitizable in Ω, Ω is a domain as Ω, Ω ⊂ Ω, and T0 is an Ω -minimal representing relation for τ (see Theorem 2.6), then the sign types of τ and T0 are d-compatible in Ω .

3. Boundary value problems with locally definitizable functions in the boundary condition 3.1. Boundary value spaces and Weyl functions associated with symmetric relations in Krein spaces Let (K, [·, ·]) be a separable Krein space, let J be a corresponding fundamental

symmetry and let A ∈ C(K) be a closed symmetric relation in K. We say that A is of defect m ∈ N ∪ {∞}, if both deficiency indices n± (JA) = dim ker((JA)∗ − λ),

λ ∈ C± ,

of the symmetric relation JA in the Hilbert space (K, [J·, ·]) are equal to m. Here ∗ denotes the Hilbert space adjoint. We remark, that this is equivalent to the fact that there exists a selfadjoint extension of A in K and that each selfadˆ joint extension Aˆ of A in K satisfies dim A/A = m. We shall use the so-called boundary value spaces for the description of the selfadjoint extensions of closed symmetric relations in Krein spaces. The following definition is taken from [5]. Definition 3.1. Let A be a closed symmetric relation in the Krein space (K, [·, ·]). We say that {G, Γ0 , Γ1 } is a boundary value space for A+ if (G, (·, ·)) is a Hilbert space and there exist mappings Γ0 , Γ1 : A+ → G such that Γ = ΓΓ01 : A+ → G × G is surjective, and the relation [f , g] − [f, g ] = (Γ1 fˆ, Γ0 gˆ) − (Γ0 fˆ, Γ1 gˆ) f g holds for all fˆ = f , gˆ = g ∈ A+ . In the following we recall some basic facts on boundary value spaces which can be found in, e.g., [4] and [5]. For the Hilbert space case we refer to [11], [7] and [8]. Let A be a closed symmetric relation in K and let λ be a point of regular type of A. We denote by Nλ,A+ := ker(A+ − λ) = ran (A − λ)[⊥] the defect subspace of A at the point λ and we set ' & fλ % % ˆλ,A+ = N λfλ fλ ∈ Nλ,A+ .

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When no confusion can arise we write Nλ and Nˆλ instead of Nλ,A+ and Nˆλ,A+ . If there exists a selfadjoint extension A of A such that ρ(A ) = ∅ then we have A+ = A

.

Nˆλ

for all λ ∈ ρ(A ).

In this case there exists a boundary value space {G, Γ0 , Γ1 } for A+ such that ker Γ0 = A (cf. [5]). Let in the following A, {G, Γ0 , Γ1 } and Γ be as in Definition 3.1. It follows that the mappings Γ0 and Γ1 are continuous. The selfadjoint extensions A0 := ker Γ0

and

A1 := ker Γ1

of A are transversal, i.e., A0 ∩ A1 = A and A0 A1 = A+ . The mapping Γ induces, via & '

AΘ := Γ−1 Θ = fˆ ∈ A+ | Γfˆ ∈ Θ , Θ ∈ C(G), (3.1) a bijective correspondence Θ → AΘ between the set of all closed linear relations

C(G) in G and the set of closed extensions AΘ ⊂ A+ of A. In particular (3.1) gives a one-to-one correspondence between the closed symmetric (selfadjoint) extensions of A and the closed symmetric (resp. selfadjoint) relations in G. If Θ is a closed operator in G, then the corresponding extension AΘ of A is determined by AΘ = ker Γ1 − ΘΓ0 . (3.2) Assume that ρ(A0 ) = ∅ and denote by π1 the orthogonal projection onto the first component of K × K. For every λ ∈ ρ(A0 ) we define the operators ˆλ )−1 ∈ L(G, K) and M (λ) = Γ1 (Γ0 |Nˆλ )−1 ∈ L(G). γ(λ) = π1 (Γ0 |N The functions λ → γ(λ) and λ → M (λ) are called the γ-field and Weyl function corresponding to A and {G, Γ0 , Γ1 }. γ and M are holomorphic on ρ(A0 ) and the relations γ(ζ) = (1 + (ζ − λ)(A0 − ζ)−1 )γ(λ)

(3.3)

M (λ) − M (ζ)∗ = (λ − ζ)γ(ζ)+ γ(λ)

(3.4)

and

hold for all λ, ζ ∈ ρ(A0 ) (cf. [5]).

If Θ ∈ C(G) and AΘ is the corresponding extension of A (see (3.1)), then a point λ ∈ ρ(A0 ) belongs to ρ(AΘ ) if and only if 0 belongs to ρ(Θ − M (λ)). For λ ∈ ρ(AΘ ) ∩ ρ(A0 ) the well-known resolvent formula −1 γ(λ)+ (AΘ − λ)−1 = (A0 − λ)−1 + γ(λ) Θ − M (λ) holds (for a proof see, e.g., [5]).

(3.5)


63

3.2. Locally definitizable functions as Weyl functions of symmetric relations Let, as in Section 2.2, Ω be a domain in C symmetric with respect to the real axis such that Ω ∩ R = ∅ and the intersections of Ω with the upper and lower open half-planes are simply connected. In the next proposition we consider boundary value spaces and Weyl functions associated with symmetric relations in Krein spaces which have the additional property that there exists a locally definitizable selfadjoint extension. We restrict ourselves to the case of defect one. Proposition 3.2. Let A be a closed symmetric relation of defect one in the Krein space K and assume that there exists a selfadjoint extension A0 of A which is definitizable over Ω. Let {C, Γ0 , Γ1 } be a boundary value space for A+ such that A0 = ker Γ0 . Then the corresponding Weyl function M is definitizable in Ω. If ∆ is an open subset of Ω ∩ R which is of positive (negative) type with respect to A0 , then ∆ is of positive (resp. negative) type with respect to M . Proof. As A0 is a selfadjoint relation which is definitizable over Ω it follows that the Weyl function M corresponding to A and {C, Γ0 , Γ1 } is piecewise meromorphic in Ω\R and no point of Ω ∩ R is an accumulation point of the non-real poles of M . Let λ0 ∈ ρ(A0 ). Making use of (3.3) and (3.4) we obtain that M is symmetric with respect to the real axis, Im M (λ0 ) = (Im λ0 )γ(λ0 )+ γ(λ0 ) and M (λ) = M (λ0 ) + (λ − λ0 )γ(λ0 )+ γ(λ) = Re M (λ0 ) − i (Im λ0 )γ(λ0 )+ γ(λ0 ) (3.6) + γ(λ0 )+ (λ − λ0 ) + (λ − λ0 )(λ − λ0 )(A0 − λ)−1 γ(λ0 ) = Re M (λ0 ) + γ(λ0 )+ (λ−Reλ0 ) + (λ− λ0 )(λ− λ0 )(A0 − λ)−1 γ(λ0 ) holds for all λ ∈ ρ(A0 ). Since A0 belongs to the class S ∞ (Ω ∩ R) it follows that M fulfils the second condition in Definition 2.5. Let µ ∈ Ω∩R and let Iµ ⊂ Ω∩R be an open connected neighborhood of µ in R such that both components of Iµ \{µ} are of definite type with respect to A0 . By [16, Theorem 3.18] a component of Iµ \{µ} is of positive (negative) type with respect to A0 if and only if it is of positive (resp. negative) type with respect to the function λ → (λ − Re λ0 ) + (λ − λ0 )(λ − λ0 )(A0 − λ)−1 . Now it follows from (3.6) that both components of Iµ \{µ} are of the same sign type with respect to A0 and M and therefore the Weyl function M is definitizable in Ω. The same argument shows that an open subset ∆ ⊂ Ω∩R which is of positive (negative) type with respect to A0 is also of positive (negative) type with respect to M . The next theorem is a variant of [3, Theorem 3.3]. For the convenience of the reader we sketch the proof. Theorem 3.3. Let τ be a complex-valued locally definitizable function in Ω and assume that τ is not identically equal to a constant. Let Ω be a domain with the

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J. Behrndt

same properties as Ω and Ω ⊂ Ω. Then there exists a Krein space H, a closed symmetric relation T of defect one in H and a boundary value space {C, Γ0 , Γ1 } for T + such that τ coincides with the corresponding Weyl function on Ω and T0 := ker Γ0 is an Ω -minimal representing relation for τ . Sketch of the proof of Theorem 3.3. Let τ be represented with an Ω -minimal selfadjoint relation T0 in a Krein space H as in Theorem 2.6. Let γ ∈ L(C, H) be as in Theorem 2.6 and fix some λ0 ∈ h(τ ) ∩ Ω . For all λ ∈ h(τ ) ∩ Ω we define γ (λ) := 1 + (λ − λ0 )(T0 − λ)−1 γ ∈ L(C, H). The linear functional γ (λ)c → c defined on ran γ (λ) is denoted by γ (λ)(−1) . The closed symmetric relation . % f % ∈ T0 % [g − µf, γ (µ) 1] = 0 T := g has defect one and does not depend on the choice of µ ∈ h(τ ) ∩ Ω . For some fixed f fλ λ ∈ h(τ ) ∩ Ω we write the elements fˆ ∈ T + in the form fˆ = f0 + λf , where λ 0 f0 ∈ T0 and fλ ∈ ran γ (λ) = Nλ,T + . As in the proof of [3, Theorem 3.3] (see f0 also [7, Theorem 1]) one verifies that {C, Γ0 , Γ1 }, where Γ0 fˆ := γ (λ)(−1) fλ , Γ1 fˆ := γ (λ)+ (f0 − λf0 ) + τ (λ)γ (λ)(−1) fλ , is a boundary value space for T + and the corresponding Weyl function coincides with τ on Ω . Remark 3.4. Let the function τ be definitizable in Ω and assume that τ is not identically equal to a constant. Let Ω be a domain with the same properties as Ω, Ω ⊂ Ω, and assume that τ is the restriction of a definitizable function or that the boundary of Ω is contained in h(τ ). If we choose T0 as in Remark 2.7 and T ⊂ T0 ⊂ T + as in Theorem 3.3, then the condition H = clsp {ran γ (λ) | λ ∈ ρ(T0 ) ∩ Ω } = clsp { Nλ,T + | λ ∈ ρ(T0 ) ∩ Ω } is fulfilled. In this case T is an operator. In the following proposition we use Definition 2.8. The statements will be useful in the proof of our main result in Section 3.3. Proposition 3.5. Let A be a closed symmetric relation of defect one in the Krein space K, let {C, Γ0 , Γ1 } be a boundary value space for A+ and denote by M the corresponding Weyl function. Assume that the selfadjoint relation A0 = ker Γ0 is definitizable over Ω and let τ be a complex-valued function which is definitizable in Ω such that the sign types of τ and A0 are d-compatible in Ω. Let Ω be a domain with the same properties as Ω, Ω ⊂ Ω, and let T0 be an Ω -minimal representing relation for τ in some Krein space H (see Theorem 2.6). Then the following holds:


65

× H) is definitizable over Ω and the (i) The selfadjoint relation A0 × T0 ∈ C(K sign types of A0 × T0 and the functions τ and M are d-compatible in Ω . (ii) The function M + τ is definitizable in Ω. (iii) If τ (η) = 0 and (M + τ )(η ) = 0 for some η, η ∈ Ω, then the functions −1 M (λ) −1 M (λ) 0 and λ → − (3.7) λ → −1 −τ (λ)−1 0 −τ (λ)−1 are definitizable in Ω and their sign types are d-compatible with the sign types of the selfadjoint relation A0 × T0 in Ω . Proof. (i) Since A0 and T0 belong to S ∞ (Ω ∩ R) and S ∞ (Ω ∩ R), respectively, we conclude that A0 × T0 belongs to S ∞ (Ω ∩R). Let µ ∈ Ω ∩R and let Iµ ⊂ Ω ∩R be an open connected neighborhood of µ in R such that each component of Iµ \{µ} is of the same sign type with respect to A0 and τ . As T0 is an Ω -minimal representing relation for τ both components of Iµ \{µ} are of definite type with respect to A0 ×T0 and it follows that A0 × T0 is definitizable over Ω . The assumption that the sign types of τ and A0 are d-compatible in Ω implies that the sign types of τ and A0 × T0 as well as the sign types of M and A0 × T0 are d-compatible in Ω. (ii) For µ ∈ Ω ∩ R we choose an open connected neighborhood Iµ ⊂ Ω ∩ R of µ such that both components of Iµ \{µ} are of the same sign type with respect to A0 and τ . By Proposition 3.2 the sign types of M are the same as of A0 and therefore both components of Iµ \{µ} are of the same sign type with respect to M + τ . The growth properties of M and τ imply that M + τ fulfils the second condition in Definition 2.5 and therefore M + τ is definitizable in Ω. (iii) By [1, Theorem 2.5] the function −τ −1 is definitizable in Ω and it follows from the proof of [1, Theorem 2.5] that each point µ ∈ Ω ∩ R has an open connected neighborhood Iµ ⊂ Ω ∩ R such that both components of Iµ \{µ} are of the same sign type with respect to −τ −1 and τ . Therefore the sign types of −τ −1 and A0 are d-compatible in Ω. Now it is easy to see that the first function in (3.7) is definitizable in Ω and its sign types are d-compatible with the sign types of A0 ×T0 in Ω . As the function M (λ) −1 λ → −1 −τ (λ)−1 is also definitizable in Ω another application of [1, Theorem 2.5] shows that the second function in (3.7) is definitizable in Ω and its sign types are d-compatible with the sign types of A0 × T0 in Ω . 3.3. The main result In this section we investigate the spectral properties of linearizations of a class of abstract eigenvalue dependent boundary value problems with locally definitizable functions in the boundary condition. Similar problems with a local variant of

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J. Behrndt

generalized Nevanlinna functions in the boundary condition have been considered in [3]. The main feature in Theorem 3.6 below is that the linearization turns out to be locally definitizable. Theorem 3.6. Let Ω be a domain as in the beginning of Section 3.2 and let A be a closed symmetric operator of defect one in the Krein space K such that K = clsp { Nλ,A+ | λ ∈ Ω} holds. Assume that there exists a selfadjoint extension A0 of A which is definitizable over Ω. Let {C, Γ0 , Γ1 } be a boundary value space for A+ , A0 = ker Γ0 , and denote by γ and M the corresponding γ-field and Weyl function, respectively. Let τ be a nonconstant function which is definitizable in Ω, let Ω be a domain as Ω, Ω ⊂ Ω, choose H, T ⊂ T0 ⊂ T + and {C, Γ0 , Γ1 } as in Theorem 3.3 and assume that the condition H = clsp { Nλ,T + | λ ∈ Ω } is fulfilled. Let the sign types of τ and A0 be d-compatible in Ω, assume that the function M + τ is not identically equal to zero and define h0 := h(M ) ∩ h(τ ) ∩ h τ −1 ∩ h (M + τ )−1 . Then the relation (& ) % '

= A fˆ1 , fˆ2 ∈ A+ × T + % Γ1 fˆ1 − Γ1 fˆ2 = Γ0 fˆ1 + Γ0 fˆ2 = 0

(3.8)

is a selfadjoint extension of A in K × H which is definitizable over Ω and the

are d-compatible with the sign types of τ and M in Ω . The set sign types of A Ω \(R ∪ h0 ) is finite. For every k ∈ K and every λ ∈ h0 ∩ Ω the unique solution of the eigenvalue dependent boundary value problem f1 ˆ ˆ ˆ ∈ A+ , f1 − λf1 = k, τ (λ)Γ0 f1 + Γ1 f1 = 0, f1 = (3.9) f1 is given by

− λ)−1 {k, 0} = (A0 − λ)−1 k − γ(λ) M (λ) + τ (λ) −1 γ(λ)+ k, f1 = PK (A

f1 = λf1 + k.

(3.10)

in K × H with the Proof. 1. In this step we construct the selfadjoint relation A

help of the coupling method from [6, §5.2], we show that h0 ∩ Ω belongs to ρ(A) and that (3.10) is the unique solution of the boundary value problem (3.9). We follow the lines of [3, Proof of Theorem 4.1]. As the functions M , τ and M + τ are definitizable in Ω (see Proposition 3.5) [1, Theorem 2.3] implies that −τ −1 and −(M + τ )−1 are also definitizable in Ω. Let Ω and h0 be as in the assumptions of the theorem. As the non-real poles of the functions M , τ , τ −1 and (M + τ )−1 in Ω do not accumulate to Ω ∩ R we conclude from Ω ⊂ Ω that the set Ω \(R ∪ h0 ) is finite. Let H, T ⊂ T + and {C, Γ0 , Γ1 } be as in Theorem 3.3. Then τ is the corresponding Weyl function and the selfadjoint relation T0 = ker Γ0 is definitizable over Ω . We denote the γ-field corresponding to {C, Γ0 , Γ1 } by γ and we set


67

T1 := ker Γ1 . As {C, Γ1 , −Γ0 } is a boundary value space for T + with corresponding γ-field and Weyl function λ → γ (λ)τ (λ)−1 and λ → −τ (λ)−1 , λ ∈ h(τ ) ∩ h τ −1 ∩ Ω , (3.11)

0 , Γ

1 }, where Γ

0 and Γ

1 are respectively, it follows without difficulty that {C2 , Γ + + 2 mappings from A × T into C defined by & ' & ' Γ0 fˆ1 Γ1 fˆ1 ˆ ˆ ˆ ˆ

and Γ1 f1 , f2 := , Γ0 f1 , f2 := Γ1 fˆ2 −Γ0 fˆ2 {fˆ1 , fˆ2 } ∈ A+ × T + , is a boundary value space for A+ × T + with corresponding γ-field γ(λ) 0 λ → γ

(λ) = , λ ∈ h(M ) ∩ h(τ ) ∩ h(τ −1 ) ∩ Ω , (3.12) 0 γ (λ)τ (λ)−1 and Weyl function /(λ) = λ → M

M (λ) 0 , 0 −τ (λ)−1

λ ∈ h(M ) ∩ h(τ ) ∩ h(τ −1 ) ∩ Ω .

in K × H corresponding to Θ = The selfadjoint relation A and (3.2) is given by

0 1 10

∈ L(C2 ) via (3.1)

= ker(Γ

1 − ΘΓ

0 ) A (& ) % ' = fˆ1 , fˆ2 ∈ A+ × T + % Γ1 fˆ1 − Γ1 fˆ2 = Γ0 fˆ1 + Γ0 fˆ2 = 0 .

can be written as For λ ∈ h0 ∩ Ω the resolvent of A 0 (A0 − λ)−1 −1 /(λ) −1 γ

+ γ (λ) Θ − M (A − λ) =

(λ)+ , 0 (T1 − λ)−1

(3.13)

(3.14)

/(λ))−1 one verifies that the compressed resolvent (see (3.5)). Calculating (Θ − M

of A onto K is given by

− λ)−1 |K = (A0 − λ)−1 − γ(λ) M (λ) + τ (λ) −1 γ(λ)+ , λ ∈ h0 ∩ Ω . PK (A

− λ)−1 {k, 0} and f2 := PH (A

− λ)−1 {k, 0}. Then For k ∈ K we set f1 := PK (A {f1 , f2 }

⊂ A+ × T + ∈A {λf1 + k, λf2 } f2 ˆλ,T + . From (3.13) and since τ is the ∈N and fˆ1 := λff11+k ∈ A+ , fˆ2 := λf 2 Weyl function corresponding to {C, Γ0 , Γ1 } we get Γ1 fˆ1 = Γ1 fˆ2 = τ (λ)Γ0 fˆ2 = −τ (λ)Γ0 fˆ1 , and it follows that fˆ1 ∈ A+ is a solution of (3.9).

λ ∈ h0 ∩ Ω ,

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J. Behrndt

+ ˆ Let that the vector verify +that this solution f1 ∈ A is unique. Assume gus 1 gˆ1 = λg1 +k ∈ A is also a solution of (3.9), λ ∈ h0 ∩ Ω . Then fˆ1 − gˆ1 belongs to Nˆλ,A+ and 0 = τ (λ)Γ0 (fˆ1 − gˆ1 ) + Γ1 (fˆ1 − gˆ1 ) = τ (λ) + M (λ) Γ0 (fˆ1 − gˆ1 )

implies fˆ1 −ˆ g1 ∈ A0 ∩ Nˆλ,A+ as τ (λ)+M (λ) = 0. Therefore fˆ1 = gˆ1 since λ ∈ h(M ).

is definitizable over Ω and that the sign types 2. It remains to prove that A

of A are d-compatible with the sign types of the functions τ and M in Ω . In this step we show that for every point µ ∈ Ω ∩ R there exists an open connected neighborhood Iµ of µ in Ω ∩R such that both components of Iµ \{µ} are of definite

type with respect to A. As the sign types of τ and A0 are d-compatible in Ω, Proposition 3.5 implies that the selfadjoint relation A0 × T0 is definitizable over Ω . It is straightforward 0 , Γ 1 }, where to check that {C2 , Γ

1 − ΘΓ

0 , 0 := Γ Γ

1 := −Γ

0, Γ

0 = A.

The corresponding γ-field is a boundary value space for A+ × T + with ker Γ * are defined on ρ(A)

and for λ ∈ h0 ∩ Ω they are given by γ and Weyl function M /(λ) − Θ −1 λ → γ (λ) = γ (λ) M (3.15) and

−1 −1 *(λ) = −(M /(λ) − Θ)−1 = − M (λ) λ → M , −1 −τ (λ)−1 respectively. In particular *(λ) = Re M *(λ0 ) + γ (λ0 )+ (λ − Re λ0 ) M

− λ)−1 γ (λ0 ) + (λ − λ0 )(λ − λ0 )(A

(3.16)

holds for a fixed λ0 ∈ h0 ∩ Ω and all λ ∈ h0 ∩ Ω (see the proof of Proposition 3.2). * is definitizable in Ω and the sign types of M * By Proposition 3.5 the function M and A0 × T0 are d-compatible in Ω . Let µ ∈ Ω ∩ R and assume, e.g., that a one-sided open connected neighborhood ∆+ of µ in R, ∆+ ⊂ Ω ∩ R, is of positive type with respect to A0 × T0 . * and A0 × T0 are d-compatible in Ω , it is no restriction to As the sign types of M *. Since A0 × T0 and A

assume that ∆+ is also of positive type with respect to M are both selfadjoint extensions of the symmetric relation A × T in K × H we have

− λ)−1 − ((A0 × T0 ) − λ)−1 ≤ 2 dim ran (A for all λ ∈ h0 ∩Ω . Let Ω∆+ be a domain with the same properties as Ω, Ω∆+ ⊂ Ω ,

is definitizable such that Ω∆+ ∩ R = ∆+ . It follows from [2, Corollary 2.5] that A over Ω∆+ and that for every finite union δ of open connected subsets in ∆+ , δ ⊂ ∆+ , such that the spectral projection EA (δ) is defined the space EA (δ)(K×H)


69

equipped with the inner product [·, ·]K×H is a Pontryagin space with finite rank of negativity. Let Ω be a domain with the same properties as Ω such that Ω ⊂ Ω and ∆+ ⊂ Ω ∩ R. By Theorem 2.6 there exists an Ω -minimal representing relation S *, that is, S is a selfadjoint relation in some Krein space G which is definitizable for M *) ∩ Ω , and with a suitable Λ ∈ L(C2 , G) we have over Ω , ρ(S) ∩ Ω = h(M *(λ) = Re M *(λ0 ) + Λ+ (λ − Re λ0 ) + (λ − λ0 )(λ − λ0 )(S − λ)−1 Λ (3.17) M for a fixed λ0 ∈ ρ(S) ∩ Ω and all λ ∈ ρ(S) ∩ Ω . The spectral function of S on Ω ∩ R will be denoted by ES (comp. (2.3)). In the following we will assume that the point λ0 in (3.16) and (3.17) belongs

∩ ρ(S) ∩ Ω . This is no restriction. From (3.16) and (3.17) we obtain to ρ(A)

− λ)−1 (λ0 ) = Λ+ Λ and γ (λ0 )+ (A γ (λ0 ) = Λ+ (S − λ)−1 Λ γ (λ0 )+ γ

∩ ρ(S) ∩ Ω . Therefore the relation for all λ ∈ ρ(A) . % #n −1

∩ Ω )Λxk % λk ∈ ρ(S) ∩ ρ(A) k=1 (1 + (λk − λ0 )(S − λk ) # V := % n −1

) γ (λ0 )xk xk ∈ C2 , k = 1, . . . , n k=1 (1 + (λk − λ0 )(A − λk ) is isometric. The assumptions K = clsp { Nλ,A+ | λ ∈ Ω} imply K = clsp and H = clsp

( (

and H = clsp { Nλ,T + | λ ∈ Ω }

∩ Ω ran γ(λ) | λ ∈ ρ(S) ∩ ρ(A)

)

)

∩ Ω , ran γ (λ) | λ ∈ ρ(S) ∩ ρ(A)

respectively. From (3.12) and (3.15) we obtain ( )

∩ Ω , x ∈ C2 K × H = clsp γ (λ)x | λ ∈ ρ(S) ∩ ρ(A) and therefore ran V is dense in K × H. This implies that V is an isometric operator and the same holds for its closure V . Let δ be a finite union of open connected subsets in ∆+ , δ ⊂ ∆+ , such

Then that the boundary points of δ in R are of definite type with respect to A. (EA (δ)(K × H), [·, ·]K×H ) is a Pontryagin space with finite rank of negativity. As ∆+ is of positive type with respect to S the spectral projection ES (δ) is defined and ES (δ)G equipped with the inner product from G is a Hilbert space. Writing

respectively, one ES (δ) and EA (δ) as strong limits of the resolvents of S and A, verifies that V is reduced by ES (δ)G × EA (δ)(K × H). Then V δ := V ∩ ES (δ)G × EA (δ)(K × H) is a closed isometric operator from the Hilbert space ES (δ)G with dense range in the Pontryagin space EA (δ)(K×H). As in the proof of [12, Theorem 6.2] one verifies that V δ is bounded and from this we conclude dom V δ = ES (δ)G and ran V δ = EA (δ)(K × H). The isometry of V δ implies that EA (δ)(K × H) is a Hilbert space.

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J. Behrndt

and choose a sequence uv n ∈ A

with un = 1 and Let ξ ∈ δ ∩ σ(A) n vn − ξun → 0 for n → ∞. From

∩ (I − E (δ))(K × H) 2 − ξ −1 ∈ L (I − E (δ))(K × H) A A A and lim (I − EA (δ))(vn − ξun ) = 0

n→∞

we obtain (I − EA (δ))un → 0 and EA (δ)un → 1 for n → ∞. As EA (δ)(K × H) is a Hilbert space we have lim inf [un , un ]K×H = lim inf [EA (δ)un , EA (δ)un ]K×H > 0, n→∞

n→∞

If ∞ belongs to δ∩

a similar reathat is, ξ is of positive type with respect to A. σ (A)

soning shows that ∞ is of positive type with respect to A. Therefore δ is of positive

As this is true for every finite union δ of open connected type with respect to A. subsets in ∆+ , δ ⊂ ∆+ , such that the boundary points of δ in R are of definite type

we conclude that ∆+ is also of positive type with respect to A. with respect to A Analogously one verifies that a one-sided open connected neighborhood ∆− * is of µ in R, ∆− ⊂ Ω ∩ R, which is of negative type with respect to A0 × T0 and M

of negative type with respect to A. We have shown that for every point µ ∈ Ω ∩ R there is an open connected neighborhood Iµ in R such that both components of

Iµ \{µ} are of the same sign type with respect to A0 × T0 and A.

belongs to S ∞ (Ω ∩ R). For this we use the 3. It remains to verify that A relation (3.14). We show first that the selfadjoint relation T1 = ker Γ1 in H is definitizable over Ω . As the function −τ (λ)−1 = Re −τ (λ0 )−1 + τ (λ0 )−1 γ (λ0 )+ (λ − Re λ0 ) + (λ − λ0 )(λ − λ0 )(T1 − λ)−1 γ (λ0 )τ (λ0 )−1 (see (3.11) and the proof of Proposition 3.2) is definitizable in Ω and the selfadjoint relation T0 is definitizable over Ω the same considerations as in step 2 of the *, A0 × T0 and A

show that every proof applied to −τ −1 , T0 and T1 instead of M point µ ∈ Ω ∩ R has an open connected neighborhood Iµ in R such that both components of Iµ \{µ} are of definite type with respect to T1 . By (3.5) we have (T1 − λ)−1 = (T0 − λ)−1 + γ (λ) −τ (λ)−1 γ (λ)+ for all λ ∈ h(τ ) ∩ h(τ −1 ) ∩ Ω . Since −τ −1 is definitizable in Ω the non-real spectrum of T1 in Ω does not accumulate to points in Ω ∩ R. The growth properties of −τ −1 (see Definition 2.5) and the resolvent of T0 imply T1 ∈ S ∞ (Ω ∩ R) and therefore T1 is definitizable over Ω . As the sign types of −τ −1 and A0 are d-compatible in Ω (see the proof of Proposition 3.5 (iii)) it follows that A0 × T1 is definitizable over Ω . The relation

− λ)−1 = (A0 × T1 − λ)−1 + *(λ) (A γ (λ)M γ (λ)+ ,

λ ∈ h0 ∩ Ω ,


71

* and the resolvent of A0 × T0 show (cf. (3.14)) and the growth properties of M ∞

A ∈ S (Ω ∩ R). This completes the proof of Theorem 3.6. Remark 3.7. Let Ω, Ω , A ⊂ A0 , τ and T0 be as in Theorem 3.6 and let ∆ be an open connected subset in R, ∆ ⊂ Ω ∩ R, which is of positive (negative) type with respect to A0 and τ . As T0 is an Ω -minimal representing relation for τ it follows that ∆ is of positive (negative) type with respect to the selfadjoint relation A0 × T0 in K × H. From [2, Corollary 2.5] we obtain that for every finite union δ of open connected subsets in R, δ ⊂ ∆, such that the spectral projection EA (δ)

in (3.8) and the set δ is defined, (E (δ)(K × H), [·, ·]K×H ) is corresponding to A A a Pontryagin space with finite rank of negativity (positivity). This can also be deduced from [3, Theorem 4.1]. Remark 3.8. The assumption H = clsp { Nλ,T + | λ ∈ Ω } in Theorem 3.6 implies

in (3.8) satisfies the minimality condition that the selfadjoint extension A ' &

− λ)−1 ){k, 0} | k ∈ K, λ ∈ ρ(A)

∩ Ω (3.18) K × H = clsp (1 + (λ − λ0 )(A

∩ Ω . This can be verified as in [3, Proof of Theorem 4.1]. for some fixed λ0 ∈ ρ(A)

is a selfadjoint extension Let A ∈ C(K) be as in Theorem 3.6 and assume that B of A in some Krein space K × H which is definitizable over Ω such that the

− λ)−1 |K

onto K yields a solution of (3.9). Then PK (B compressed resolvent of B −1

− λ) |K coincide. If B

fulfils the minimality condition (3.18) with and PK (A

∩ Ω , respectively, and we

K × H and ρ(A) ∩ Ω replaced by K × H and ρ(B)

choose λ0 ∈ ρ(A) ∩ ρ(B) ∩ Ω , then , + # n

− λi )−1 ){ki , 0} %% λi ∈ ρ(A)

∩ ρ(B)

∩ Ω , (1 + (λi − λ0 )(A i=1 #n W := % −1

ki ∈ K, i = 1, 2, . . . , n ){ki , 0} i=1 (1 + (λi − λ0 )(B − λi ) is a densely defined isometric operator in K×H with dense range in K×H and the

and B

same holds for its closure W . We denote the local spectral functions of A by EA and EB , respectively. Let ∆ be an open connected subset in R, ∆ ⊂ Ω ∩ R, such that EA (∆) is defined. Then also EB (∆) is defined and W is reduced by EA (∆)(K × H) × EB (∆)(K × H ). The closed isometric operator W∆ := W ∩ EA (∆)(K × H) × EB (∆)(K × H ) intertwines the resolvents of

1 := A

∩ E (∆)(K × H) 2 A A

1 := B

∩ E (∆)(K × H ) 2 , and B B

1 ) ∩ ρ(B

1 ) ∩ Ω and x ∈ dom W∆ we have i.e., for λ ∈ ρ(A

1 − λ)−1 x = (B

1 − λ)−1 W∆ x. W∆ (A In particular, the ranks of positivity and negativity of the inner products on the subspaces EA (∆)(K × H) and EB (∆)(K × H ) coincide.

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If, in addition to the assumptions above, (EA (∆)(K × H), [·, ·]K×H ) is a Pontryagin space, then EB (∆)(K × H ) equipped with the inner product from K × H is also a Pontryagin space and by [12, Theorem 6.2] the operator W∆ is an iso 1 and B

1 are metric isomorphism of EA (∆)(K × H) onto EB (∆)(K × H ), i.e., A isometrically equivalent. The case that the function τ is a real constant is excluded in Theorem 3.6. In this case we have the following theorem. Theorem 3.9. Let Ω be a domain as in the beginning of Section 3.2 and let A be a closed symmetric operator of defect one in the Krein space K such that K = clsp { Nλ,A+ | λ ∈ Ω} holds. Assume that there exists a selfadjoint extension A0 of A which is definitizable over Ω. Let {C, Γ0 , Γ1 } be a boundary value space for A+ , A0 = ker Γ0 , denote by γ and M the corresponding γ-field and Weyl function, respectively, and let τ be a real constant. Then the relation A−τ = ker Γ1 + τ Γ0 is a selfadjoint extension of A in K which is definitizable over Ω. The sign types of M and A−τ are d-compatible in Ω. For every k ∈ K and every λ ∈ h(M ) ∩ h((M + τ )−1 ) ∩ Ω a solution of the boundary value problem f (3.19) f1 − λf1 = k, τ Γ0 fˆ1 + Γ1 fˆ1 = 0, fˆ1 = 1 ∈ A+ , f1 is given by

−1 γ(λ)+ k, f1 = (A−τ − λ)−1 k = (A0 − λ)−1 k − γ(λ) M (λ) + τ

f1 = λf1 + k.

Proof. The proof of Theorem 3.9 is a modification of the proof of Theorem 3.6. Note first that the relation (3.4) and K = clsp {ran γ(λ) | λ ∈ Ω∩ρ(A0 )} imply that the Weyl function M is not identically equal to a constant. Here it is obvious that the resolvent of A−τ yields a solution of the boundary value problem (3.19) (compare (3.1), (3.2) and (3.5)). As in step 2 and step 3 of the proof of Theorem 3.6 the function λ → −(M (λ)+τ )−1 , which by [1, Theorem 2.3] is definitizable in Ω, can be ˆ 1} ˆ 0, Γ regarded as the Weyl function corresponding to a boundary value space {C, Γ + ˆ for A with A−τ = ker Γ0 . Now the same arguments as in the proof of Theorem 3.6 show that A−τ is definitizable over Ω. The details are left to the reader.

References [1] T.Ya. Azizov, P. Jonas, On Locally Definitizable Matrix Functions, Preprint 21-2005, Institute of Mathematics, Technische Universität Berlin. [2] J. Behrndt, P. Jonas, On Compact Perturbations of Locally Definitizable Selfadjoint Relations in Krein Spaces, Integral Equations Operator Theory 52 (2005), 17–44. [3] J. Behrndt, P. Jonas, Boundary Value Problems with Local Generalized Nevanlinna Functions in the Boundary Condition, to appear in Integral Equations Operator Theory.


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[4] V.A. Derkach, On Weyl Function and Generalized Resolvents of a Hermitian Operator in a Krein Space. Integral Equations Operator Theory 23 (1995), 387–415. [5] V.A. Derkach, On Generalized Resolvents of Hermitian Relations in Krein Spaces. J. Math. Sci. (New York) 97 (1999), 4420–4460. [6] V.A. Derkach, S. Hassi, M.M. Malamud, H.S.V. de Snoo, Generalized Resolvents of Symmetric Operators and Admissibility. Methods Funct. Anal. Topology 6 (2000), 24–53. [7] V.A. Derkach, M.M. Malamud, Generalized Resolvents and the Boundary Value Problems for Hermitian Operators with Gaps. J. Funct. Anal. 95 (1991), 1–95. [8] V.A. Derkach, M.M. Malamud, The Extension Theory of Hermitian Operators and the Moment Problem. J. Math. Sci. (New York) 73 (1995), 141–242. [9] A. Dijksma, H.S.V. de Snoo, Symmetric and Selfadjoint Relations in Krein Spaces I. Oper. Theory Adv. Appl. 24, Birkh¨ auser Verlag Basel (1987), 145–166. [10] A. Dijksma, H.S.V. de Snoo, Symmetric and Selfadjoint Relations in Krein Spaces II, Ann. Acad. Sci. Fenn. Math. 12, 1987, 199–216. [11] V.I. Gorbachuk, M.L. Gorbachuk, Boundary Value Problems for Operator Differential Equations. Kluwer Academic Publishers, Dordrecht (1991). [12] I.S. Iohvidov, M.G. Krein, H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric. Mathematical Research 9 (1982), AkademieVerlag Berlin. [13] P. Jonas, On a Class of Unitary Operators in Krein Space. Oper. Theory Adv. Appl. 17, Birkh¨ auser Verlag Basel (1986), 151–172. [14] P. Jonas, A Class of Operator-Valued Meromorphic Functions on the Unit Disc, Ann. Acad. Sci. Fenn. Math. 17 (1992), 257-284. [15] P. Jonas, Operator Representations of Definitizable Functions. Ann. Acad. Sci. Fenn. Math. 25 (2000), 41–72. [16] P. Jonas, On Locally Definite Operators in Krein Spaces. in: Spectral Theory and Applications, Theta Foundation, (2003). [17] P. Jonas, On Operator Representations of Locally Definitizable Functions, Oper. Theory Adv. Appl. 162, Birkh¨ auser Verlag, Basel (2005), 165–190. [18] H. Langer, Spectral Functions of Definitizable Operators in Krein Spaces. Functional Analysis Proceedings of a Conference held at Dubrovnik, Yugoslavia, November 2-14 (1981), Lecture Notes in Mathematics 948, Springer Verlag Berlin-Heidelberg-New York (1982), 1–46. [19] H. Langer, A. Markus, V. Matsaev, Locally Definite Operators in Indefinite Inner Product Spaces, Math. Ann. 308 (1997), 405–424. Jussi Behrndt Institut f¨ ur Mathematik, MA 6-4 Technische Universit¨ at Berlin Straße des 17. Juni 136 D-10623 Berlin Germany e-mail: [email protected]


Riesz Bases of Root Vectors of Indefinite Sturm-Liouville Problems with Eigenparameter Dependent Boundary Conditions, I ´ Paul Binding and Branko Curgus Abstract. We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigenparameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space. Mathematics Subject Classification (2000). Primary 34L10, 34B24, 34B09, 47B50. Keywords. Sturm-Liouville equations, indefinite weight functions, Riesz bases.

1. Introduction We consider a regular indefinite Sturm-Liouville boundary eigenvalue problem of the form −(p f ) + q f = λ r f on [−1, 1]. (1.1) The coefficients 1/p, q, r in (1.1) are assumed to be real and integrable over [−1, 1], p(x) > 0, and x r(x) > 0 for almost all x ∈ [−1, 1]. We impose two boundary conditions on (1.1) (only one of which is λ-dependent): L b(f ) = 0,

Mb(f ) = λ Nb(f ).

(1.2)

where L, M and N are 1 × 4 non-zero (row) matrices and the boundary mapping b is defined for all f in the domain of (1.1) by T b(f ) = f (−1) f (1) (pf )(−1) (pf )(1) . We shall utilize an operator theoretic framework developed in [3]. Under Condition 2.1 below, a self-adjoint operator A in the Krein space L2,r (−1, 1) ⊕ C∆ can be associated with the eigenvalue problem (1.1), (1.2). Here ∆ is a non-zero

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´ P. Binding and B. Curgus

real number which is determined by M and N – see Section 2 for details. We remark that the topology of this Krein space is that of the corresponding Hilbert space L2,|r| (−1, 1) ⊕ C|∆| . (In the rest of the paper we abbreviate L2,r (−1, 1) to L2,r and L2,|r| (−1, 1) to L2,|r| .) Our main goal in this paper is to provide sufficient conditions on the coefficients in (1.1), (1.2) under which there is a Riesz basis of the above Hilbert space consisting of the union of bases for all the root subspaces of the above operator A. This will be referred to for the remainder of this section as the Riesz-basis property of A. Completeness and expansion theorems with a stronger topology, but in a smaller space corresponding to the form domain of the operator A, have been considered by many authors – see [3] (and the references there) and [12]. Although the topology of the Krein space L2,r ⊕ C∆ is weaker than the topology of the form domain, which in our case is a Pontryagin space, the expansion question turns out to be much more challenging mathematically. Indeed, even for the case when the boundary conditions are λ-independent this problem is nontrivial. In our notation, this case corresponds to L being a nonsingular 2 × 4 matrix, with the second equation in (1.2) suppressed. The Rieszbasis property of the operator corresponding to A, now defined in L2,r , has been discussed by several authors, e.g., in [2, 6, 9, 14, 15]. The first general sufficient condition for this was given by Beals [2], who required the weight function r to behave like a power of the independent variable x in an open neighborhood of the turning point x = 0, although his method does allow more general weight functions. Refinements of Beals’s method in [9] and [15] show that a “one-sided” condition on r (i.e., in only a half-neighborhood of x = 0 ) is enough to guarantee the Riesz-basis property. That some extra condition on r is indeed necessary follows from [15] where Volkmer showed that weight functions r exist for which the corresponding SturmLiouville problem (1.1), under the conditions used here, does not have the Rieszbasis property. Explicit examples of such weight functions were given in [1] and [10]. Recently, Parfyonov [13] has given an explicit necessary and sufficient condition for the Riesz basis property in the case p = 1, q = 0 with odd weight function r. Here, and in most of the above references, Dirichlet boundary conditions were imposed. General self-adjoint (perhaps non-separated, but still λ-independent) bound´ ary conditions were treated by Curgus and Langer [6]. They showed that if the essential boundary conditions, i.e., those not including derivatives, were separated, then a Beals-type condition in a neighborhood of x = 0 was sufficient for the Riesz-basis property. But if some of the non-separated boundary conditions were essential then [6] established the Riesz-basis property only by imposing extra restrictions on the weight function in (half-)neighborhoods of both endpoints of the interval [−1,1]. Again, some extra restriction is necessary, since in [4] we gave an explicit example of (1.1) under the conditions used here, satisfying a Beals-type condition at x = 0, but without the Riesz-basis property. Of course at least one (in fact one, in this antiperiodic case) boundary condition was essential and nonseparated. In some sense, then, the boundary ±1 behaves as a turning point under such boundary conditions.

Riesz Bases of Root Vectors, I

77

In summary, the Riesz-basis property is quite subtle, and depends significantly on the nature of the boundary conditions even when they are independent of λ. In this paper and its sequel, we shall examine the analogous situation for the cases of one and two λ-dependent boundary conditions, where the possibilities for the (λ-dependent) boundary conditions are much greater. As in the λ-independent case, a condition on the weight function is needed near the turning point x = 0 to ensure the Riesz-basis property of A. We shall develop such a condition (which is implied by the ones discussed above) in Section 4. Depending on the nature of the boundary conditions (1.2), we may also need a condition near the boundary, and this is discussed in Section 5. It should be remarked that for the case of exactly one λ-dependent boundary condition treated here we need only one such condition, near either x = −1 or x = 1, and this can be viewed as a “one-sided” condition at ±1. In the case of two λ-dependent boundary conditions we shall also need a condition involving both boundary points x = −1 and x = 1. It turns out that all the above conditions have a common core. This is not immediately obvious, since there are differences between the “turning points” 0 and ±1. For example, when the boundary conditions are separated, the values of f and f are equal at 0 but are independent at −1 and 1. The common core, which will also be needed in Part II, involves the notion of smoothly connected half-neighborhoods, and this is defined and studied in Section 3. In order to apply the above conditions, we use a criterion in Theorem 2.2, equivalent to the Riesz-basis property of A, involving a positive homeomorphism of L2,r ⊕ C∆ with the form domain of A as an invariant subspace. This, together with certain mollification arguments, is used for our main results, which are detailed in Section 6. To paraphrase these, we recall that a λ-independent boundary condition is essential if it does not include derivatives. Similarly, a λ-dependent boundary condition will be called essential if it does not include derivatives in the λ-terms. In Theorem 6.1 we discuss situations when a condition on r near x = 0 suffices for the Riesz-basis property of A. For example this holds when the first (λ-independent) boundary condition in (1.2) is either non-essential, or essential and separated, and the second (λ-dependent) one is non-essential. If the latter condition is essential instead, then the same result holds if a sign condition is also satisfied, and this includes a result of Fleige [11], which is the only reference we know where the Riesz-basis property of A has been studied for λ-dependent boundary conditions. In Theorems 6.2 and 6.3 we consider those cases of (1.2) which are not covered by Theorem 6.1. Then we require a condition near just one of the boundary points ±1, not both as in [6]. The choice of the boundary point is arbitrary in Theorem 6.2 which deals with the case when the boundary conditions in (1.2) are, respectively, essential non-separated and non-essential. In Theorem 6.3, however, this choice is not arbitrary but depends on the sign of the number ∆ used in defining the inner product on L2,r ⊕ C∆ .


78

2. Operators associated with the eigenvalue problem The maximal operator Smax in L2,r (−1, 1) = L2,r associated with (1.1) is defined by 1 Smax : f → (f ) := −(pf ) + qf , f ∈ D(Smax ), r where ' & D(Smax ) = Dmax = f ∈ L2,r : f, pf ∈ AC[0, 1], (f ) ∈ L2,r . We define the boundary mapping b by T b(f ) := f (−1) f (1) (pf )(−1) (pf )(1) , and the concomitant matrix Q corresponding to ⎡ 0 0 −1 ⎢0 0 0 Q := i ⎢ ⎣1 0 0 0 −1 0

f ∈ D(Smax ),

b by ⎤ 0 1⎥ ⎥. 0⎦ 0

We notice that Q = Q−1 . Integrating by parts we easily calculate that 1 1 Smax f g r − f Smax g r = i b(g)∗ Qb(f ), f, g ∈ D(Smax ). −1

−1

Throughout, we shall impose the following non-degeneracy and self-adjointness conditions on the boundary data. Condition 2.1. The row vectors L, M and N in (1.2) satisfy: ⎡ ⎤ L (1) the 3 × 4 matrix ⎣M⎦ has rank 3, N (2) LQL∗ = MQM∗ = NQN∗ = LQM∗ = LQN∗ = 0, (3) i MQ−1 N∗ is a non-zero real number and we define i . ∆=− MQ−1 N∗

(2.1)

Clearly the boundary value problem (1.1)–(1.2) will not change if row reduction is applied to the coefficient matrix 7 6 L 0 . (2.2) M N In what follows we will assume that the 2 × 8 matrix in (2.2) is row reduced to row echelon form (starting the reduction at the bottom right corner). After the row reduction, we write the row vectors L and N as L = L e Ln , N = Ne Nn . (2.3) If either of the 1 × 2 matrices Ln , Nn is non-zero, the corresponding boundary condition is called “non-essential”. In any case these matrices do not appear in the representation of the form domain of A, discussed below, but they will play an

Riesz Bases of Root Vectors, I

79

important role in our conditions for Riesz bases in Section 6. The 1 × 2 matrices Le and Ne represent the “essential” boundary conditions if the non-essential parts Ln and Nn are zero matrices. Next we define a Krein space operator associated with the problem (1.1)– (1.2). We consider the linear space L2,|r| ⊕ C, equipped with the inner product 6 7 1 g f := , f gr + w∆z , f, g ∈ L2,|r| , z, w ∈ C. w z −1 Then L2,|r| ⊕ C, [ · , · ] is a Krein space, which we denote by L2,r ⊕ C∆ . A fundamental symmetry on this Krein space is given by 7 6 J0 0 , (2.4) J := 0 sgn ∆ where sgn ∆ ∈ {−1, 1} and J0 : L2,r → L2,r is defined by x ∈ [−1, 1].

(J0 f )(x) := f (x) sgn(r(x)),

Then [J · , · ] is a positive definite inner product which turns L2,r ⊕ C∆ into a Hilbert space L2,|r| ⊕ C|∆| . The topology of L2,r ⊕ C∆ is defined to be that of L2,|r| ⊕ C|∆| , and a Riesz basis of L2,r ⊕ C∆ is defined as a homeomorphic image of an orthonormal basis of L2,|r| ⊕ C|∆| . We define the operator A in the Krein space L2,r ⊕ C∆ on the domain ⎧ ⎫ ⎨6 7 L2,r ⎬ f ∈ ⊕ : f ∈ D Smax , L b(f ) = 0, z = Nb(f ) D(A) = (2.5) ⎩ z ⎭ C∆ by

6 A

7 7 6 Smax f f , := Mb(f ) Nb(f )

6

7 f ∈ D(A). Nb(f )

(2.6)

Using [3, Theorems 3.3 and 4.1] we see that this operator is self-adjoint in L2,r ⊕ C∆ and in particular: (i) A is quasi-uniformly positive [7] (and therefore definitizable) in L2,r ⊕ C∆ . (ii) A has a discrete spectrum. (iii) The root subspaces corresponding to real distinct eigenvalues of A are mutually orthogonal in the Krein space L2,r ⊕ C∆ . (iv) All but finitely many eigenvalues of A are semisimple and real. For further properties of A, we refer the reader to [3, Theorem 3.3]. From (i), (ii) and the characterization of the regularity of the critical point infinity for definitizable operators in Krein spaces given in [5, Theorem 3.2], we then obtain the following, which is our central tool. Theorem 2.2. Let F (A) denote the form domain of A. There exists a basis for each root subspace of A, so that the union of all these bases is a Riesz basis of L2,|r| ⊕ C|∆| if and only if there exists a bounded, boundedly invertible, positive operator W in L2,r ⊕ C∆ such that W F (A) ⊂ F(A).

80


In order to apply this result, we need to characterize the form domain F (A). To this end, let Fmax be the set of all functions f in L2,r , absolutely continuous 1 on [−1, 1], such that −1 p |f |2 < +∞. On Fmax we define the essential boundary mapping be : Fmax → C2 by T be (f ) := f (−1) f (1) , f ∈ Fmax . Clearly be is surjective. By [3, Theorem 4.2], there are four possible cases for the form domain F (A) of A: If Ln = 0 and Nn = 0, then ⎫ ⎧ ⎬ ⎨6 7 L2,r f ∈ ⊕ : f ∈ Fmax , z ∈ C . (2.7) F(A) = ⎭ ⎩ z C∆ If Ln = 0 and Nn = 0, then ⎧ ⎫ ⎨6 7 L2,r ⎬ f ∈ ⊕ : f ∈ Fmax , Le be (f ) = 0, z ∈ C . F (A) = ⎩ z ⎭ C∆ If Ln = 0 and Nn = 0, then ⎧ ⎫ 7 L2,r ⎨6 ⎬ f ∈ ⊕ : f ∈ Fmax . F (A) = ⎩ Ne be (f ) ⎭ C∆ If Ln = 0 and Nn = 0, then ⎧ ⎫ 7 L2,r ⎨6 ⎬ f ∈ ⊕ : f ∈ Fmax , Le be (f ) = 0 . F (A) = ⎩ Ne be (f ) ⎭ C∆

(2.8)

(2.9)

(2.10)

To construct an operator W as in Theorem 2.2 we need to impose conditions on the coefficients p and r in (1.1). In all cases we need Condition 4.1 in a neighborhood of 0, and in some cases we also need one of two conditions, 5.1 or 5.2, on r in neighborhoods of −1 or 1. These will be discussed in Sections 4 and 5 respectively.

3. Smooth connection and associated operator To prepare the ground for the conditions mentioned above (and in Part II), we develop the concept of smoothly connected half-neighborhoods. A closed interval of non-zero length is said to be a left half-neighborhood of its right endpoint and a right half-neighborhood of its left endpoint. Let ı be a closed subinterval of [−1, 1]. By Fmax (ı) we denote the set of all functions f in L2,r (ı) which are absolutely continuous on ı and such that ı p |f |2
0 and decomposed as the orthogonal sum L(N m > 0, then the elements of L(q) are the polynomials of degree < n and the elements of L(M ) are 2-vector functions with polynomial entries. Unless stated otherwise we assume n > 0 and m > 0. If n = 0 or m = 0, then L(q) = {0} or L(M ) = {0} and the formulas simplify; we consider these cases separately. in the space L(N

). For In this section we give minimal models for N and N this we introduce the vector function v(z) = c(z) c(z) 1 c(z)(N0 (z) + q(z)) and the following ⎛ 1 0 ⎜0 1 Aα = ⎜ ⎝0 0 0 0

4 × 4 matrices ⎞ ⎛ 0 0 0 ⎜0 0 0⎟ ⎟, B = −⎜ ⎝0 α 0⎠ 1 0 1

0 0 0 1

0 0 1 0

⎞ 1 1⎟ ⎟, 0⎠ 0

⎛ 0 ⎜0 B = − ⎜ ⎝0 1

0 0 0 1

0 0 0 0

⎞ 1 1⎟ ⎟. 0⎠ 0

Theorem 4.1. Assume the conditions (1)–(4) hold. in L(N

) are given by the triplets (i) The minimal models of N and N

(B, KN ( · , z ∗ )v(z)N (z), S)

and

K ( · , z ∗ )v(z), S),

(B, N

where & '

) | ∃h ∈ C4 : g (ζ) − ζ f (ζ) = (A0 + N

(ζ)B)h , B = {f , g } ∈ L(N

(4.2)

' &

(ζ)B)h = {f , g } ∈ L(N

) | ∃h ∈ C4 : g(ζ) − ζ f (ζ) = (IC4 + N , B

(4.3)

and & '

) | ∃h ∈ C4 with h3 = 0 :

(ζ)B)h S = {f , g } ∈ L(N g(ζ) − ζ f (ζ) = (A0 + N .

108

A. Dijksma, A. Luger and Y. Shondin

) is (ii) S has defect (1, 1) and the family of all its self-adjoint extensions in L(N

α given by B and B , α ∈ R, where B α

= =

with

B + α · , w

L(N ) w

& '

) | ∃h ∈ C4 :

(ζ)B)h (4.4) {f , g} ∈ L(N g(ζ) − ζ f (ζ) = (Aα + N w

= 0 0

1

). 0 ∈ L(N

= B ∞ , the limit of B α in the resolvent sense Moreover, B = B 0 and B by letting α → ∞. Note that S has defect (1, 1) shows that it does not coincide with SN in the

in L(N

), which has defect (4, 4) if n = 0 and (3, 3) canonical representation of N if n = 0. On account of (1)–(4) the four equivalent statements in Theorem 2.1 hold For the Pontryagin space P we take the reproducing kernel Pontryafor N and N. ). For P we take L(N ) but we identify it with L(N ) via the unitary gin space L(N map N defined by (3.2). Since ∞ is a generalized zero of N , we have N ∈ L(N ) (see [10, Corollary 2.3(iii)]). It follows that (2.1) holds with w = N and A = AN ) we have w = w1 = 1 and A = A in P and in the identification of P with L(N N and so N (z) = (AN − z)−1 1, 1L(N) . The expansion (2.2) for N implies that the functions w1 , w2 , . . . , wm+n (or, what amounts to the same, w, Aw, . . . , Am+n−1 w) are given by 1, ζ, . . . , ζ m+n−1 (see [3, Lemma 5.2]). Here we use that the moments sj are zero for 0 ≤ j ≤ 2m + n − 2. in statement (iii) of Theorem 2.1 holds with A = A The representation for N N and u = KN ( · , z0∗ ). (4.5) Notice that u, w1 L(N ) = 1 by the reproducing property of the kernel. Since (AN − z)−1 1 = Rz 1 = 0, we see directly that AN is a relation with a nontrivial multi-valued part: 1 ∈ AN (0). In Section 3 we showed that the triplets , K ( · , z ∗ )N (z), S ) (A N N N

and (AN , KN ( · , z ∗ ), SN )

in L(N ) and that A is the limit in the resolvent are minimal models of N and N N

α , and (3.6) and (3.7). The main ; see (3.3), Theorem 3.1 applied to N sense of A N are isomorphic copies of idea of the proof of the theorem is that B α and B

α A and A . The isomorphism is given in Lemma 4.4 below. We begin with two N

N

technical lemmas. h1 ∈ L(M ) and h2 = 0, then deg h1 < m. Lemma 4.2. If h2

Minimal Models for Nκ∞ -functions

109

Proof. Since deg p0 < 2m, the space L(M ) is spanned by the 2m linearly independent 2-vector functions # j p0 j c R0 , R0 , j = 1, 2, . . . , m, c 0 where R0 is the difference-quotient operator given by (3.1) with z = 0. If h2 = 0, then # h1 j c | j = 1, 2, . . . , m}, ∈ span {R0 h2 0 which implies deg h1 < m. Lemma 4.3. Assume N0 ∈ N0 satisfies the conditions in (1.3). If h1 and h2 are polynomials and N0 h1 + h2 ∈ L(N0 ), then h1 = h2 = 0. Proof. Whenever defined we have Rw0 (f g)(ζ) = Rw0 (f )(ζ)g(w0 ) + f (ζ)Rw0 (g)(ζ). Let h1 and h2 be polynomials such that N0 h1 + h2 ∈ L(N0 ). Assume that h1 and h2 are not both identically equal to 0. If we apply Rw0 with w0 ∈ D(N0 ) a number of times to the function N0 h1 +h2 ∈ L(N0 ) and use the above formula and that Rw0 (N0 )(ζ)c ∈ L(N0 ), c ∈ C, we find that there is pair of complex numbers (c1 , c2 ) = (0, 0) such that N0 c1 + c2 ∈ L(N0 ). If N0 c1 + c2 = 0, then N0 is a real constant and therefore, on account of the last equality in (1.3) equal to zero. But this is in contradiction with the first equality in (1.3). If N0 c1 + c2 = 0, then, by Theorem 3.1(iii) applied to N0 , ∗ {0, 0} = {0, N0 c1 + c2 } ∈ SN , 0

which implies that the minimal operator SN0 in L(N0 ) is not densely defined. This is in contradiction with the first two equalities in (1.3). These contradictions imply that h1 and h2 are identically equal to 0.

) → L(N ) defined by (Vf )(ζ) = v # (ζ) f (ζ) is Lemma 4.4. The mapping V : L(N unitary. The corresponding mapping in [10, Lemma 3.1], where realizations of N related to its basic factorization (1.1) are considered, is a partial isometry but not necessarily injective. Proof of Lemma 4.4. A straightforward calculation shows v # (ζ)KN (ζ, z)v(z ∗ ) = KN (ζ, z).

(4.6)

We claim that the number of negative squares of the kernels KN and KN coincide. Indeed, the first number is the sum of the number of negative squares of the scalar function q and the matrix function M . The first of which is equal to (n + 1)/2 if n is odd and the leading coefficient of the polynomial q is negative, and otherwise it is n/2 . The kernel of M has m negative squares since z0 is the only zero of M in the closed upper half-plane and its multiplicity is m. By Theorem 2.1, the sum of these numbers equals κ, which is the number of negative squares of KN . Hence

110


[4, Theorem 1.5.7.] implies that V is a surjective partial isometry. We show that it is in fact a unitary mapping, that is, ker V is trivial. Assume there is an element f (ζ) = f (ζ) a(ζ) b(ζ) d(ζ) ∈ ker V. Then c# (ζ)f (ζ) + c# (ζ)a(ζ) + b(ζ) + c# (ζ)(N0 (ζ) + q(ζ))d(ζ) ≡ 0.

(4.7)

Since N0 is holomorphic at the point z0∗ , we have that f ∈ L(N0 ) is holomorphic at z0∗ also and the equality (4.7) implies that the polynomial b has a zero of order at least m at z0∗ . So we have b(ζ) = (ζ − z0∗ )m b1 (ζ) for some polynomial b1 . Equality (4.7) implies N0 h1 + h2 = −f ∈ L(N0 ) with polynomials h1 = d and h2 = a + b1 + qd. By Lemma 4.3, h1 = h2 = 0, that is, d = 0 and a + b1 = 0. Now we use Lemma 4.2: Since b d ∈ L(M ) and d = 0, the lemma yields that deg b < m, which implies b1 = 0 and hence b = 0 and a = 0. Finally, on account of (4.7), we have f = 0. We conclude that f = 0, that is, ker V = {0}. and S by the formulas Proof of Theorem 4.1. We first define B, B, V, B = V−1 A N

= V−1 A V, B N

= V−1 S V S = B ∩ B N

and claim that they coincide with the relations in part (i) of the theorem. Assuming the claim is true, we have, according to (4.6), that under the unitary mapping V the

) is the isomorphic copy of the element K ( · , z ∗ ) in element KN ( · , z ∗ )v(z) in L(N N ). Hence the triplets in (i) are isomorphic copies of the minimal models (3.3) L(N in L(N

). It remains to prove the claim. It and (AN , KN ( · , z ∗ ), SN ) for N and N is easy to see that & '

)2 | ∃d ∈ C : v # (ζ)( (ζ)d , B = {f , g} ∈ L(N g (ζ) − ζ f (ζ)) = N & ' =

)2 | ∃c ∈ C : v # (ζ)( B {f , g} ∈ L(N g (ζ) − ζ f (ζ)) = c . First we prove formula (4.2). Denote by BA0 ,B the relation defined by the right hand side of (4.2). If {f , g} ∈ BA0 ,B and h = h1 h2 h3 h4 then

(ζ)B)h = −N (ζ)h3 , g(ζ) − ζ f (ζ) = v # (ζ)(A0 + N (4.8) v # (ζ) hence BA0 ,B ⊂ B. Since BA0 ,B and B are self-adjoint operators, see Theorem 3.3, equality holds. In the same way, if {f , g} ∈ BI 4 ,B , the operator defined by the C right-hand side of (4.3), then

(ζ)B)h = h3 , v # (ζ)( g (ζ) − ζ f (ζ)) = v # (ζ)(IC4 + N

(4.9)

and equality prevails because both self-adjoint relations have ⊂ B nonempty resolvent sets. The formula for S follows from (4.8) and (4.9). This completes the proof of part (i). As to (ii) we first define the operators B α by hence BI

C4 , B

) α V, B α = V−1 (A N

α ∈ R.


111

On account of (3.6), we have

L(N ) w,

B α = B + α · , w where

w

= V−1 1 = 0 0

0 .

1

(4.10)

We now show (4.4). From (2.4), applied to this situation, we have (B α − z)−1 = (B − z)−1 −

L(N ) · , (B − z ∗ )−1 w N (z) + 1/α

(B − z)−1 w

and, by Theorem 3.3, −1

(BAα ,B − z)−1 = (AN − z)−1 − ΓNz Ez ,

B(Aα + N (ζ)B)

where Ez = Γ∗N z∗ is the operator of evaluation at the point z. For α = 0, the last equality yields

(ζ)B)−1 Ez (B − z)−1 = (AN − z)−1 − ΓN z B(A0 + N and, on account of (4.10), (B − z)−1 w

= N (z)ΓNz

v(z). Combining these relations we find (B α − z)−1 − (BAα ,B − z)−1 6 −1

= ΓN z −B(A0 + N(ζ)B) −

7 N (z)2 # −1

Ez . v(z)v (z) + B(Aα + N (ζ)B) N (z) + α1

A straightforward calculation shows that the expression in square brackets vanishes, which implies (4.4) for all α ∈ R. = B ∞ , where the relation on the Clearly, B = B 0 and, because of (3.7), B right-hand side is obtained via infinite coupling of B and w,

that is, by taking the limit of B α in the resolvent sense by letting α → ∞. Finally, the statement concerning S and its extensions follow from the corresponding results for SN and the unitarity of V. The following corollary is an immediate consequence of Theorem 3.3. We set ⎛ ⎞ c(z)c# (z) c(z)c# (z) c(z) α − p0 (z) ⎜c(z)c# (z) c(z)c# (z) ⎟ c(z) α − p0 (z) ⎟ Kα (z) = ⎜ # # ⎝ c# (z) c (z) 1 (N0 (z) + q(z))c (z) ⎠ α − p0 (z) α − p0 (z) (N0 (z) + q(z))c(z) (N0 (z) + q(z))(α − p0 (z)) and

⎛ 0 ⎜0 1 K(z) = limα→∞ Kα (z) = ⎜ ⎝0 α 1

0 0 0 1

0 0 0 0

⎞ 1 ⎟ 1 ⎟. ⎠ 0 N0 (z) + q(z)

112


are given by Corollary 4.5. The resolvents of B α and B N (z) Γ Kα (z)Ez , (B α − z)−1 = (AN − z)−1 + 1 + αN (z) N z − z)−1 = (A − z)−1 + Γ K(z)E (B z, N

Nz

(4.11) (4.12)

), where (AN − z)−1 is the difference-quotient operator in the space L(N ∗ ∗ ∗ ΓNz

= diag {KN0 ( · , z ), Kq ( · , z ), KM ( · , z )},

and Ez = (ΓN z∗ )∗ is the evaluation operator at the point z. From (4.11) it readily follows that

w

= (B α − z)−1 w,

N (z) , 1 + αN (z)

α ∈ R ∪ {∞},

which is consistent with (2.6). It remains to discuss the simplifications if n = 0 or m = 0. The case n = 0 and m > 0: Then q(z) = q0 with q0 ∈ R, hence Kq (ζ, z) = 0 and L(q) = {0}; as to the 2 × 2 matrix function M (z): deg p0 (z) < 2m and the space

(z) becomes the 3 × 3 matrix function L(M ) is nontrivial. Now N

) = L(N0 ) ⊕ L(M ).

(z) = diag {N0 (z) + q0 , M (z)} and L(N N

(z) + q0 ) the operator V With v(z) = c(z) 1 c(z)(N 0 : L(N ) → L(N ) of

in multiplication by v # (z) = c# (z) 1 c# (z)(N0 (z) + q0 ) is unitary and w −1 (4.10) becomes w

:= V 1 = 0 1 0 . Theorem 4.6. Assume n = 0 and m > 0. Then Theorem 4.1 and Corollary 4.5 remain true provided in all 4 × 4 matrices the 2-nd row and the 2-nd column are S,

and B α the space C4 and the entry h3 deleted and in the formulas for B, B, 3 are replaced by C and h2 . The case n > 0 and m = 0: Now c(z) = 1, q(z) is a nonconstant real (z) = N0 (z)+q(z). polynomial, and the irreducible representation (1.2) becomes N

From N (z) = diag {N0 (z), q(z)} it follows that L(N ) = L(N0 ) ⊕ L(q). With the

) → L(N ) of multiplication 1 1 the mapping V : L(N vector function v(z) = # by v (z) = 1 1 is unitary. Theorem 4.7. Assume n > 0 and m = 0 and for α ∈ R define the operators & '

) | ∃h ∈ C2 : g (ζ) − ζ f (ζ) = (Aα + N

(ζ)B)h , g } ∈ L(N B α = {f , 1 0 , −1 α

where Aα =

B=−

0 0

1 . 1

= A = B ∞ , and S = B ∩ B, (i) Theorem 4.1 holds provided B = B 0 , B N which takes the form & S = {f , g } ∈ AN | ∃h ∈ C : ( g (ζ) − ζ f (ζ)) = h 1 −1 }.


113

(ii) Corollary 4.5 becomes: For α ∈ R, (B

α

−1

− z)

1 1 1 Ez , Γ − N0 (z) + q(z) − α N z 1 1

−1

= (AN − z)

), where (AN − z)−1 is the difference-quotient operator in the space L(N ΓN z = diag {KN0 ( · , z ∗ ), Kq ( · , z ∗ )}, and Ez = (ΓN z∗ )∗ is point evaluation at z.

5. A decomposition of L(N) In this section we choose a basis in L(q) ⊕ L(M ) and determine the associated

see Lemma 5.1 below. In the next section we identify L(q)⊕L(M ) Gram matrix G; n m

and exhibit with C ⊕ C ⊕ Cm equipped with an inner product determined by G

α and the relation B. As in [12] we the matrix representations of the operators B

) the linearly independent elements define in L(N

vj = (B − z0∗ )j−1 w,

j = 1, . . . , m, m + 1, . . . , m + n, ∗

and with ϕ(z) = ϕ( · , z) = KN ( · , z )v(z) 1 d j−1 − z0 )−j+1 ϕ(z0 ) = ϕ(z) |z=z0 , uj = (B (j − 1)! d z

j = 1, . . . , m.

Here, on account of (4.6), u1 = V−1 u, where u is given by (4.5). Moreover, we introduce the three subspaces L0 = span {v1 , . . . , vm }, L = span {vm+1 , . . . , vm+n }, M = span {u1 , . . . , um }. at ∞ is the direct sum L +L ˙ 0 and According to Theorem 2.1, the root space of B 0 L is its isotropic part. ˙ Lemma 5.1. We have L(q) = L and L(M ) = L0 +M. The basis elements for L(q) and L(M ) can be written more explicitly as vm+j (ζ) = 0 (ζ − z0∗ )j−1 0 0 , j = 1, . . . , n, and vj (ζ)

=

uj (ζ)

=

0 0 0 0

(ζ − z0∗ )j−1 Rzj 0 p0 (ζ)

0 , (ζ − z0 )m−j ,

j = 1, . . . , m, j = 1, . . . , m,

where Rz stands for the difference-quotient operator at z. The Gram matrix asso˙ ciated with this basis for the space L(q) ⊕ (L0 +M) is given by ⎛ ⎞ Gq 0 0

=⎝0 0 ICm ⎠ , G 0 ICm Gp0

114


in which Gq = (qi,j )ni,j=1 has entries qi,j = vm+j , vm+i L(N ) =

m+j−1 m+i−1 k=0

l=0

m+i−1 m+j −1 (−z0∗ )m+j−1−k (−z0 )m+i−1−l sk+l , l k

where sk+l are the moments of N in (2.2), and Gp0 = (pi,j )m i,j=1 has entries pi,j = uj , ui L(N ) =

(5.1)

1 d j−1 d i−1 p0 (z) − p0 (w∗ ) %% 1 . % (j − 1)! (i − 1)! dz dw∗ z − w∗ z=w=z0

The formulas for the basis element uj (ζ) and the entry pi,j of the Gram matrix Gp0 in this lemma are independent of the way the polynomial p0 (z) is # 0 k written. If we write p0 (ζ) = k=0 pk (ζ − z0 ) and set pk = 0 if k > 0 , then 0 # Rzj 0 p0 (ζ) = pk (ζ − z0 )k−j and straightforward calculations yield k=j

pi,j =

2m−j k=i

k−1 pk+j−1 (z0∗ − z0 )k−i ; i−1

so, in particular, if z0 ∈ R then pi,j = pi+j−1 . After the proof of the lemma we give some other formulas for the Gram matrix Gq as well. Proof of Lemma 5.1. Since v1 = w,

the element v1 is of the given form. We calculate vj = (B − z0∗ )vj−1 for j = 2, . . . , m + n. Write Bvj−1 as Bvj−1 (ζ) = f (ζ) a(ζ) b(ζ) d(ζ) ∈ L(N0 ) ⊕ L(q) ⊕ L(M ). Then, by (4.4), there exists a vector h = h1 h2 h3 h4 ∈ C4 such that ⎛ ⎞ ⎞ ⎛ h1 − N0 (ζ)h4 f (ζ) ⎜ ⎟ ⎟ ⎜ a(ζ) h2 − q(ζ)h4 ⎜ ⎟ ⎟ ⎜ (5.2) = ⎜ ⎜ ⎟. ⎟ ⎝b(ζ) − ζ(ζ − z0∗ )j−2 ⎠ ⎝−c# (ζ)(h1 + h2 ) − p0 (ζ)h3 ⎠ d(ζ)

−c(ζ)h3 + h4

By Lemma 4.3, h1 = h4 = 0. Since d is a polynomial of degree less than m also h3 = 0. Thus b is the first component of an element in L(M ) whose second component d = 0, therefore, see Lemma 4.2, the degree of b is less than m. The equality between the third components of the vectors in (5.2) now reads as b(ζ) − ζ(ζ − z0∗ )j−2 = −c# (ζ)h2 .

(5.3)

For 2 ≤ j ≤ m, a comparison of the degrees of the polynomials on both sides, yields h2 = 0. Thus b(ζ) = (ζ − z0∗ )j−1 + z0∗ (ζ − z0∗ )j−2 and hence we have vj (ζ) = 0 0 (ζ − z0∗ )j−1 0 , j = 1, . . . , m.


115

If j = m + 1 then (5.3) implies h2 = 1 and hence we find vm+1 (ζ) = 0 1 0 0 . Now the formula for vj , j = m+ 2, . . . , n, can be checked in a similar way as above. It is easy to see that the element u1 = KN ( · , z0∗ )v(z0 ) has the stated form. By (4.12), we have for 2 ≤ j ≤ m, ⎛ ⎞ ⎛ ⎞ 0 0 ⎜ ⎟ ⎜ ⎟ 0 0 − z0 )−1 uj−1 (ζ) = (A − z0 )−1 ⎜ ⎟=⎜ j ⎟. uj (ζ) = (B N ⎝ Rzj−1 ⎠ ⎝ ⎠ R p (ζ) p (ζ) 0 z0 0 0 (ζ − z0 )m−j+1

(ζ − z0 )m−j

the zeros come from the facts that L(q) ⊥ L(M ) As to the Gram matrix G, 0 and L is neutral. The formula for Gq follows from expanding

(B − z0∗ )m+i−1 w

L(N ) qi,j = (B − z0∗ )m+j−1 w, in terms of B l w,

B k w

L(N ) N )k 1 = AlN N, AkN N L(N ) = Al w, Ak wP = sk+l . = (AN )l 1 , (A L(N)

are obtained from the reproducing kernel property of The entries ICm in G KN (ζ, z): % 1 d j−1 % KN ( · , z ∗ )v(z), vi L(N ) % uj , vi L(N ) = (j − 1)! dz z=z0 % j−1 d 1 % = (z − z0 )i−1 % = δij , 1 ≤ i, j ≤ m. (j − 1)! dz z=z0 Finally, the formula for pi,j in the lemma readily follows from 1 1 (j − 1)! (i − 1)! % d j−1 d i−1 % ∗ ∗ K ( · , z )v(z), K ( · , w )v(w) . × %

N N L( N ) dz d w∗ z=w=z0

uj , ui L(N ) =

We claim that the Gram matrix Gq = (vm+j , vm+i L(N ) )ni,j=1 in Lemma 5.1 is lower diagonal with respect to the second diagonal. To see this we use the equality vm+j , vm+i L(N ) = vm+j+1 , vm+i−1 L(N ) + (z0∗ − z0 )vm+j , vm+i−1 L(N ) , (5.4) which readily follows from the relation vm+i = (B − z0∗ )vm+i−1 . Since L(q) is orthogonal to L0 the recurrence relation (5.4) implies vm+j , vm+1 L(N ) = 0,

j = 1, . . . , n − 1,

and hence, again with (5.4), also the lower triangular form of Gq . Furthermore, (5.4) also implies that the entries on the second diagonal: vm+d , vm+n+1−d L(N )

116


are independent of d = 1, . . . , n. Note that Gq is not a Hankel matrix in general, however, it is if z0 = z0∗ . In the following two propositions we present two other formulas for Gq . The first one is in terms of the real coefficients τj of q: q(z) = τn z n + τn−1 z n−1 + · · · + τ1 z + τ0 ,

τn = 0.

The Gram matrix Sq associated with the standard basis {1, ζ, . . . , ζ n−1 } in L(q) is equal to ⎞−1 ⎛ ⎛ ⎞ τ1 τ2 · · · τn−1 τn 0 0 ··· 0 σn−1 ⎜ τ2 τ3 · · · ⎜ 0 τn 0⎟ 0 · · · σn−1 σn ⎟ ⎟ ⎜ ⎜ ⎟ Sq = ⎜ . =⎜ . ⎟ . . . . .. ⎟ . .. .. .. .. .. ⎠ ⎝ .. ⎝ .. . ⎠ 0 0 σn−1 σn · · · σ2n−3 σ2n−2 τn 0 · · · where the real numbers σj are defined by the expansion 1 σn−1 σ2n−2 1 = n + · · · + 2n−1 + O( 2n ), q(z) z z z

z = iy, y ↑ ∞

(5.5)

(see, for example, [10, Theorem 3.4]). Notice that τ0 does not play a role. Since Gq is the Gram matrix of the basis vm+1 , vm+2 , . . . , vm+n , we express this basis in terms of the standard basis via the n × n matrix H: 1 ζ − z0∗ (ζ − z0∗ )2 . . . (ζ − z0∗ )n−1 = 1 ζ ζ 2 . . . ζ n−1 H. and then we have Gq = H∗ Sq H. If H = (hi,j )ni,j=1 then the entries in the jth column are given by ⎧ ⎨ j−1 (−z0∗ )j−i , i = 1, . . . j, i−1 hi,j = ⎩ 0, i = j + 1, . . . n. The connection with the moments sj for N given by (2.2) can be obtained from the fact that the asymptotics of N (z) = −1/N(z) in Theorem 2.1(ii) is the same as the asymptotics of the function −1/(c# (z)q(z)c(z)): Write for |z| > |z0 |, ∞ 1 tm+k−1 k m+k−1 (5.6) = , t = z m+k−1 0 m−1 c(z) z m+k k=0

(note that tm−1 = 1) and set ⎛ tm−1 tm ⎜ 0 t m−1 ⎜ T=⎜ . .. ⎝ .. . 0 0

··· ···

tm+n−3 tm+n−4 .. .

···

0

⎞ tm+n−2 tm+n−3 ⎟ ⎟ .. ⎟ . . ⎠ tm−1


117

Using (5.5) and (5.6) to calculate the asymptotics of −1/(c#(z)q(z)c(z)) and comparing it with (2.2) we find that ⎛ ⎞ 0 0 ··· 0 s2m+n−1 ⎜ 0 0 · · · s2m+n−1 s2m+n ⎟ ⎜ ⎟ MN = ⎜ ⎟ = T∗ Sq T. .. .. .. .. ⎝ ⎠ . . . . s2m+n−1

s2m+n

···

s2m+2n−3

s2m+2n−2

Hence we have proved the following proposition. Proposition 5.2. With H, Sq , T and MN as defined above we have Gq = H∗ T−∗ MN T−1 H = H∗ Sq H. The first equality follows from the formula for Gq given in Lemma 5.1. The triangular forms of the matrices H with 1 on the diagonal and Sq with σn−1 = 1/τn on the second diagonal yield the triangular form of Gq with 1/τn on the second diagonal.

) To derive yet another formula for Gq , we identify the elements vm+j ∈ L(N ∗ j−1 with the functions (ζ − z0 ) ∈ L(q), j = 1, . . . , n. Also for later use, we introduce the vector polynomial sq (z) = (si (z))ni=1 ,

si (z) = Rzi 0∗ q(z),

(5.7)

where Rz is the difference-quotient operator at z. The kernel Kq ( · , z) can be expressed in the basis {vm+1 , . . . , vm+n } as n Kq ( · , z) = vm+i si (z ∗ ). (5.8) i=1

For this, write q as q(z) =

n #

qk (z−z0∗ )k , then si (z) =

#n

k=0

Kq (ζ, z) = =

n k=1 n

and hence

n k (ζ − z0∗ )k − (z ∗ − z0∗ )k qk = qk (ζ − z0∗ )i−1 (z ∗ − z0∗ )k−i ζ − z∗ i=1

(ζ − z0∗ )i−1

i=1

∗ k−i k=i qk (z−z0 )

k=1

n k=i

qk (z ∗ − z0∗ )k−i =

n

vm+i (ζ) si (z ∗ ).

i=1

For the next proposition, we choose n distinct points z1 , . . . , zn ∈ C, denote by V the n × n Vandermonde matrix ⎛ ⎞ 1 z1 − z0∗ . . . (z1 − z0∗ )n−1 ⎜1 z2 − z0∗ . . . (z2 − z0∗ )n−1 ⎟ ⎜ ⎟ V = ⎜. ⎟, .. .. ⎝ .. ⎠ . . ∗ ∗ n−1 1 zn − z0 . . . (zn − z0 ) and define the n × n matrix S = (si,j )ni,j=1 by si,j = si (zj∗ ). Proposition 5.3. With the above notation Gq = S−∗ V.

118


# Proof. If S−1 = (ti,j )ni,j=1 , then on account of (5.8), vm+i = nk=1 Kq ( · , zk )tk,i , i, j = 1, . . . , n, and by the reproducing kernel property, n n vm+j , vm+i L(q) = t∗k,i vm+j , Kq ( · , zk )L(q) = t∗k,i vm+j (zk ) =

k=1 n

k=1

(S−∗ )i,k (zk − z0∗ )j−1 = (S−∗ V)i,j .

k=1

6. Minimal models in the space (K; G) in the In this section we construct minimal models for the functions N and N n m m orthogonal sum K = L(N0 )⊕C ⊕C ⊕C , where L(N0 ) is the reproducing kernel Hilbert space associated with the Nevanlinna function N0 in the representation . The inner product on Cm will be denoted by (x, y)m = y∗ x, x, y ∈ Cm ; (4.1) of N the index m in the inner product will be omitted when it is clear from the context. We denote by (K; G) the linear space K equipped with the indefinite inner product G · , · K defined by the Gram matrix ⎛ ⎞ IL(N0 ) 0 0 0 ⎜ 0 Gq 0 0 ⎟ ⎟, G=⎜ (6.1) ⎝ 0 0 0 ICm ⎠ 0 0 ICm Gp0 where Gq and Gp0 are given in Lemma 5.1. Because of Lemma 4.3 we have N0 ∈ L(N0 ). But since the element Rw0 N0 = KN0 ( · , w0∗ ),

w0 ∈ D(N0 ),

belongs to L(N0 ), we see that N0 is a generalized element belonging to the space L(N0 )−1 defined in the Introduction. Thus the pairing f0 , N0 between an element f0 ∈ dom AN0 and N0 is well defined: f0 , N0 = (AN0 − w0 )f0 , KN0 ( · , w0 )L(N0 ) = g0 (w0 ) − w0 f0 (w0 ),

(6.2)

where g0 = AN0 f0 and the right-hand side is independent of w0 ∈ D(N0 ). In this connection we write χ−1 for N0 ∈ L(N0 )−1 and we define ϕ0 (z) = (AN0 − z)−1 χ−1 = KN0 ( · , z ∗ ),

z ∈ D(N0 ).

By Theorem 3.1, the triplet (AN0 , ϕ0 (z), SN0 ) is a minimal model for N0 in L(N0 ). To keep the formulation of the next theorem short, we introduce the following notation. We write 0 n q(z) = qj (z − z0∗ )j , p0 (z) = pk (z − z0 )j , (6.3) j=0

j=0

set pk = 0 for k > 0 , and define the column vectors ∗ q = q0 · · · qn−1 ∈ Cn , p = p0 · · · pm−1 ∈ Cm .


119

We use the column vector polynomials m

m

sq (z) = (sj (z))nj=1 , tp0 (z) = (tj (z))m j=1 , r1p0 (z) = (r1j (z))j=1 , r2p0 (z) = (r1j (z))j=1 . Here sq (z) is as in (5.7), tj (z) = Rzj 0 p0 (z), and the entries of the last two vectors are the coefficients in the expansions Rz Rzm0 p0 (ζ) =

m

r1j (z)(ζ − z0∗ )j−1 ,

j=1

and 0

Rzj 0 p0 (ζ)(z

− z0 )

j−1

=

j=m+1

m

r2j (z)(ζ − z0∗ )j−1 .

j=1

Furthermore, we write b(z) = (z − z0∗ )m−1 (z − z0∗ )m−2 · · · 1 , d(z) = 1 (z − z0 ) · · · (z − z0 )m−1 . We denote by Jm (z0 ) the m × m Jordan block matrix at z0 with Jm (z0 )em,1 = z0 em,1 , where for j = 1, 2, . . . , m, em,j stands for the jth element in the standard orthogonal basis of Cm . Theorem 6.1. Assume the conditions (1)–(4). (i) For α ∈ R, let C α be the set of all pairs of the form

⎧⎛ ⎞ ⎛ ⎞⎫ ⎪ ⎪ g0 + w0 λχ0 ⎪ ⎪ ⎪ ⎜ f0 +λχ0 ⎟ ⎜ ⎟⎪ ⎪ ⎪ ⎪ ∗ ⎟ ⎜−f , χ ⎟⎪ ⎬ ⎨⎜ a e − λ(N (w )e + q) + J (z ) a + (b, e )e ⎜ ⎟ ⎜ 0 −1 n,1 0 0 n,1 n 0 m,m n,1 ⎟ ⎜ ⎟ ,⎜ ⎟ , ∗ ⎟ ⎜ ⎟⎪ ⎪⎜ b −λG e + J (z ) b + (d, αe − p)e ⎪ p0 m,m m 0 m,1 m,1 ⎪ ⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ d λem,m + Jm (z0 )d

with χ0 = ϕ0 (w0 ), {f0 , g0 } ∈ AN0 , λ ∈ C, a ∈ Cn such that an = λqn , and b, d ∈ Cm , where w0 is a fixed point in D(N0 ). Then C α is the graph of a self-adjoint operator (also denoted by C α ) in the space (K; G). be the set of all pairs of the form (ii) Let C ⎧⎛ ⎞ ⎛ ⎪ f0 +λχ0 ⎪ ⎪ ⎪⎜ ⎟ ⎜ ⎪ ⎨⎜ ⎜ a ⎟ ⎜ ⎟ ⎜−f0 , χ−1 en,1 ⎜ ⎟ ,⎜ ⎜ ⎟ ⎜ ⎪ b ⎪ ⎪ ⎝ ⎠ ⎝ ⎪ ⎪ ⎩ d

⎞⎫

⎪ g0 + w0 λχ0 ⎪ ⎪ ⎟⎪ ⎪ ⎬ − λ(N0 (w0 )en,1 + q) + Jn (z0 )∗ a + (b, em,m )en,1 ⎟ ⎟ , ⎟ ⎟⎪ −λGp0 em,m + Jm (z0 )∗ b + µem,1 ⎪ ⎠⎪ ⎪ ⎪ ⎭ λem,m + Jm (z0 )d

with χ0 = ϕ0 (w0 ), {f0 , g0 } ∈ AN0 , λ, µ ∈ C, a ∈ Cn such that an = λqn , b ∈ Cm , and d ∈ Cm such that d1 = 0, where w0 is a fixed point in D(N0 ). is a self-adjoint relation in (K; G). Then C in the space (K; G) are given by the triplets (iii) The minimal models of N and N (C, N (z)Γz , S)

and

Γz , S), (C,

120

A. Dijksma, A. Luger and Y. Shondin and where C = C 0 , S = C ∩ C, ⎛

⎞ ϕ0 (z)c(z) ⎜ ⎟ sq (z)c(z) ⎟ Γz = ⎜ ⎝r2p0 (z) + b(z)c(z)(N0 (z) + q(z))⎠ . d(z)

(iv) The family of all self-adjoint extensions of S in (K; G) is given by C α , Moreover, C = C ∞ , the limit in the resolvent sense of C α α ∈ R, and C. as α → ∞. Note that C α is of the form (2.3): ⎛ ⎞ ⎛ ⎞ 0 0 < ; ⎜ 0 ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎟ C α = C + α · , ⎜ ⎝em,1 ⎠ ⎝em,1 ⎠ . 0 0 is multi-valued: 0 0 em,1 Note also C

0 ∈ C(0), S can also be written as

⎧⎧⎛ ⎞ ⎛ ⎞⎫⎫∗ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨⎨⎜ ⎟ ⎜ ⎟⎬⎬ 0 0

α ⎜ ⎟,⎜ ⎟ ∩ S=C ⎝0⎠ ⎝em,1 ⎠⎪⎪ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩⎩ ⎭⎪ 0 0 and that

C α

⎧⎧⎛ ⎞ ⎛ ⎞⎫⎫ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨⎜ ⎨⎪ ⎟ ⎜ ⎟⎬⎬ 0 0 ⎜ ⎟,⎜ ⎟ = S + span 0 (z0 ))em,1 ⎠⎪⎪ , ⎪⎝ 0 ⎠ ⎝(α − N ⎪ ⎪ ⎪ ⎩ ⎭ ⎩⎪ ⎭⎪ em,1 z0 em,1

in particular, the domain of C α is dense and independent of α ∈ R. In the proof of Theorem 6.1 we identify – according to the basis discussed in

) with K = L(N0 )⊕Cn ⊕Cm ⊕Cm and the relations B α , Section 5 – the space L(N and S defined in the theorem. To explain the identification, and S with C α , C, B,

) be given as let the vector function f ∈ L(N f (ζ) = f (ζ) a(ζ)

b(ζ)

d(ζ)

,

where L(q) a(ζ) =

n j=1

n aj vm+j (ζ) 2 = aj (ζ − z0∗ )j−1 j=1

(6.4)

Minimal Models for Nκ∞ -functions and

˙ L0 +M

=

m

b(ζ) d(ζ)

121

bj vj (ζ) 3,4 + dj uj (ζ) 3,4

j=1

⎛# ⎞ m djRzj 0 p0 (ζ)+bj (ζ − z0∗ )j−1 ⎜j=1 ⎟ ⎟. =⎜ m # ⎝ ⎠ dj (ζ − z0 )m−j j=1

Then f will be identified with the element ⎛ ⎞ f ⎜a⎟ ⎜ ⎟ ∈ K, ⎝b⎠ d where a = a1 . . . an , etc. We write f f w

in (4.10) we have w

0 0 em,1 0 .

b d . For example, for

a

) given by Proof of Theorem 6.1. Identify the vector functions f , g ∈ L(N

= f (ζ) a1 (ζ) b1 (ζ) d1 (ζ) , g (ζ) = g(ζ) a2 (ζ) b2 (ζ) d2 (ζ) f(ζ) with the elements

f f

b2 d2 in K. Here for i = 1, 2, the entries of the vectors ai = ai,1 . . . ai,n , etc., appear as coefficients in the representations ⎛# ⎞ m j ∗ j−1 d R p (ζ)+bi,j (ζ − z0 ) n ⎜j=1 i,j z0 0 ⎟ bi (ζ) ⎟. =⎜ ai (ζ) = ai,j (ζ − z0∗ )j−1 , m # ⎝ ⎠ di (ζ) m−j di,j (ζ − z0 ) j=1 a1

b1

d1

,

g g

a2

j=1

According to Theorem 4.1 we have

g (ζ)} ∈ B α ⇐⇒ ∃ h1 {f(ζ), ⎛

h2

h3

h4

∈ C4 :

⎞ h1 − N0 (ζ)h4 ⎜ ⎟ h2 − q(ζ)h4 ⎟ g (ζ) − ζ f (ζ) = ⎜ ⎝−c# (ζ)(h1 + h2 ) + (α − p0 (ζ))h3 ⎠ . −c(ζ)h3 + h4

(6.5)

Comparison of the fourth components on both sides of this equality yields h3 = d1,1 and d2,j = z0 d1,j + d1,j+1 , j = 1, . . . , m − 1;

d2,m = z0 d1,m + h4 .

122


In the same way the second components in (6.5) yield h4 = a1,n /qn and a1,n , a2,1 = z0∗ a1,1 + h2 − q0 qn a1,n , j = 2, . . . , n. a2,j = z0∗ a1,j + a1,j−1 − qj−1 qn We now consider the third components in (6.5). Inserting the expressions for d2,j already obtained we find that m

d2,j

j=1

reduces to

0

pk (ζ − z0 )k−j −

0 m

d1,j pk (ζ − z0 )k−j+1 + z0 (ζ − z0 )k−j

j=1 k=j

k=j m

d1,j pj−1 − h3 p0 (ζ) + h4

j=1

0

pk (ζ − z0 )k−m .

k=m

In the last term we may replace 0 by 2m − 1, because 0 < 2m and pk = 0 if k > 0 . Using the relation (5.1) the last term can now be rewritten as m # h4 um , uj L(N ) (ζ − z0∗ )j−1 . Then in the third components there only remain j=1

powers of ζ − z0∗ and comparing these we find h1 + h2 = b1,m and b2,1 = z0∗ b1,1 + αd1,1 −

m

pj−1 d1,j − h4 um , u1 L(N ) ,

j=1

b2,j = z0∗ b1,j + b1,j−1 − h4 um , uj L(N ) ,

j = 2, . . . , m.

If we rewrite N0 (ζ) in the first component of (6.5) as (ζ − w0 )KN0 (ζ, w0∗ ) + N0 (w0 ) we find g(ζ) − w0 h4 KN0 (ζ, w0∗ ) − ζ f (ζ) − h4 KN0 (ζ, w0∗ ) = h1 − h4 N0 (w0 ). = = >? @ >? @ =:g0 (ζ)

=:f0 (ζ)

So {f0 , g0 } ∈ AN0 and with λ = a1,n /qn = h4 we have f = f0 + λKN0 ( · , w0∗ ), g = g0 + w0 λKN0 ( · , w0∗ ).

(6.6)

Because N0 ∈ L(N0 ), we have that KN0 ( · , w0∗ ) ∈ dom AN0 , hence the decomposition (6.6) is unique. Furthermore, we have that h1 = f0 , N0 + λN0 (w0 ). Indeed, since {f0 , g0 } ∈ AN0 , the difference g0 (ζ) − ζf0 (ζ) is identically equal to a constant and hence, on account of (6.2), h1 − λN0 (w0 ) = h1 − h4 N0 (w0 ) = g0 (ζ) − ζf0 (ζ) = g0 (w0 ) − w0 f0 (w0 ) = f0 , N0 . Hence h2 = b1,m − h1 = b1,m − f0 , N0 − λN0 (w0 ). Together these formulas show that {f (ζ), g (ζ)} ∈ B α can be identified with a pair of elements in K of the form described in the theorem. Hence under the identification B α coincides with C α .


123

can be identified with B can be proved in a similar way and therefore (ii) That C the details are omitted.

) and K, the Γ-field K ( · , z ∗ )v(z) in Theo(iii) In the identification between L(N N rem 4.1 coincides with the Γ-field Γz in (iii). (z) = (z 2 + 1)N0 (z) + γ3 z 3 + γ2 z 2 + γ1 z + γ0 , where N0 is Example. Consider N a Nevanlinna function satisfying (1.3) and the γj ’s are real numbers with γ3 = 0. in the form (4.1) and (6.3): We rewrite N (z) = (z + i){N0 (z) + q1 (z + i) + q0 }(z − i) + p1 (z − i) + p0 N (i) = γ0 − γ2 + i(γ1 − γ3 ). with q1 = γ3 , q0 = γ2 − iγ3 , p1 = γ1 − γ3 and p0 = N ∈ N ∞ , where κ = 2 if γ3 < 0 and κ = 1 if γ3 > 0. The Then m = 1, n = 1, N κ is K = L(N0 ) ⊕ C ⊕ C2 equipped with the indefinite inner state space for N and N product G · , · K in which the Gram matrix is given by 1 0 1 . , G = diag IL(N0 ) , 1 γ1 − γ3 γ3 We take w0 = i, and recall χ0 (z) = KN0 ( · , z ∗ ) ∈ L(N0 ) and χ−1 = N0 ∈ L(N0 )−1 . We find that the graph of the self-adjoint operator C α is the set of all elements of the form ⎧⎛ ⎞ ⎛ ⎞⎫ ⎪ ⎪ g0 + iλχ0 ⎪ ⎪ ⎪ ⎜ f0 + λχ0 ⎟ ⎜ ⎪ ⎪ ⎟⎪ ⎪ ⎜ ⎬ ⎨⎜ ⎟⎪ λγ3 ⎟ ⎜ ⎟ ⎜ −f0 , χ−1 − λ(N0 (i) + γ2 ) + b ⎟ ⎜ ⎟ ,⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎪ ⎪ ⎪ b ⎪ ⎪ ⎠ ⎝ −λ(γ1 − γ3 ) − ib + (α − N (i))d⎠ ⎪ ⎪ ⎪⎝ ⎪ ⎪ ⎭ ⎩ d λ + id is the set of all elements with {f0 , g0 } ∈ AN0 , λ, b, d ∈ C. The self-adjoint relation C of the form ⎧⎛ ⎞ ⎛ ⎞⎫ ⎪ ⎪ f0 + λχ0 g0 + iλχ0 ⎪ ⎪ ⎪ ⎜ ⎪ ⎪ ⎟ ⎜ ⎟⎪ ⎪ ⎬ ⎨⎜ ⎟ ⎜ −f , χ ⎟⎪ λγ − λ(N (i) + γ ) + b 3 0 −1 0 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ,⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎪ ⎪ µ b ⎪ ⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ λ 0 with {f0 , g0 } ∈ AN0 , λ, b, µ ∈ C. From Corollary 4.5 we obtain the following theorem. We set S1 (z; z0 ) = 0 and if n ≥ 2, ⎞ ⎛ 0 1 (z − z0 ) . . . (z − z0 )n−2 ⎟ ⎜ .. .. .. ⎟ ⎜ . . . ⎟ ⎜ ⎟ ⎜ . . .. .. Sn (z; z0 ) = ⎜ (z − z0 ) ⎟ ⎟ ⎜ ⎟ ⎜ .. ⎠ ⎝ . 1 0 and we recall that the definitions of the matrix functions Kα (z) and K(z) are given just before Corollary 4.5.

124


in Theorem 6.2. The resolvents of the operators C α , α ∈ R, and the relation C K are given by Sm (z; z0∗ ) r1p0 (z)b# (z) (C α −z)−1 = diag (AN0 − z)−1 , Sn (z; z0∗ ), # 0 Sm (z; z0∗) N (z) r2p0 (z) b(z) Kα (z) diag ϕ0 (z), sq (z), + d(z) 0 1 + αN (z) # d (z) tp0 (z) , × diag · , ϕ0 (z ∗ )L(N0 ) , d# (z), 0 b# (z) where ϕ0 (z) = KN0 ( · , z ∗ ), and Sm (z; z0∗ ) r1p0 (z)b# (z) −1 −1 ∗ (C−z) = diag (AN0 − z) , Sn (z; z0 ), # 0 Sm (z; z0∗ ) r2p0 (z) b(z) K(z) +diag ϕ0 (z), sq (z), d(z) 0 # d (z) tp0 (z) ∗ # × diag · , ϕ0 (z )L(N0 ) , d (z), . 0 b# (z) As to the proof of the theorem we only mention the following identifications between the elements and operators in L(q) and Cn : −1 Kq ( · , z ∗ ) sq (z) = (Jn (z0 )∗ − z (q − q(z)en,1 ), · , Kq ( · , z) Gq · , sq (z ∗ ) = d# (z), (Aq − z)−1 Sn (z; z0∗ ) = (I + (z0∗ − z)Jn (0))−1 Jn (0). The first identification follows from (5.7) and (5.8) which show that sq (z) is the vector representation of the element Kq ( · , z ∗ ) ∈ L(q) relative to the basis {vm+i }ni=1 . The linear functional · , Kq ( · , z ∗ )L(q) on L(q) can be identified with (Gq · , sq (z ∗ )) = d# (z) viewed as a mapping from Cn to C. This also follows from (6.4). In the theorem r1p0 (z)b# (z) is an m × m matrix polynomial of rank 1. Also, the matrix r2p0 (z) b(z) d(z) 0 whose entries are column m-vectors should be viewed as a mapping from C2 to C2m , whereas the matrix # d (z) tp0 (z) 0 b# (z) whose entries are row m-vectors should be seen as a mapping from C2m to C2 . The matrices are related via the formula # # r2p0 (z) b(z) 0 ICm d (z) tp0 (z) , = d(z) 0 ICm Gp0 0 b# (z) which corresponds to a part of the identity Γ∗N z∗ = Ez .


125

We now consider the analogs of Theorem 6.1 in the cases n = 0 and m = 0. The case n = 0 and m > 0: Here K = L(N0 ) ⊕ Cm ⊕ Cm and the Gram matrix takes the form ⎛ ⎞ IL(N0 ) 0 0 0 ICm ⎠ . G=⎝ 0 0 ICm Gp0 Theorem 6.3. Assume n = 0 and m > 0. Then Theorem 6.1 holds if we delete the second component in all 4-vectors in the formulas, omit in (i) and (ii) the statement “a ∈ Cn such that an = λqn ”, add in (i) and (ii) the statement “bm = f0 , χ−1 + λ(N0 (w0 ) + q0 )”, and set q(z) = q0 in Γz in (iii). The case n > 0 and m = 0: Now K = L(N0 ) ⊕ Cn and G = diag {IL(N0 ) , Gq }. Theorem 6.4. Assume n > 0 and m = 0. Then Theorem 6.1 holds if C α is the set of all pairs of the form ⎧⎛ ⎨ f ⎝ 0 ⎩

⎞⎫

⎞ ⎛

⎬ + λχ0 ⎠ ⎝ g0 + w0 λχ0 ⎠ , ∗ a −f0 , χ−1 en,1 − λ(N0 (w0 )en,1 + q + αen,1 ) + Jn (z0 ) a ⎭ (6.7) with χ0 = ϕ0 (w0 ), {f0 , g0 } ∈ AN0 , λ ∈ C, and a ∈ Cn such that an = λqn , where w0 is a fixed point in D(N0 ), if = AN0 ⊕ {{a, Jn (z0 )∗ a + µen,1 } | a ∈ Cn , an = 0, µ ∈ C}, C and if Γz = ϕ0 (z), sq (z) .

Next we give the formulas for the compressions of the resolvents (C α −z)−1 , − z)−1 to the subspaces L(N0 ) and L(N0 ) ⊕ Cn of (K; G) in the α ∈ R, and (C case n > 0 and m > 0; similar formulas can be obtained in the other two cases. We denote by P0 and P1 the orthogonal projections in (K; G) onto L(N0 ) and L(N0 ) ⊕ Cn . Theorem 6.5. (i) For α ∈ R, P0 (C α − z)−1 |L(N0 ) = (AN0 − z)−1 − with parameter Tα (z) = q(z) +

1 · , ϕ0 (z ∗ )L(N0 ) ϕ0 (z) N0 (z) + Tα (z)

p0 (z) − α , and for α = ∞, c# (z)c(z)

− z)−1 |L(N ) = (AN0 − z)−1 . P0 (C 0 (ii) For α ∈ R, P1 (C

α

−1

− z)

|L(N0 )⊕Cn

(AN0 − z)−1 = 0

1 − N0 (z) + Tα (z)

0 Sn (z; z0∗ )

· , ϕ0 (z ∗ )L(N0 ) ϕ0 (z) ϕ0 (z)d# (z) · , ϕ0 (z ∗ )L(N0 ) sq (z)

sq (z)d# (z)

,

126

A. Dijksma, A. Luger and Y. Shondin and for α = ∞, P1 (C

∞

−1

− z)

|L(N0 )⊕Cn =

(AN0 − z)−1 0

0

Sn (z; z0∗ )

.

The resolvent formula (Aτ − z)−1 = (AN0 − z)−1 −

1 · , ϕ0 (z ∗ )ϕ0 (z) N0 (z) + τ

describes the family of all self-adjoint extensions Aτ of SN0 in L(N0 ) in terms of the parameter τ ∈ R ∪ {∞}. If the number τ is replaced by a generalized Nevanlinna function τ (z) the formula describes the minimal self-adjoint extensions which act in spaces containing L(N0 ). This is called Krein’s resolvent formula. In part (i) the parameter describing C α with α ∈ R is explicitly given by τ (z) = Tα (z) ∈ Nκ . Related to Krein’s formula here are the references [21, Theorem 4.7], [13, Theorem 4.2], and [7, Theorem 3.5]. with It is of interest to compare the compressed resolvents of C α and C the compressions of these operators/relation themselves. Recall Stenger’s lemma (see [15, Theorem 3.3 and a remark after the theorem]) that if A is a self-adjoint

and H is a Hilbert or Pontryagin relation with ρ(A) = ∅ in a Pontryagin space H

such that dim H

H < ∞, then the compression of A to H, that subspace of H, is, the linear relation PH A |H = {{f, PH g} | {f, g} ∈ A, f ∈ H},

onto H, is self-adjoint in H. In our where PH is the orthogonal projection in H case L(N0 ) is a Hilbert subspace and L(N0 ) ⊕ Cn is a Pontryagin subspace of the Pontryagin space K and both have a finite codimension. Therefore the compressions just mentioned are self-adjoint. The following theorem follows directly from Theorem 6.1 and its versions for the special cases n = 0 or m = 0. Theorem 6.6. (i) If n > 0 and m ≥ 0, then L(N ) = AN0 , P0 C α |L(N0 ) = P0 C| 0 and if n = 0 and m > 0, then (in graph notation) L(N ) P0 C α |L(N0 ) = P0 C| 0 = {{f0 + λχ0 , g0 + w0 λχ0 }|{f0 , g0 } ∈ AN0 , f0 , χ−1 + λ(N0 (w0 ) + q0 ) = 0}. (ii) If n > 0 and m > 0, then L(N )⊕Cn P1 C α |L(N0 )⊕Cn = P1 C| 0 and their graphs coincide with the set of all pairs of the form (6.7) with α = 0.


127

7. Block operator matrix models in the space (K; H) Changing the basis we have considered in the previous sections we can write C α in Theorem 6.1 with w0 = z0∗ in a block operator matrix form, which is not and C possible with the basis used so far. Set ⎛ ⎞ 1 IL(N0 ) ( · , en,n )χ0 0 0 ⎜ qn ⎟ ⎜ ICn 0 0 ⎟ T =⎜ 0 ⎟, ⎝ 0 0 I m 0 ⎠ C

0

0

0

ICm

where χ0 = ϕ0 (z0∗ ), and define the operators D α = T −1 C α T , , and the Gram matrix = T −1 CT relation D ⎛ 1 IL(N0 ) ( · , en,n )χ0 ⎜ qn ⎜1 h0 ⎜ ( · , en,n )en,n + Gq H = T ∗ GT = ⎜ · , χ0 L(N0 ) en,n qn2 ⎜ qn ⎝ 0 0 0

0

α ∈ R, the linear ⎞ 0 0 0 ICm

0

⎟ ⎟ 0 ⎟ ⎟, ⎟ ICm ⎠ Gp0

where h0 = χ0 , χ0 L(N0 ) = KN0 (z0 , z0 ). Since T −1 w

= w,

we have

K w,

D α = D + αH · , w

D = T −1 CT.

The space K equipped with the indefinite inner product H · , · K will be are self-adjoint in (K; H). The relation denoted by (K; H). Clearly, D α and D can be obtained via infinite coupling of D and w, D

that is, as limit of D α in the resolvent sense by letting α → ∞. The following theorem shows that D α can be expressed by means of block operator matrices. We use the notation and D explained directly above Theorem 6.1. be as defined above. Then: Theorem 7.1. Let D α , α ∈ R, and D (i) dom D α = (dom AN0 ) ⊕ Cn ⊕ Cm ⊕ Cm and on this domain D α has the block matrix form ⎞ ⎛ AN0 D12 0 0 ⎟ ⎜ ⎟ ⎜ D21 D22 ( · , em,m )en,1 0 ⎟ ⎜ ⎟ ⎜

α 1 D =⎜ ⎟, − ( · , en,n )Gq em,m Jm (z0 )∗ ( · , αem,1 − p)em,1 ⎟ ⎜ 0 ⎟ ⎜ qn ⎠ ⎝ 1 0 ( · , en,n )em,m 0 Jm (z0 ) qn where with χ0 = ϕ0 (z0 ) qn−1 1 ( · , en,n ) − ( · , en,n−1 ) χ0 , D12 = 2 qn qn

D21 = − · , χ−1 en,1 ,

128

A. Dijksma, A. Luger and Y. Shondin and D22 = Jn (z0 )∗ −

1 ( · , en,n ) N0 (z0∗ )en,1 + q . qn

= (dom AN0 ) ⊕ Cn ⊕ Cm ⊕ (Cm {em,1 }) and (ii) dom D = {{f , ∃µ ∈ C : D g}|f ∈ dom D, g = Df + 0 0 µem,1

0

},

where D = D 0 .

z , S),

(D, Γ

z , S)

are minimal models for N and N in (iii) The triplets (D, N (z)Γ

(K; H), where S = D ∩ D and ⎛ ⎞ (ϕ0 (z) − ϕ0 (z0 ))c(z) ⎜ ⎟ sq (z)c(z)

z = ⎜ ⎟ Γ ⎝r2p0 (z) + b(z)c(z)(N0 (z) + q(z))⎠ . d(z) The theorem follows directly from Theorem 6.1 with w0 = z0∗ , the definitions and Γ

z = T −1 Γz . Note that Γ

z0 = 0 0 0 em,1 . of D α and D, From the theorems in Section 6 one can easily obtain formulas for the resol We vents, the compressions of the resolvent and the compressions of D α and D. leave the details to the reader.

8. Examples We give two examples and discuss an approximation problem taken from [11], [17], and [14] to which we refer for details and proofs. They are related to the Bessel differential expression ν 2 − 1/4 y(x) x2 on (0, 1] with a self-adjoint boundary condition at the regular endpoint x = 1 and on (0, ∞), which is limit point at x = ∞. We recall the series expansion of the Bessel function ∞ z 2k z ν (−1)k Jν (z) = , (8.1) 2 k!Γ(k + ν + 1) 2 ν y(x) = −y (x) +

k=0

where the series on the right converges absolutely, and uniformly in any bounded domain of z and ν. We denote by 1 Yν (z) (Jν (z)cos (νπ) − J−ν (z)), = sin (νπ) 1 (1) (J−ν (z) − Jν (z)e−iνπ ), Hν (z) = isin (νπ) π (1) Kν (z) = i π2 ei 2 ν Hν (iz) the Neumann function of order ν, the first Hankel function of order ν, and the Basset (or MacDonald) function of order ν, respectively; see, for example, [18].


129

Example. See [11]. We consider ν on the interval (0, 1] and impose the boundary condition y(1) = 0 at the regular endpoint x = 1 on all functions y. (i) First assume 0 < ν < 1. Then the minimal realization S of ν in H0 = L2 (0, 1) is a symmetric operator with defect indices (1, 1). We denote by A0 the self-adjoint operator extension of S with graph A0 = {{y, ν y} ∈ S ∗ |limx↓0 xν−1/2 y(x) = 0}. The function π z ν/2 x1/2 ϕ(x, z) = − 2 sin πν

√ √ √ J−ν ( z) √ Jν (x z) − J−ν (x z) Jν ( z)

(8.2)

(8.3)

belongs to ker (S ∗ − z) and is a defect function for S and A0 with corresponding Q-function √ π ν J−ν ( z) √ . N (z) = − z (8.4) 2 sin πν Jν ( z) Thus the following relations hold: ϕ(z) = (I + (z − z0 )(A0 − z)−1 )ϕ(z 0 ),

(z) − N (w)∗ N = ϕ(z), ϕ(w) 0 . (8.5) z − w∗

(ii) Now assume ν > 1, ν = 2, 3, . . .. Then the results are quite different from those in (i): The minimal realization of ν in H0 is self-adjoint, the function ϕ( · , z) in in (8.4) is now (8.3) is well defined but it does not belong to H0 , and the function N a generalized Nevanlinna function with κ = [(ν + 1)/2] negative squares. Thus the model for this function involves a self-adjoint operator or relation in a Pontryagin ∈ N ∞ : Let zn , n = 1, 2 . . ., space with κ negative squares. In [11] we show that N √κ −ν/2 Jν ( z) in increasing order. be the enumeration of the zeros of the function z admits (The zeros are positive and, since the function is entire, countable.) Then N the decomposition (z) = z 2κ (N0 (z) + q0 ) + p0 (z), N where N0 (z) =

∞ n=1

q0 =

zn 1 − 2 zn − z zn + 1

2znν−2κ , Jν (zn )2

∞ 1 (2κ) 2znν−2κ−1 (0) − , N (2κ)! (zn2 + 1)Jν (zn )2 n=1

and p0 (z) =

2κ−1 j=0

pj z j ,

pj =

1 (j) N (0). j!

∈ N ∞. The Nevanlinna function N0 satisfies the relations (1.3) and hence N κ κ Theorem 6.3 with m = κ > 0, z0 = 0, and Gp0 = (pi+j−1 )i,j=1 yields the descrip and N = −1/N. tion of the models for N

130


Example. See [17]. We now consider ν on (0, ∞). The endpoint x = ∞ is limit point so we do not need to impose a condition at this endpoint. Where possible we use the same notation as in the previous example. (i) First assume 0 < ν < 1. Then the minimal realization S of v in H0 = L2 (0, ∞) is a symmetric operator with defect indices (1, 1). The self-adjoint extension A0 of S defined by formula (8.2) is uniquely determined by the facts that its spectrum σ(A0 ) = √ [0, ∞) is absolutely continuous and that the functions y(x, λ) = c(λ)x1/2 Jν (x λ), λ ∈ [0, ∞), form a complete set of generalized eigenfunctions of A0 , where c(λ) is some normalizing factor. The function √ √ ν ϕ(x, z) = x(−z) 2 Kν (x −z) (8.6) belongs to ker (S ∗ − z) and is a defect function for S and A0 with corresponding Q-function (z) = − π N (−z)ν . (8.7) 2 sin πν is Thus the relations (8.5) are also valid in this case. It follows from (8.7) that N a Nevanlinna function, which satisfies the limit conditions in (1.3). (ii) Now assume ν > 1 and ν = 2, 3, . . .. Then, as in the previous example, the minimal realization of ν in H0 is self-adjoint, the function ϕ( · , z) in (8.6) does in (8.7) is a generalized Nevanlinna function not belong to H0 , and the function N with κ = [(ν + 1)/2] negative squares. Here the branch of (−z)ν is chosen so that (−z)ν = rν e iν(θ−π) if z = re iθ , 0 < θ < 2π. For any z0 ∈ (−∞, 0) the function N admits the decomposition (z) = (z − z0 )2κ (N0 (z) + q0 ) + p0 (z), N where

N0 (z) = 0

∞

t 1 − 2 t−z t +1

tν dt, 2(t − z0 )2κ

q0 = −

and p0 (z) =

2κ−1 j=0

pj (z − z0 ) , j

1 (j) π(−1)j+1 pj = N (z0 ) = j! 2 sin πν

π , 4 sin πν 2

ν (−z0 )ν−j . j

Since N0 is a Nevanlinna function which satisfies the relations (1.3), we have that ∈ N ∞ . Hence Theorem 6.3 with m = κ > 0 applies and provides the description N κ (z) and N (z) = −1/N (z). Since z0 is real, the Gram matrix of the models for N κ Gp0 is given by Gp0 = (pi+j−1 )i,j=1 . Inspired by [29] and [28], we discuss an approximation problem in Nκ∞ ; for details we refer to the paper [14] in preparation. In the context of the discussion around (1.4), the problem is to approximate strongly singular perturbations by ∈ N ∞ (κ > 0) with irreducible smoother perturbations. Consider a function N κ representation (4.1): (z) = (z − z0 )2κ (N0 (z) + q0 ) + p0 (z), N


131

j ∈ N ∞ with irreducible representation (4.1): and a sequence of functions N κ j (z) = N0j (z) + qj (z), N

j = 1, 2, . . . ,

where N0 and all N0j are Nevanlinna functions satisfying (1.3), z0 is a real number belonging to their common domain D of holomorphy, q0 ∈ R, p0 (z) and qj (z) are real polynomials with deg p0 ≤ 2κ − 1, and n = deg qj is either 2κ or 2κ ± 1: if n is odd and the leading coefficient of qj (z) is negative, then n = 2κ − 1; if n is odd and the leading coefficient of qj (z) is positive, then n = 2κ + 1. Assume that, as j converges to N uniformly on compact subsets of D. The approximation j → ∞, N problem with variable spaces then is to describe this convergence in terms of the j and N and of the corresponding state spaces. We rewrite N j in the models of N following form j (z) = (z − z0 )2κ (M0j (z) + q0j ) + p0j (z) + qj,2κ+1 (z − z0 )2κ+1 , N

(8.8)

where M0j (z) is a Nevanlinna function, p0j (z) is a real polynomial with deg p0j ≤ 2κ − 1, and q0j , qj,2κ+1 ∈ R with qj,2κ+1 ≥ 0 (if qj,2κ+1 > 0 then it is the leading coefficient of qj (z)). The convergence assumption is equivalent to the convergence of M0j (z) + q0j to N0 (z) + q0 uniformly on compact subsets of D, the pointwise convergence of the polynomials p0j (z) to p0 (z), and the convergence qj,2κ+1 → 0, j (z) need not be irreducible, and so as j → ∞. The representation (8.8) of N models will have to be constructed, which fall outside the scope of this paper. Approximation of operators with variation of the space in which they act has been considered in [22, pp. 512, 513]; for such approximations in an indefinite setting, see [27] and [26]. The application we have in mind is related to ν and the last example. In [14] in (8.7) with ν > 1, ν = 2, 3, . . . , can be approximated we show that the function N by functions of the form δ (z) = N δ (z) + q δ (z) N (8.9) 0

by letting δ ↓ 0. Here q δ (z) is some real polynomial of degree [ν] with coefficients depending on δ, which we will not further specify here, and the function N0δ is obtained as follows. Consider the family of regularized differential expressions lν,δ y(x) = −y (x) +

ν 2 − 1/4 y(x) (x+δ)2

on (0, ∞), where the parameter δ varies over some interval (0, δ0 ), δ0 > 0. Let Sδ be the minimal operator associated with lν,δ in the Hilbert space H0 = L2 (0, ∞); it is symmetric and its defect indices are (1, 1). Each self-adjoint extension of Sδ can be obtained as the restriction of the maximal operator Sδ ∗ by the boundary condition y (0) = αy(0) with α ∈ R ∪ {∞}. We denote by Aδ the extension corresponding to α = ∞. The function √ 1/2 Kν ((x+δ) −z) √ , γ = 2ν−1 Γ(ν), ϕδ (x, z) = γ(x+δ) δ ν Kν (δ −z)

132


is a defect function for Sδ and Aδ . The function considered in (8.9) is by definition the function √ √ Kν (δ −z) δ 2 1−2ν √ N0 (z) = γ δ . (8.10) −z Kν (δ −z) It satisfies the relation N0δ (z) − N0δ (w)∗ = ϕδ (z), ϕδ (w)0 z − w∗ and hence is a Q-function for Sδ and Aδ . It follows that N0δ is a Nevanlinna function with integral representation ∞ t 1 δ − 2 dσδ (t) + Re N0δ (i), N0 (z) = t−z t +1 0 where, for t ≥ 0, dσδ (t) =

1 2γ 2 1 √ dt. Im N0δ (t + i0) dt = 2 2ν 2 √ π π δ Jν (δ t) + Yν2 (δ t)

(8.11)

δ in (8.9) belongs to N ∞ with It will be shown (in [14]) that the function N κ in (8.7) uniformly on compact subsets κ = [(ν + 1)/2] and, if δ ↓ 0, converges to N ). Note that the representation (8.9) of N δ is irreducible and corresponds of D(N whose irreducible to (4.1) with m = 0. This in contrast with the limit function N representation corresponds to (4.1) with m = κ > 0. We conclude the paper with a final remark. Remark 8.1. From the beginning up to and including Section 7 we may replace the factor c(z) = (z − z0 )m by c(z) = (z − z1 ) · · · (z − zm ) with zj ∈ D(N0 ) to obtain similar but more general models as in [13, Sections 6 and 7].

References [1] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable models in quantum mechanics, Springer, 1988. [2] S. Albeverio and P. Kurasov, Singular perturbations of differential operators, London Mathematical Society Lecture Notes 271, Cambridge Univ. Press, 2000. [3] D. Alpay, A. Dijksma, and H. Langer, Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions, Linear Algebra Appl. 387C (2004), 313–342. [4] D. Alpay, A. Dijksma, J. Rovnyak, and H. de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Operator Theory: Adv. Appl. 96, Birkh¨ auser Verlag, Basel, 1997. [5] V.A. Derkach and S. Hassi, A reproducing kernel space model for Nκ -functions, Proc. Amer. Math. Soc. 131 (12) (2003), 3795–3806. [6] V. Derkach, S. Hassi, and H. de Snoo, Operator models associated with singular perturbations, Methods Funct. Anal. Topology 7 (3) (2001), 1–21.


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[26] H. Langer and B. Najman, Perturbation theory for definizable operators in Krein spaces, J. Operator Theory 9(1983), 297–317. [27] B. Najman, Perturbation theory for selfadjoint operators in Pontrjagin spaces, Glasnik Mat. 15(35) (1980), 351–371. [28] Yu. Shondin, On approximation of high order singular perturbations, J. Phys. A: Math. Gen. 38(2005), 5023–5039. [29] O.Yu. Shvedov, Approximations for strongly singular evolution equations, J. Funct. Anal. 210(2) (2004), 259–294. Aad Dijksma Department of Mathematics University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands e-mail: [email protected] Annemarie Luger Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstrasse 8–10 A-1040 Vienna Austria e-mail: [email protected] Yuri Shondin Department of theoretical Physics State Pedagogical University Str. Ulyanova 1 Nizhny Novgorod 603950 Russia e-mail: [email protected]


Generalized Friedrichs Extensions Associated with Interface Conditions for Sturm-Liouville Operators Andreas Fleige, Seppo Hassi, Henk de Snoo and Henrik Winkler Dedicated to Heinz Langer on the occasion of his retirement

Abstract. For a class of Sturm-Liouville operators with an interface condition at an interior point all selfadjoint realizations are determined. This result is obtained via a description of the selfadjoint extensions of the coupling of two symmetric operators. The (generalized) Friedrichs extension, when it exists, is determined. Sufficient conditions for the (generalized) Friedrichs extension to exist are given. Mathematics Subject Classification (2000). Primary 47A10, 47B25; Secondary 34B05, 34B24. Keywords. Symmetric operator, selfadjoint extension, (generalized) Friedrichs extension, boundary triplet, Weyl function, interface condition.

1. Introduction Let −DpD + q be a Sturm-Liouville expression with real coefficients on the subset [−b, 0)∪(0, b] of R. It is assumed that there are fixed separated boundary conditions at −b and b and that there is a selfadjoint interface condition at 0 of the form (pu )(0+) = τ (u(0+) − u(0−)),

(pu )(0+) = (pu )(0−),

when τ ∈ R, and of the form u(0+) = u(0−),

(pu )(0+) = (pu )(0−),

when τ = ∞. For τ ∈ R ∪ {∞} these interface conditions describe all selfadjoint extensions of the symmetric operator which is given by the interface condition u(0+) = u(0−),

(pu )(0+) = (pu )(0−) = 0,

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A. Fleige, S. Hassi, H.S.V. de Snoo and H. Winkler

together with the fixed separated boundary conditions at −b and b. The purpose of this note is to indicate conditions which show the existence of a generalized Friedrichs extension (cf. [5], [6]) and to determine the corresponding value of τ ∈ R ∪ {∞} under mild conditions on the singularity at 0. Note that the conditions on the coefficients p and q at 0 can be further relaxed along the lines of, e.g., [10]. It is clear that interface conditions at an interior point can be easily described via the orthogonal coupling of symmetric operators. The selfadjoint extensions for such orthogonal couplings are best expressed in terms of the corresponding boundary triplets; see the extension results in [1]. The description of generalized Friedrichs extensions can also be given in terms of sesquilinear forms along the lines of [3], which will be done in [4].

2. Preliminaries 2.1. Boundary triplets Let S be a closed symmetric operator (or relation) with equal defect numbers in a Hilbert space H with inner product (·, ·). Recall that a relation in H is a linear subspace of the Cartesian product H × H, and its elements are denoted as ordered pairs. Let Nλ = ker (S ∗ − λ) be the defect subspace of S, and denote λ := { fλ = {fλ , λfλ } : fλ ∈ Nλ }, N

λ ∈ C,

λ ⊂ S ∗ . A boundary triplet Π = {H, Γ0 , Γ1 } of S ∗ consists of a Hilbert so that N space H and the boundary mappings Γj , j = 0, 1, from S ∗ to H such that Γ : f → { Γ0 f, Γ1 f} from S ∗ into H × H is surjective and such that the identity (f , g) − (f, g ) = Γ1 f, Γ0 g − Γ0 f, Γ1 g (2.1) H

H

holds for all f = {f, f }, g = {g, g } ∈ S ∗ . The mappings Γi define two selfadjoint extensions Ai of S via Ai = ker Γi , i = 0, 1. The mapping Γ : f → {Γ0 f, Γ1 f} induces a one-to-one correspondence between the closed extensions H of S which are intermediate (i.e., which satisfy S ⊂ H ⊂ S ∗ ) and the closed linear relations τ in H, via H := { f ∈ S ∗ : Γf ∈ −τ −1 }. (2.2) In particular, this correspondence (2.2) is one-to-one between all selfadjoint extensions H of S and all selfadjoint relations τ in H, cf. [1], [2]. When S is densely defined, then often the mappings Γ0 and Γ1 are interpreted as being defined on dom S ∗ instead of on S ∗ , i.e., in that case one speaks of Γ0 f and Γ1 f when f = {f, f } ∈ S ∗ . This convention will be followed in the present note. Associated to the boundary triplet Π are two operator functions: the Weyl function M (λ), defined by λ } M (λ) = { {Γ0 fλ , Γ1 fλ } : fλ ∈ N

λ ∈ ρ(A0 ),

(2.3)

Generalized Friedrichs Extensions

137

the graph of a bounded linear operator in H, and the γ-field γ(λ) defined by λ }, γ(λ) = { {Γ0 fλ , fλ } : fλ ∈ N

λ ∈ ρ(A0 ),

(2.4)

the graph of a bounded linear operator from H to Nλ . Both functions are holomorphic on ρ(A0 ). The relation between the Weyl function M (λ) and the γ-field γ(λ) is given by M (λ) − M (µ)∗ = γ(µ)∗ γ(λ). (2.5) λ−µ ¯ The identity (2.5) implies that M (λ) is the Q-function of the pair {S, A0 }. Each Weyl function belongs to the class N of Nevanlinna functions, i.e., is holomorphic ¯ and Im M (λ) ≥ 0 for λ ∈ C+ . Moreover, on C \ R, and satisfies M (λ)∗ = M (λ), M (λ) is strict, i.e., 0 ∈ ρ(Im M (λ)) for all λ ∈ C \ R. In general, the Weyl function M (λ) determines up to unitary isomorphisms, a model for the symmetric operator S and its selfadjoint extension A0 . 2.2. Generalized Friedrichs extensions For the purposes of this paper it is sufficient to consider the case of a symmetric operator S with defect numbers (1, 1), in which case the Weyl function M (λ) is a scalar function. The Weyl function M (λ) is said to belong to the Kac class N1 if ∞ Im M (iy) dy < ∞. y 1 In this case there is a real limit γ = lim M (iy) ∈ R.

(2.6)

y→∞

It follows from (2.2) (identifying the selfadjoint relations in R with R ∪ {∞}) that there is a one-to-one correspondence between all selfadjoint extensions of S in H and all numbers in R ∪ {∞} via A(τ ) = { f ∈ S ∗ : Γ0 f = −τ Γ1 f},

τ ∈ R ∪ {∞}.

(2.7)

Recall that the selfadjoint extension A(τ ) is determined by τ ∈ R∪{∞} via Kre˘ın’s formula: (A(τ ) − z)−1 = (A0 − z)−1 − γ(z)(M (z) + 1/τ )−1 (·, γ(¯ z )),

z ∈ C \ R.

Note that A(0) corresponds with A0 = ker Γ0 and that A(∞) corresponds with A1 = ker Γ1 . The Weyl functions Mτ (λ) of the selfadjoint extensions A(τ ) of S in (2.7) are related by Mτ (λ) =

M (λ) − τ , 1 + τ M (λ)

τ ∈ R ∪ {∞}.

(2.8)

If M (λ) belongs to the Kac class N1 then all Mτ (λ), τ ∈ R ∪ {∞}, belong to N1 , except for τ = −1/γ where γ is given by (2.6). The value τ = −1/γ gives rise to an exceptional selfadjoint extension of S, the generalized Friedrichs extension, cf. [5], [6].

138


Closely related to the Kac class N1 is the class S of Stieltjes functions. A Nevanlinna function M (λ) is said to belong to the class S if both M (λ) and λM (λ) are Nevanlinna functions. Equivalently, M (λ) belongs to S if and only if M (λ) belongs to N1 and is analytic on C \ [0, ∞), such that lim M (x) ≥ 0,

x→−∞

cf. [8]. Analogously, a Nevanlinna function M (λ) is said to belong to the class S− if both M (λ) and λM (−λ) are Nevanlinna functions. Now M (λ) belongs to S− if and only if M (λ) belongs to N1 and is analytic on C \ (−∞, 0], such that lim M (x) ≤ 0.

x→∞

2.3. Coupling of symmetric operators with defect numbers (1, 1) Consider two closed symmetric operators S+ and S− in the Hilbert spaces H+ and H− and assume that their defect numbers are (1, 1). Form the orthogonal sum H = H+ ⊕H− and define in H the closed symmetric operator S = S+ ⊕S− . Clearly, + ∗ ∗ S ∗ = S+ ⊕ S− and the defect numbers of S are (2, 2). Let Π+ = {H, Γ+ 0 , Γ1 } and − − ∗ ∗ Π− = {H, Γ0 , Γ1 } be boundary triplets for S+ and S− with γ-fields γ+ (λ) and γ− (λ). Then − − Γ 0 = Γ+ Γ 1 = Γ+ (2.9) 0 ⊕ Γ0 , 1 ⊕ Γ1 , forms a boundary triplet for the orthogonal sum S ∗ . Observe that the corresponding γ-field and Weyl function are of the form γ+ (λ) ⊕ γ− (λ),

M+ (λ) ⊕ M− (λ),

(2.10) A+ 0

where γ± (λ) and M± (λ) correspond to the selfadjoint extensions = ker Γ+ 0 − − and A0 = ker Γ0 of S+ and S− , respectively. It is also of interest to consider one-dimensional symmetric extensions S of S, cf. [1], [5]. + − − Proposition 2.1. Let Π+ = {H, Γ+ 0 , Γ1 } and Π− = {H, Γ0 , Γ1 } be boundary ∗ ∗ triplets for S+ and S− with γ-fields γ+ (λ), γ− (λ) and Weyl functions M+ (λ), M− (λ), respectively. Then the linear relation S defined by S = { f = f+ ⊕ f− ∈ S ∗ ⊕ S ∗ : Γ+ f+ = Γ− f− = Γ+ f+ + Γ− f− = 0 }, +

−

0

0

1

1

is closed and symmetric in H = H+ ⊕ H− and has defect numbers (1, 1). Its adjoint S ∗ is given by S ∗ = { f = f+ ⊕ f− ∈ S ∗ ⊕ S ∗ : Γ+ f+ = Γ− f− , }, +

−

0

0

and the defect spaces of S are of the form Nλ (S ∗ ) = ker (S ∗ − λ) = { γ+ (λ) ⊕ γ− (λ) h : h ∈ H }. A boundary triplet for S ∗ is given by + −

∗

∗ Π = { H, Γ+ 0 S , (Γ1 + Γ1 ) S },

and the corresponding Weyl function is given by M+ (λ) + M− (λ).


139

All selfadjoint extensions A(τ ) of S in H are in one-to-one correspondence with τ ∈ R ∪ {∞} via A(τ ) = { f = f+ ⊕ f− ∈ S ∗ ⊕ S ∗ : Γ+ f+ = Γ− f− , Γ+ f+ = −τ (Γ+ f+ + Γ− f− ) }. +

−

0

0

0

1

1

In particular, − + − ∗ ∗ A(∞) = { f = f+ ⊕ f− ∈ S+ ⊕ S− : Γ+ 0 f+ = Γ0 f− , Γ1 f+ + Γ1 f− = 0 }.

Now assume that the Weyl functions M+ (λ) and M− (λ) corresponding to the + − − selfadjoint extensions A+ 0 = ker Γ0 and A0 = ker Γ0 , respectively, each belong to N1 . Clearly, the sum M+ (λ)+M− (λ) then also belongs to the class N1 . Therefore, the symmetric extension S of S+ ⊕ S− has a generalized Friedrichs extension. In fact, if limy→∞ M+ (iy) = 0 and limy→∞ M− (iy) = 0, then the value τ = ∞ corresponds to the generalized Friedrichs extension.

3. Sturm-Liouville expressions and interface conditions 3.1. Sturm-Liouville operators Let p and q be real-valued functions on an interval (0, b], b > 0, such that 1/p and q are integrable on (0, b], and consider the Sturm-Liouville expression L+ := −DpD + q on (0, b]. The following lemma can be checked directly. Lemma 3.1. Let L+,max be the maximal differential operator in L2 (0, b) associated with L+ on (0, b], b > 0. Then: (i) the restriction S+ of L+,max defined by S+ = { u ∈ dom L+,max : u(0) = (pu )(0) = u(b) = 0 } is a closed, densely defined, and symmetric operator with defect numbers (1, 1); (ii) the adjoint of S+ is given by ∗ S+ = { u ∈ dom L+,max : u(b) = 0 }; ∗ (iii) a boundary triplet for S+ is defined by the boundary mappings Γ+ 0 u = −(pu )(0),

Γ+ 1 u = u(0).

Now let p and q be real-valued functions on an interval [−b, 0), b > 0, such that 1/p and q are integrable on [−b, 0), and consider the Sturm-Liouville expression L− = −DpD + q on the interval [−b, 0). The following analog of Lemma 3.1 is immediate. Lemma 3.2. Let L−,max be the maximal differential operator in L2 (−b, 0) associated with (3.1) on [−b, 0), b > 0. Then: (i) the restriction S− of L−,max defined by S− = { u ∈ dom L−,max : u(0) = (pu )(0) = u(−b) = 0 } is a closed, densely defined, and symmetric operator with defect numbers (1, 1);

140


(ii) the adjoint of S− is given by ∗ S− = { u ∈ dom L−,max : u(−b) = 0 }; ∗ (iii) a boundary triplet for S− is defined by the boundary mappings Γ− 0 u = −(pu )(0),

Γ− 1 u = −u(0),

The corresponding Weyl functions can be calculated by fixing a fundamental system for the equation −(pu ) + qu = λu,

λ ∈ C.

(3.1)

Let ϕ(·, λ) and ψ(·, λ) denote the fundamental solutions of (3.1) on (0, b] which satisfy the initial conditions ϕ(0, λ) = 1, (pϕ )(0, λ) = 0, (3.2) ψ(0, λ) = 0, (pψ )(0, λ) = −1. Introduce the function M+ (λ) by M+ (λ) = −

ψ(b, λ) , ϕ(b, λ)

(3.3)

and define a solution of L+ u = λu by χ+ (·, λ) = ψ(·, λ) + M+ (λ)ϕ(·, λ). Then ∗ χ+ (x, λ) satisfies χ+ (b, λ) = 0 and thus it spans ker (S+ − λ). Furthermore, M+ (λ) is the Weyl function corresponding to the boundary triplet in (iii) of Lemma 3.1, + since Γ+ 0 χ+ (·, λ) = 1, Γ1 χ+ (·, λ) = M+ (λ), cf. (2.3). Likewise it can be shown that χ+ (·, λ) is the corresponding γ-field, cf. (2.4). Observe that the functions ϕ(x, λ) and ψ(x, λ) in (3.2) can also be seen as a fundamental system for the equation (3.1) on the interval [−b, 0). Introduce the function M− (λ) by ψ(−b, λ) M− (λ) = , (3.4) ϕ(−b, λ) and define a solution of L− u = λu by χ− (·, λ) = ψ(·, λ) − M− (λ)ϕ(·, λ). Then ∗ χ− (x, λ) satisfies χ− (−b, λ) = 0 and thus it spans ker (S− − λ). Completely analogous to the previous case, M− (λ) is the Weyl function corresponding to the bound− ary triplet in (iii) of Lemma 3.2, since Γ− 0 χ− (·, λ) = 1, Γ1 χ− (·, λ) = M− (λ). Furthermore, χ− (·, λ) is the corresponding γ-field. 3.2. Coupling of Sturm-Liouville operators ∗ ∗ in L2 [0, b] and S− in L2 [−b, 0] are used to define a The two differential operators S+ 2 differential operator in L [−b, b] by means of an interface conditions at the origin. ∗ ∗ ∗ ∗ For this purpose define the orthogonal sum S ∗ = S+ ⊕ S− of S+ and S− in 2 L [−b, b], so that S = S+ ⊕ S− has defect numbers (2, 2). A boundary triplet for S ∗ is obtained from Lemmas 3.1 and 3.2 and the construction in (2.9), which means for the present situation −(pu )(0+) u(0+) , u ∈ dom S ∗ . Γ0 u = , Γ1 u = (3.5) −(pu )(0−) −u(0−)


141

Here the domain dom S ∗ is given by dom S ∗ = { u ∈ dom L+,max ⊕ L−,max : u(−b) = u(b) = 0 }, i.e., by the set of all u ∈ L2 [−b, b] which satisfy u, pu ∈ AC([−b, b] \ {0}), (pu ) ∈ L2 [−b, b], u(−b) = u(b) = 0.

(3.6)

The notation u(0±) = limt→±0 u(t) is used to indicate a reference to the underlying ∗ operator S± . Observe that the corresponding γ-field and Weyl function are of the form (2.10), where γ± (λ) and M± (λ) correspond to the selfadjoint extensions A+ 0 and A− 0 determined by the boundary condition (pu )(0+) = 0 and (pu )(0−) = 0, respectively. The following proposition is an immediate consequence of Proposition 2.1 and Lemmas 3.1 and 3.2. ∗ ∗ Proposition 3.3. Let S ∗ = S+ ⊕ S+ be the differential operator in L2 [−b, b], where ∗ ∗ S+ and S− are as in Lemma 3.1 and Lemma 3.2, respectively, and let {C2 , Γ0 , Γ1 } be a boundary triplet for S ∗ determined by the boundary mappings Γ0 , Γ1 in (3.5), so that the γ-field and the Weyl function are of the form (2.10). Then: (i) the linear relation S defined by

S = { u ∈ dom S ∗ : u(0+) = u(0−),

(pu )(0+) = (pu )(0−) = 0 }

(3.7)

is a closed symmetric extension of S in L2 [−b, b] with defect numbers (1, 1); (ii) the adjoint of S is given by S ∗ = { u ∈ dom S ∗ : (pu )(0+) = (pu )(0−) := (pu )(0) };

(3.8)

0, Γ

1 } for S ∗ is determined by (iii) a boundary triplet {C, Γ

0 u = −(pu )(0), Γ

1 u = u(0+) − u(0−); Γ

(iv) the corresponding Weyl function is equal to M+ (λ) + M− (λ). Moreover, the selfadjoint extensions A(τ ) of S in L2 [−b, b] are given for τ ∈ R by dom A(τ ) = { u ∈ dom S ∗ : (pu )(0+) = τ (u(0+) − u(0−)), (pu )(0+) = (pu )(0−) }, and for τ = ∞ by dom A(∞) = { u ∈ dom S ∗ : u(0+) = u(0−), (pu )(0+) = (pu )(0−) }.

4. Generalized Friedrichs extensions and interface conditions The Sturm-Liouville expressions in Section 3 satisfy conditions which guarantee the interpretation as differential operators in the spaces L2 [0, b] and L2 [−b, 0] with Weyl functions belonging to the general Nevanlinna class N. Under additional conditions it can be shown that the Weyl functions belong to the Kac class N1 .

142


Lemma 4.1. Assume that the coefficient p in the Sturm-Liouville expression L+ = −DpD + q satisfies p(x) > 0 for 0 < x ≤ b and b dx < ∞, (4.1) p(x) 0 and that the coefficient q(x) ≥ c, c ∈ R, is bounded from below and integrable on (0, b]. Then the Weyl function M+ (λ) in (3.3) belongs to the Kac class N1 , it is analytic on C \ [c, ∞), and lim M+ (λ) = 0.

(4.2)

λ→−∞

In particular, if q(x) ≥ 0 on (0, b], then M+ (λ) belongs to the class S of Stieltjes functions. Proof. It suffices to show that the spectral measure σ of M+ (λ) satisfies the relation supp σ ⊂ [c, ∞), and that (4.2) holds. It follows from the relation (3.1) that x x ϕ(x, λ) = 1 + ϕ (t, λ)dt, p(x)ϕ (x, λ) = (q(x) − λ)ϕ(t, z)dt, (4.3) 0

0

and these relations imply for 0 < x ≤ b and λ ≤ c that ϕ (x, λ) ≥ 0, and further that the functions ϕ(x, λ) and p(x)ϕ (x, λ) are nondecreasing with respect to x. In particular, ϕ(b, λ) has no zeros off [c, ∞), and according to (3.3) the relation supp σ ⊂ [c, ∞) is shown. The Wronskian identity and the initial conditions (3.2) imply that ψ(x, λ) 1 − = , ϕ(x, λ) p(x)ϕ(x, λ)2 and hence, with the relation (3.3), that b M+ (λ) = 0

1 dt . ϕ(t, λ)2 p(t)

(4.4)

As ϕ(x, λ) ≥ 1 for 0 < x ≤ b and λ ≤ c, it follows from the relations (4.3) that lim p(x)ϕ (x, λ) = ∞,

λ→−∞

lim ϕ(x, λ) = ∞.

λ→−∞

By dominated convergence, the relation (4.4) implies the relation (4.2). If q(x) ≥ 0 then supp σ ⊂ [0, ∞) and it follows from [8] that M+ (λ) ∈ S. The following result can be established in a completely analogous fashion. Lemma 4.2. Assume that the coefficient p in the Sturm-Liouville expression L− = −DpD + q on [−b, 0) satisfies p(x) < 0 for −b ≤ x < 0 and 0 dx , (4.5) −∞ < −b p(x)


143

and that the coefficient q(x) ≤ c, c ∈ R, is bounded from above and integrable on [−b, 0). Then the Weyl function M− (λ) in (3.3) belongs to the Kac class N1 , it is analytic on C \ (−∞, c], and lim M− (λ) = 0.

(4.6)

λ→∞

In particular, if q(x) ≤ 0 on [−b, 0), then M− (λ) belongs to the class S− . A combination of Lemmas 4.1 and 4.2 in conjunction with the remarks following Proposition 2.1 now leads to a description of the generalized Friedrichs extension in the situation of Proposition 3.3. Proposition 4.3. Assume that the coefficients p and q of the Sturm-Liouville expression −DpD + q on [−b, b] satisfy the assumptions of Lemma 4.1 on (0, b], and the assumptions of Lemma 4.2 on [−b, 0). Then the symmetric operator S in (3.7) has a generalized Friedrichs extension and it corresponds to τ = ∞. It is also possible to obtain a similar result for the case when the sign of the coefficient p is not fixed on (0, b] or on [−b, 0). Define for this purpose the following functions x 1 P0 (x) = dt, 0 |p(t)| x |t| P1 (x) = dt, 0 |p(t)| x 1 P01 (x) = dt. 0 P0 (t) The following result is inspired by [7] (and could be stated in a slightly more general fashion, similar to the formulations in [7]). Proposition 4.4. Assume that one of the following functions P01 (δ) P1 (δ)|p(δ)|

or

P0 (δ) δ

(4.7)

is locally integrable in a neighborhood of 0. Then the symmetric operator S in (3.7) has a generalized Friedrichs extension corresponding to τ = ∞. Proof. First consider the case of the interval (0, b]. It suffices to recall the inequality [7, Theorem 4.1 and (4.8)]: δ(y) 1 + y 0 P0 (t) dt |M+ (iy)| ≤ C , (4.8) yδ(y) where δ(y) is a monotonically decreasing function with δ(y) → 0,

yδ(y) → ∞,

y → ∞,

144


which satisfies further conditions, cf. [7, Lemma 3.2]. Hence, if for some y0 > 0, the following inequality δ(y) ∞ 1 + y 0 P0 (t) dt dy < ∞ (4.9) y 2 δ(y) y0 is satisfied, then the function M+ (λ) ∈ N1 and limy→∞ M+ (iy) = 0. Now if one of the functions in (4.7) is locally integrable near 0 then there exists a monotonically decreasing, absolutely continuous function δ(y) : [y0 , ∞) → (0, 1) with the required properties such that the inequality (4.9) holds, cf. [7]. Completely similar to the case of the interval (0, b], the case of the interval [−b, 0) can be treated.

References [1] V.A. Derkach, S. Hassi, M.M. Malamud, and H.S.V. de Snoo, Generalized resolvents of symmetric operators and admissibility, Methods of Functional Analysis and Topology, 6 (2000), 24–55. [2] V.A. Derkach, S. Hassi, M.M. Malamud, and H.S.V. de Snoo, Boundary relations and their Weyl families, Trans. Amer. Math. Soc., to appear. [3] A. Fleige, S. Hassi, and H.S.V. de Snoo, A Kre˘ın space approach to representation theorems and generalized Friedrichs extensions, Acta Sci. Math. (Szeged), 66 (2000), 633–650. [4] A. Fleige, S. Hassi, H.S.V. de Snoo, and H. Winkler, Sesquilinear forms corresponding to a non-semibounded Sturm-Liouville operator, in preparation. [5] S. Hassi, M. Kaltenb¨ ack, and H.S.V. de Snoo, Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass N1 of Nevanlinna functions, J. Operator Theory, 37 (1997), 155–181. [6] S. Hassi, H. Langer, and H.S.V. de Snoo, Selfadjoint extensions for a class of symmetric operators with defect numbers (1, 1), 15th OT Conference Proceedings, (1995), 115–145. [7] S. Hassi, M. M¨ oller, and H.S.V. de Snoo, Sturm-Liouville operators and their spectral functions, J. Math. Anal. Appl., 282 (2003), 584–602. [8] I.S. Kac and M.G. Kre˘ın, R-functions-analytic functions mapping the upper halfplane into itself, Supplement II to the Russian edition of F.V. Atkinson, Discrete and continuous boundary problems, Mir, Moscow, 1968 (Russian) (English translation: Amer. Math. Soc. Transl., (2) 103 (1974), 1–18. [9] I.S. Kac and M.G. Kre˘ın, On the spectral functions of the string, Supplement II to the Russian edition of F.V. Atkinson, Discrete and continuous boundary problems, Mir, Moscow, 1968 (Russian) (English translation: Amer. Math. Soc. Transl., (2) 103 (1974), 19–102). [10] H.-D. Niessen and A. Zettl, Singular Sturm-Liouville problems: The Friedrichs extension and comparison of eigenvalues, Proc. London Math. Soc., 64 (1992), 545–578.

Generalized Friedrichs Extensions Andreas Fleige Am S¨ udwestfriedhof 27 44137 Dortmund Deutschland e-mail: [email protected] Seppo Hassi Department of Mathematics and Statistics University of Vaasa P.O. Box 700 65101 Vaasa Finland e-mail: [email protected] Henk de Snoo Department of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen Nederland e-mail: [email protected] Henrik Winkler Department of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen Nederland e-mail: [email protected]

145


Spectral Properties of Operator Polynomials with Nonnegative Coefficients Karl-Heinz Förster and Béla Nagy Dedicated to Professor Dr. Heinz Langer

Abstract. We study properties of the polynomial Q(λ) = λm I − S(λ) where S(λ) = λl Al +· · ·+λA1 +A0 and 1 ≤ m < l. The coefficients Al , . . . , A0 are in the positive cone of an ordered Banach algebra or are positive operators on a complex Banach lattice E and I is the identity. We study the properties of the spectral radius of S(λ) if λ is a nonnegative real number, and its connection with the existence of spectral divisors with nonnegative coefficients in the considered sense. We prove factorization results for nonnegative elements in an ordered decomposing Banach algebra with closed normal algebra cone and in the Wiener algebra. Earlier results on monic (nonnegative) operator polynomials are applied to the operator polynomial class studied here. Mathematics Subject Classification (2000). MSC(2000): Primary 47A56; Secondary 46H99, 47B65 . Keywords. Operator polynomials, positive coefficients, factorization, ordered Banach algebras, spectral radius.

1. Introduction In the present paper we consider the operator polynomial Q(λ) = λm I − λl Al − · · · − λA1 − A0 , where Al , . . . , A0 are nonnegative (= positive) operators on a complex Banach lattice E, Al = 0 and I is the identity operator on E. Most of our results will be valid and proved for polynomials of the type above with coefficients in the cone of an ordered Banach algebra. This work was completed with partial support of the Hungarian National Science Grants OTKA Nos T-030042 and T- 047276 and partial support of the DAAD and the Technical University of Berlin.

148

K.-H. Förster and B. Nagy

Concerning the used concepts of Banach algebras we refer to the monographs T.W. Palmer[18] and I. Gohberg, S. Goldberg and M. A. Kaashoek [10], whereas concerning the spectral theory of operator polynomials we refer to the monographs A.S. Markus [16] and L. Rodman [19], concerning the theory of nonnegative operators on Banach lattices we refer to the monographs P. Meyer-Nieberg [17] and H.H. Schaefer [23]. We will adopt several results on operator polynomials for our more general case, if no essential differences in their proofs appear. Monic operator polynomials (i.e., m > l) with nonnegative operator coefficients Al , . . . , A0 have been considered in [4], [13], [15] and [19]. Here we investigate mainly the case 1 ≤ m < l. We change the notation and consider in the following polynomials q(λ) = λm e − λl al − · · · − λa1 − a0 ,

λ ∈ C,

(1.1)

where e is the unit element of a complex Banach algebra A and the coefficients aj belong to a cone C in A for j = 0, 1, . . . , l; we say also, they are nonnegative (with respect to this cone). We define the polynomial s(λ) = λl al + · · · + λa1 + a0 .

(1.2)

By (s(λ)) we denote the spectral radius of s(λ). One of our main results is the following (see Section 5) : Let 1 ≤ m < l and (s(r0 )) < r0m for some r0 > 0. Then q(·) has a monic (right) spectral divisor of degree m with nonnegative coefficients and a comonic (left) spectral divisor with nonnegative coefficients. In Section 4 we prove factorization results for nonnegative elements in an abstract ordered decomposing Banach algebra with closed normal algebra cone. As corollaries we obtain results on factorizations in the Wiener algebra. In Section 2 we recall the notation of an ordered Banach algebra with closed (normal) cone, [20], and collect some spectral properties of elements in such cones. In Section 3 we study the function s : [0, ∞[−→ R+ with s (r) = (s(r)).

(1.3)

An important property of this function is its log-log convexity (for this notion see the text below). In the last section we consider operator polynomials and apply known results on monic operator polynomials to our case here.

2. Ordered Banach algebras In this section we consider ordered Banach algebras in the sense of [20]. We assume the reader to be familiar with the definition and elementary properties of Banach algebras. Throughout this section A will denote a (real or complex) Banach algebra with unit e and zero element 0. As in [20, §3] we call a subset C ⊂ A an algebra cone if C satisfies the following conditions:

Spectral Properties of Operator Polynomials 1. C + C ⊂ C, 3. C · C ⊂ C,

149

2. λ C ⊂ C for all λ ≥ 0, 4. e ∈ C.

C is called a proper cone, if −C ∩C = {0}. Any cone in A induces an ordering “≤” on A in the following way: a≤b if and only if b−a∈C (for every a, b ∈ A). It is well known that this is a partial ordering on A, i.e., ≤ is reflexive and transitive; ≤ is antisymmetric if and only if C is proper. If C is an algebra cone in A, then the induced partial ordering ≤ satisfies for a, b ∈ A and a scalar λ : 1 . 0 ≤ a, 0 ≤ b =⇒ 0 ≤ a + b, 3 . 0 ≤ a, 0 ≤ b =⇒ 0 ≤ a · b,

2 . 0 ≤ a, 0 ≤ λ =⇒ 0 ≤ λa 4 . 0 ≤ e.

Conversely, if ≤ is a partial ordering on A such that 1 .–4 . hold, then C = {a ∈ A : 0 ≤ a} is an algebra cone which induces ≤. If A is ordered by an algebra cone, then we call A an ordered Banach algebra. An algebra cone C of A is said to be normal if there exists a constant γ > 0 such that 0 ≤ a ≤ b in A implies a ≤ γ b. It is easy to see that a normal algebra cone is proper. In applications the Banach algebra L(E) of all linear bounded operators in a (complex) Banach lattice E with cone L(E)+ is an important example for an ordered Banach algebras with a closed normal algebra cone. Especially the algebra Cn×n of all complex n × n matrices with the cone of all entrywise nonnegative matrices is such an algebra. In the next proposition we collect some properties of an ordered Banach algebra with a normal cone. Proposition 2.1. Let A be an ordered Banach algebra with a closed normal algebra cone C. Then 1. The spectral radius is a monotone function on C; i.e., if 0 ≤ a ≤ b, then (a) ≤ (b). 2. For all a ∈ C its spectral radius (a) belongs to its spectrum σ(a); i.e., (a) ∈ σ(a) for a ∈ C. 3. Let a ∈ C and let λ ∈ C. Then λ ∈ / σ(a) and (λe − a)−1 ∈ C if and only if λ is real and (a) < λ. Proof. The first two assertion are proved in [20, Theorem 4.1.1 and Proposition 5.1] We will prove the third assertion. It is clear that (a) < λ is sufficient for λ ∈ / σ(a); (λe − a)−1 ∈ C follows from the expansion of (λe − a)−1 at ∞ (C. Neumann’s series). Suppose that (λe − a)−1 ∈ C for some λ ∈ / σ(a). For n = 0, 1, 2 . . . set xn = (λe − a)−n , where x0 = (λe − a)0 = e. Clearly xn ∈ C, xn = 0 and λxn = axn + xn−1 for n = 1, 2, . . .. By induction on n it follows from the last equality that λn xn ∈ C, λn−1 xn ∈ C and λn xn ≥ λn−1 xn−1 ≥ x0 = e. Now we can proceed as in the proof of [23, Appendix 2.3, p. 264] to obtain that (a) < λ.

150


3. The spectral radius of s(·) as a function on the nonnegative axis In this section we investigate the function s : [0, r0 [−→ R+ with s (r) = (s(r)). where s(·) is holomorphic on the open disc Dr0 = {z ∈ C | |z| < r0 }, so s(λ) =

∞

λj aj

for |λ| < r0 ,

j=0

and that the coefficients aj are in the cone C of an ordered complex Banach algebra A. The next lemma may be known. For sake of completeness we include a proof; it is an appropriate modification of the proof of [2, Lemma 3] Lemma 3.1. Let C be a normal closed cone in a complex Banach algebra. Then there exists a constant β > 0 such that s(λ) ≤ βs(|λ|),

(s(λ)) ≤ (s(|λ|))

for all functions s(·) which are holomorphic in Dr0 with coefficients in C and for all λ ∈ Dr0 . Proof. C is normal; i.e., there exists a positive constant γ such that x, y ∈ C x ≤ y imply x ≤ γ y . Then u ∈ C, v ∈ A and

and

−u ≤ v ≤ u imply v ≤ 2γ u . Indeed, from u − v ≥ 0 and u + v ≥ 0 there follows 2γ u = γ u + v + (u − v) ≥ max{ u + v , u − v } ≥ 12 ( u + v + u − v ) ≥ v . Let λ = reiφ and θ ∈ [0, 2π]. Then s(λ) = eiθ s(λ) = ≤ 2 sup 0≤ω≤2π

∞

∞

rk (cos(θ + kφ) + i sin(θ + kφ))ak

k=0

r cos(ω + kφ)ak . k

k=0

Now −rk ak ≤ rk cos(ω + kφ)ak ≤ rk ak , −

∞

r k ak ≤

k=0

Therefore s(λ) ≤ 4γ

∞ k=0

∞ # k=0 k

for k = 0, 1, 2, . . .. Adding, we have

rk cos(ω + kφ)ak ≤

∞

rk ak ∈ C.

k=0

rk ak = s(|λ|). The inequality of the spectral

radii follows now from (s(λ)) ≤ β(s(|λ))k for k = 0, 1, 2, . . . (note that sk (·) is also holomorphic in Dr0 and has nonnegative coefficients) and the well-known formula for the spectral radius.

Spectral Properties of Operator Polynomials

151

By a result of E. Vesentini (see [1, p. 52]) the function λ −→ log((s(λ))) (and then also the function λ −→ (s(λ)) ) is subharmonic on Dr0 . Since the coefficients aj are in C we obtain from the lemma above s (r) = max (s(λ)) for |λ|=r

all r ∈ [0, r0 [. From the theory of subharmonic functions (see [14], Theorem 2.13) we obtain Proposition 3.2. Under the conditions above the function s is continuous and not decreasing, and log(s (·)) is convex in log(r) on ]0, r0 [. log(s (·)) is convex in log(r) on ]0, r0 [ means that the function ηs : ] − ∞, log(r0 )[→ R

with

t −→ log(s (et ))

for

r1 , r2 ∈]0, r0 [,

(3.1)

is convex, or equivalently s (r1 τ r2 1−τ ) ≤ s (r1 )τ S (r2 )1−τ

τ ∈ [0, 1].

We call a function which satisfies the last functional inequality log-log convex. Fundamental properties of log-log convex functions give Proposition 3.3. Assume that the function s(·) satisfies the conditions above. Let k ∈ N0 . 1. Let 0 < r1 < r2 < r3 < r0 be such that s (rj ) = rjk for j = 1, 2, 3. Then s (r) = rk for all r ∈ [r1 , r3 ]. 2. Let r1 , r2 > 0 be such that r1 = r2 and s (rj ) = rjk for j = 1, 2, let s be differentiable in r1 , and let s (r1 ) = kr1k−1 . Then s (r) = rk for all r in the closed interval with the endpoints r1 and r2 . Proof. The function ηs is convex. Conditions 1 imply that the graphs of ηs and of the line R → R with t −→ kt have three different points in common. Therefore they coincide on the corresponding interval, this implies the assertion. Conditions 2 imply that the graphs of ηs and of the line have two different points in common and in one of the points the line is the tangent of ηs in this point. Again, they coincide on the corresponding interval, and we obtain the assertion. Proposition 3.4. Assume that the function s(·) satisfies the conditions above. Let q(λ) = λm e−s(λ) for all λ ∈ Dr0 and some m ∈ N. Assume that there exist positive numbers r1 and r2 such that 0 < r1 < r2 < r0 and s (r) < rm for r ∈]r1 , r2 [. Then σ(q(·)) ∩ {λ ∈ C | r1 < |λ| < r2 } = ∅. Proof. For λ ∈ C with r1 < |λ| < r2 we obtain from Lemma 3.1 that (s(λ)) ≤ (s(|λ|)) = s (|λ|) < |λm |. Therefore q(λ) = λm (e − λ−m s(λ)) is invertible.

152


Example 3.5. For n ∈ N and kj ∈ n × n (weighted cyclic) matrix ⎛ 0 λk1 ⎜ ⎜ 0 0 ⎜ .. .. S(λ) = ⎜ ⎜ . . ⎜ ⎝ 0 0 λkn 0

N0 = N ∪ {0} for j = 1, 2, . . . n consider the 0

···

λk2 .. .

..

··· ···

. 0 0

0 .. . .. . λkn−1 0

⎞ ⎟ ⎟ ⎟ ⎟ , λ ∈ C. ⎟ ⎟ ⎠

We assume that l := max kj ≥ 1. Then S(·) is a matrix polynomial of 1≤j≤n

degree l with entrywise nonnegative matrices as coefficients. These matrices are nonnegative operators on the Banach lattice Cn and elements in the normal cone R+ n×n of the nonnegative n × n square matrices in the Banach algebra Cn×n of the complex n × n square matrices with the spectral norm. Let λ = 0. Then σ(S(λ)) = {z ∈ C : z n = λk } where k = k1 + · · · + kn . The eigenvalues of S(λ) have simple (geometric and algebraic) multiplicities, the corresponding eigenvectors of S(λ) to z ∈ σ(S(λ)) are multiples of (1, λ−k1 z, λ−k1 −k2 z 2 , . . . , λ−k1 −···−kn−1 z n−1 ). For r > 0 the matrix S(r) is nonnegative and irreducible. S(1) is row-stochastic; i.e., S(1)1 = 1, where 1 is the vector in Cn which has all comk ponents equal to 1. We have S (r) = r n . Therefore S is not convex if k < n. For k = n we obtain S (r) = r for r > 0, therefore S (1) = S (1) = 1 and S (1) = 0 in this case. For L(λ) = λm − S(λ) we have σ(L) = {z ∈ C : z nm = z k }, especially σ(L) = C if and only if mn = k.

4. Factorization in ordered decomposing Banach algebras In this section we introduce the concept of an ordered decomposing Banach algebra and prove a factorization result on elements in the algebra cone which will be applied in the next section to polynomials with nonnegative coefficients. A Banach algebra A is called a decomposing Banach algebra if A is the direct sum of two closed subalgebras A+ and A− ; see [3], [10], [11]. Let P+ denote the bounded linear projection of A onto A+ annihilating A− . Then P− := I − P+ is the bounded linear projection of A onto A− annihilating A+ . Let A be at the same time an ordered Banach algebra with algebra cone C and a decomposing Banach algebra (with the projections P+ and P− ). We call A an ordered decomposing Banach algebra, if C is invariant under P+ and under P− ; i.e., P± C ⊂ C. The following theorem is an order theoretic version of Lemma 5.1 in Chapter 1 of [8]; cf. also Theorem 23.3 in [16]. Theorem 4.1. Let A be an ordered decomposing Banach algebra with closed normal algebra cone C. If a ∈ C and (a) < 1,


153

then e − a admits a factorization e − a = b+ (e − b− ) with b+ ∈ A+ , e − b+ ∈ A+ ∩ C, b− ∈ A− ∩ C, the elements b+ and e − b− are −1 invertible, and b−1 − e ∈ A− ∩ C and (e − b− )−1 ∈ C. + ∈ A+ ∩ C, (e − b− ) Proof. Let a ∈ C and (a) < 1. We define the bounded linear operator T+ : A → A by T+ x = P+ (xa) for all x ∈ A. Then T+ C ⊂ A+ ∩ C and 0 ≤ T+ e = P+ a ≤ a. Therefore 0 ≤ T+ 2 e ≤ T+ a = P+ a2 ≤ a2 , and then 0 ≤ T+ n e ≤ an for n = 1, 2, . . .. The normality of the cone C implies T+ n e ≤ γan for n = 1, 2, . . . and some ∞ # constant γ. Therefore from (a) < 1 it follows that x ˆ = T+ n e converges, belongs to A+ ∩ C and is a solution of the equation

n=0

x − P+ (xa) = x − T+ x = e. Set b− = e − x ˆ+x â, then b− = P− (ˆ xa) ∈ A− ∩ C. Next we consider the operator T− : A → A with T− y = P− (ay) for all y ∈ A. As above, T− C ⊂ A− ∩ C and ∞ # T− n e ≤ γan for n = 1, 2, . . . . Therefore yˆ = T− n e converges, belongs to n=0

C, and is a solution of the equation y − P− (ay) = y − T− y = e. Set b+ = e − P+ (aˆ y). Then b+ ∈ A+ (note that we assume e ∈ A+ ), e − b+ = P+ (aˆ y ) ∈ A+ ∩ C and b+ = (e − a)(e + P− (aˆ y )). The last equation implies x ˆ b+ = (e − b− )(e + P− (aˆ y )), and this is equivalent to xˆb+ − e = −b− + P− (aˆ y) − b− P− (aˆ y ). In the last equation the left-hand side belongs to A+ , while the right-hand side belongs to A− . Therefore both sides are equal to zero, consequently x ˆb+ = e = (e − b− )(e + P− (aˆ y )). Thus b+ is left invertible, and e − b− is right invertible. We employ now a standard argument to prove that these elements are invertible. We replace in the argument above a by τ a, where τ ∈ [0, 1]. Then τ a ∈ C and (τ a) < 1, and the formulae for x ˆ, yˆ, b+ and b− show that we have to replace them by ∞ ∞ # # x ˆ(τ ) = (τ T+ )m e, yˆ(τ ) = (τ T− )m e, n=0

b+ (τ ) = e − τ P+ (aˆ y (τ )),

n=0

b− (τ ) = τ P− (ˆ x(τ )a),

respectively. These functions are continuous on [0, 1], b+(τ ) is left invertible and e − b− (τ ) is right invertible for all τ ∈ [0, 1] , and b+ (0) = e − b− (0) = e is invertible. By Lemma 23.2 in [16], b+ = b+ (1) and b− = b− (1) are invertible and ˆ ∈ A+ ∩ C, (e − b− )−1 = e + P− (aˆ y ) ∈ C and (e − b− )−1 − e = P− (aˆ y) ∈ b−1 + = x A− ∩ C. We obtained above the equation b+ = (e − a)(e + P− (aˆ y)). Multiplying on the right by e − b− , we obtain b+ (e − b− ) = e − a.

154


We will apply Theorem 4.1 to the Wiener algebra W (A; T), see [11], where A is an ordered Banach algebra and T = T1 denotes the unit circle, i.e., ∞

W (A; T) = {a(·) : a(λ) =

λk ak , λ ∈ T, (an ) ∈ lZ1 (A)}

k=−∞

where lZ1 (A) = {(ak )k∈Z ⊂ A|

∞ #

ak is convergent}. Clearly, ak is the kth

k=−∞

Fourier coefficient of a(·) ∈ W (A; T). Let A# be a Banach algebra. Then W (A; T) is a Banach algebra with norm ∞ a(·)W = k=−∞ ak . W (A; T) is the direct sum of the closed subalgebras W+ (A; T) = W−,0 (A; T) =

{ a(·) { a(·)

: :

ak = 0 for k < 0}, ak = 0 for k ≥ 0}.

Obviously, W+ (A; T) is the set of those functions in W (A; T) that admit continuous extensions onto {λ ∈ C : |λ| ≤ 1} and are analytic on D1 , while W−,0 (A; T) is the set of those functions in W (A; T) that admit continuous extensions onto {λ ∈ C | |λ| ≥ 1}, are analytic on {λ ∈ C; |λ| > 1} and vanish at infinity. Clearly, the operator P+ defined on W (A; T) by (P+ a(·))(λ) =

∞

λk ak ,

λ ∈ T,

k=0

is the projection of W (A; T) onto W+ (A; T) annihilating W−,0 (A; T). eW (·) with eW (λ) ≡ e for all λ ∈ T is the identity in W (A; T). For a(·) ∈ W (A; T) we denote by W (a(·)) the spectral radius of a(·) in the Banach algebra W (A; T). Let A be an ordered Banach algebra with algebra cone C. Then W (A; T) is an ordered Banach algebra with algebra cone CW = {a(·) ∈ W (A; T) : ak ∈ C for all k ∈ Z}. If C is normal (or closed), then CW is normal (or closed, respectively). The following corollary is the reformulation of Theorem 4.1 for the Wiener algebra W (A; T); see also Theorem 23.4 in [16]. Corollary 4.2. Let A be an ordered Banach algebra with closed normal algebra cone C. Suppose a(·) ∈ W (A; T) such that a(·) ∈ CW

and W (a(·)) < 1

Then eW (·) − a(·) admits a canonical factorization with respect to the unit circle; i.e., there exist b+ (·) ∈ W+ (A; T) and b− (·) ∈ W−,0 (A; T) such that e − a(λ) = b+ (λ)(e − b− (λ))

for all λ ∈ T,

and b+ (·) ∈ W+ (A; T), b− (·) ∈ W− (A; T) ∩ CW , eW (·) − b+ (·) ∈ W+ (A; T) ∩ CW , b+ (·)−1 ∈ W+ (A; T) ∩ CW , (eW (·) − b− (·))−1 − eW (·) ∈ W−,0 (A; T) ∩ CW and (eW (·) − b− (·))−1 ∈ CW .


155

For functions in W (A; T) with only finitely many non-zero Fourier coefficients we obtain stronger results. Note that W (A; T) is a subalgebra of C(A; T) := {f | f : T → A is continuous} and a(·) ∞ := max a(λ) A ≤ a(·) W for all a(·) ∈ |λ|=1

W (A; T). In the following we sometimes write for a ∈ A for clarity aA and A (a) instead of a and (a), respectively. Lemma 4.3. Let A be a Banach algebra, and let a(·) ∈ W (A; T) be such that ak = 0 for all k ∈ Z with |k| ≥ m for some m ∈ N. Then a(·) W ≤ (2m + 1) a(·) ∞

and

W (a(·)) = ∞ (a(·)).

Proof. The inequality follows from Cauchy’s formula for the coefficients ak of a(·), see [16, p.127]. Applying this inequality to an (·), we obtain an (·) W ≤ (2nm + 1) an (·) ∞ for n = 1, 2 . . .. Using the known formula for the spectral radius, we have that W (a(·)) ≤ ∞ (a(·)). The reverse inequality follows from an (·) ∞ ≤ an (·) W for n = 1, 2 . . .. The next lemma is very similar to Lemma 3.1, and the proof of Lemma 3.1 needs only minor technical changes. Lemma 4.4. Let A be an ordered Banach algebra with closed normal algebra cone C. Then there exists a β > 0 such that a(·) ∞ ≤ β a(1) A

and

∞ (a(·)) = A (a(1))

for all a(·) ∈ CW . Theorem 4.5. Let A be an ordered Banach algebra with closed normal algebra cone C. Suppose that a(·) ∈ W (A; T) and m ∈ N is such that ak = 0 for |k| > m,

a(·) ∈ CW

and

A (a(1)) < 1.

Then eW (·) − a(·) admits a canonical factorization with respect to the unit circle and all assertions of Corollary 4.2 hold. Proof. Lemmata 4.3 and 4.4 show that W (a(·)) = ∞ (a(·)) = A (a(1)) < 1. Therefore, the assumptions of Corollary 3.4 are satisfied. The results of this section can be used to study matrix functions having values which are (in a certain sense) entrywise nonnegative matrices; for the factorization theory of general matrix functions relative to a curve we refer to [11]. In the next section we will apply the last theorem to polynomials with coefficients in the cone of an ordered Banach algebra.

5. Factorization of polynomials with nonnegative coefficients In this section we apply the results of the preceding section to obtain a factorization of polynomials with coefficients in a normal cone of an ordered Banach algebra; the next theorem is the main result of this paper.

156


Theorem 5.1. Let A be an ordered Banach algebra with closed normal algebra cone l # C. Let q(·) be a polynomial q(λ) = λm e − λj aj with 1 ≤ m < l and aj ∈ C for j=0

j = 0, 1 . . . , l, and al = 0. Suppose there exists a r0 > 0 such that (s(r0 )) < r0 m , where s(λ) =

l #

λj aj .

j=0

Then 1. The A-valued function e − a(·) with a(λ) =

l #

λj−m aj = λ−m s(λ) for λ = 0

j=0

admits a (right ) canonical factorization with respect to the circle Tr0 = {λ ∈ C | |λ| = r0 }; 2. q(·) has a monic spectral (right ) divisor c(·) of degree m with nonnegative coefficients, i.e., c(λ) = λm e −

m−1

cj λj ,

cj ∈ C

(5.1)

j=0

for j = 0, 1, 2 . . . , m − 1, and σ(c(·)) = σ(q(·)) ∩ {λ ∈ C | |λ| < r0 }. 3. Let b(·) be the (uniquely defined ) polynomial of degree l − m such that q(λ) = b(λ)c(λ) for all λ ∈ C. Then b(λ) = b0 (e −

l−m

λj bj ), b0 is invertible, b−1 0 ∈ C and bj ∈ C

(5.2)

j=1

for j = 1, 2, . . . , l − m, σ(b(·)) = σ(q(·)) ∩ {λ ∈ C : |λ| > r0 }. −1

4. c(t)

∈ C for r0 ≤ t; b(t)−1 ∈ C for 0 ≤ t ≤ r0 .

Proof. 1. The function ar0 (·) with ar0 (λ) = a(r0 λ) for λ ∈ T1 satisfies the assumptions of Theorem 4.5. Therefore we have e − ar0 (λ) = b+ (λ)(e − b− (λ)) for λ ∈ T1 with some b+ (·) ∈ W+ (A; T) and some b− (·) ∈ W−,0 (A; T). 2. Then q(r0 λ) = r0m b+ (λ)(λm e − λm b− (λ)) for λ ∈ T1 . Now b+ (·) and b−1 + (·) have analytic extensions onto D1 , and b− (·) has an analytic extension onto {z ∈ C : |z| > 1} which vanishes at infinity. Then it follows (cf. 22.11 in [16]) that c(λ) = λm e − λm b− (r0−1 λ) is a monic polynomial of degree m and c(·) is a spectral divisor of q(·) with σ(c(·)) = σ(q(·)) ∩ {z ∈ C : |z| < r0 }. Now b− (·) ∈ CW , therefore λm b− (r0−1 λ) = λm−1 cm−1 + · · · + c0 with cj ∈ C. 3. That b(·) is uniquely defined follows from Lemma 22.8 in [16]; this Lemma is proved for an algebra of operators but its proof works in our more general situation. From the proof of part 2 it follows that b(λ) = b+ (r0−1 λ) for λ ∈ C . b+ (·)−1 ∈

Spectral Properties of Operator Polynomials CW ∩ W+ (A; T) implies σ(b(·)) ⊂ C\Dr0 and b(λ)−1 =

∞ #

157

λk fk for |λ| ≤ r0 with

k=0

fk ∈ C for k = 0, 1, 2 . . . . Therefore b(t)−1 ∈ C for 0 ≤ t ≤ r0 ; especially, b0 is ∈ C. From Corollary 4.2 we know that eW (·) − b+ (·) ∈ CW . invertible and b−1 0 Then e − b0 ∈ C and bj ∈ C for j = 1, . . . , l − m. 4. From t > (c(·)) we obtain ⎛ ⎞−1 m−1 m−1 ∞ c(t)−1 = ⎝tm e − tj cj ⎠ = t−m cˆk ∈ C where cˆ = tj−m cj ∈ C. j=0

j=0

k=0

The proof for b(t)−1 ∈ C is similar.

Corollary 5.2. Let A be an ordered Banach algebra with closed normal cone C and #l let q(·) be the polynomial q(λ) = λm e − j=0 λi aj = λm e − s(λ) where 1 ≤ m < l and aj ∈ C, al = 0. Then there exist polynomials b(·) and c(·) as in (5.1) and (5.2) with q(λ) = b(λ)c(λ) for all λ ∈ C and an r0 > 0 such that part 4 of Theorem 5.1 holds if and only if (s(r0 )) < r0m . Any factorization with these properties is unique. Proof. We have to prove the “only if” part. Let r0 m e − s(r0 ) = q(r0 ) = b(r0 )c(r0 ) be such that b(r0 )−1 and c(r0 )−1 exist and belong to C. Then r0 m e − s(r0 ) has an inverse in C. It follows from Proposition 2.1.3 that (s(r0 )) < r0 m . The last assertion follows from [16, Lemma 22.8]. Theorem 5.3. Let A be an ordered complex Banach algebra with closed normal cone C and let the polynomials q(·) and s(·) satisfy the assumptions of the last theorem; hence S (r0 ) < r0m for some r0 > 0. Then 1. Either s (t) < tm for all t ∈]0, r0 ], or there exists exactly one r1 ∈]0, r0 [ such that S (r1 ) = r1 m . In the second case we have s (t) s (t)

< tm > tm

for for

r1 < t ≤ r0 , 0 < t < r1 ,

(5.3) (5.4)

# j r1 = (c(·)) ∈ σ(c(·)) and r1 m = (c+ (r1 )), here c+ (λ) = m−1 j=0 cj λ , see (5.1). 2. Either s (t) < tm for all t ∈ [r0 , ∞[, or there exists exactly one r2 ∈]r0 , ∞[ such that s (r2 ) = r2 m . In the second case we have s (t) s (t)

< tm > tm

for for

r0 ≤ t < r2 , r2 < t,

(5.5) (5.6)

r2 ∈ σ(b(·)), σ(b(·)) ⊂ {λ ∈ C : r2 ≤ |λ|} and (b+ (r2 )) = 1, here b+ (λ)) = l−m # j λ bj , see (5.2). j=1

158


Proof. 1. Assume that s (t) ≥ tm for some t ∈]0, r0 [. The function s (·) is continuous, by the intermediate-value theorem there exists an r1 ∈ [t, r0 [ such that s (r1 ) = r1m . For any such r1 we obtain from the convexity of ηs (·) (see Proposition ηs (u0 ) − mu1 (u − u1 ) + mu1 for u1 ≤ u ≤ u0 . Here uj = log rj 3.2) that ηs (u) ≤ u0 − u1 for j = 0, 1. Clearly ηs (u0 ) < mu0 implies ηs (u) < mu for u1 < u ≤ u0 . This is equivalent to s (t) < tm for t ∈]r1 , r0 ]. Now it is clear there is exactly one r ∈]0, r0 ] such that s (r) = rm . The inequalities (5.3) and (5.4) are now clear. For this r1 we have r1 m ∈ σ(s(r1 )) by Proposition 2.1.1. Therefore r1 ∈ σ(q(·)) ∩ Dr0 = σ(c(·)) by Theorem 5.1.2, thus r1 ≤ (c(·)). Set r = (c(·)). Then r ∈ [r1 , r0 ]. Further r ∈ boundary σ(c(·)) ⊂ σ(q(·)), see [16, Lemma 22.3, p.112], therefore rm ∈ σ(s(r)), and then rm ≤ (s(r)). From (5.3) we obtain r1 = r = (c(·)). For monic polynomials of degree m with coefficients in C we have (c+ (·)) = (c(·))m = rm , see [19, Proposition 2.1]. 2. Assume that s (t) ≥ tm for some t ∈]r0 , ∞[. A similar argument as in the proof of part 1 shows that there exists exactly one r2 ∈]r0 , ∞[ with s (r2 ) = r2 m and s (t) < tm for t ∈ [r0 , r[. The inequalities (5.5) and (5.6) are now clear. For a proof of the other assertions we define ˜b(·) by ˜b(λ) = λl−m b−1 0 b(1/λ) for λ = 0 and ˜b(0) = bl−m , here we use (5.2). Then ˜b(·) is the monic polynomial with nonnegative coefficients ˜b(λ) = λl−m e − (λl−m−1 b1 + · · · + bl−m ) = λl−m e − ˜b+ (λ). Note that σ(˜b(·))\{0} = { λ1 : λ ∈ σ(b(·))} = { λ1 : λ ∈ σ(q(·)), r0 ≤ |λ|} by Theorem 5.1.3. We will show that r12 = (˜b(·)). Now q(r2 ) = r2 m−l b0˜b( r12 )c(r2 ) is singular but b0 and c(r2 ) are invertible, thus ˜b( r12 ) is singular. Then r12 ∈ σ(˜b(·)), and r12 ≤ (˜b(·)). Set r˜ = (˜b(·)). Then r˜ ∈ boundary σ(˜b(·)), and b( r1˜ ) is singular. Therefore q( 1r˜ ) is not invertible, see [19, Proposition 2.1], and r0 < r1˜ ≤ r2 . Thus 1 1 1 1 1 r˜m ∈ σ(s( r˜ )), and then r˜m ≤ (s( r˜ )). From (5.5) we obtain r˜ = r2 . Now r2 ∈ { λ1 : λ ∈ σ(˜b(·)),λ = 0} = σ(b(·)) ⊂ {λ ∈ C : r2 ≤ |λ|}. By [19, Proposition 2.1] we have r2m−l = (˜b( r12 ), and then (b+ (r2 )) = (r2 l−m˜b+ ( r12 ) = 1.

6. Operator polynomials with nonnegative coefficients In this section we consider the operator polynomial Q(λ) = λm I − λl Al − · · · − λA1 − A0 = λm I − S(λ)

(6.1)

where A0 , . . . , Al are nonnegative (= positive) operators on a complex Banach lattice E with Al = 0. We assume that 1 ≤ m < l and (S(r0 )) < r0m for some positive number r0 . The results of the preceding section (Theorem 5.1) show that Q(·) can be factorized in a special way, namely Q(λ) = B0 B(λ)C(λ)

for all λ ∈ C.

(6.2)


159

Here B0 is invertible with B0−1 ∈ L(E)+ , B(·) is a comonic operator polynomial of degree l − m with nonnegative coefficients, i.e., B(λ) = I − λB1 − · · · − λl−m Bl−m = I − B+ (λ),

(6.3)

B0−1 Al

where Bj ∈ L(E)+ for j = 1, . . . , l − m and Bl−m = = 0, and C(·) is a monic operator polynomial of degree m with nonnegative coefficients, i.e., C(λ) = λm I − λm−1 Cm−1 − . . . − λ1 C1 − C0 = λm I − C+ (λ)

(6.4)

where Cj ∈ L(E)+ for j = 0, 1, . . . , m − 1. We will use results on monic (and comonic) operator polynomials with nonnegative coefficients from [4], [5] and [19] to describe spectral properties of Q(·) on the circles with radius r such that S (r) = rm . Theorem 6.1. Let E, Q(·), S(·), B(·), C(·), B+ (·), C+ (·), B0 and r0 be as above, especially (S(r0 )) < r0 m . Then for all r ∈ [0, ∞[ with S (r) = rm the following statements hold: 1. rm ∈ σ(S(r)) and r ∈ σ(Q(·)). 2. Let 0 < r < r0 . Then r is a pole of order k of Q−1 (·) and the space of all Jordan chains of Q(·) corresponding to r has a finite dimension h if and only if r is a pole of R(·, r−m+1 C+ (r)) with residue of finite rank h. 3. Let r0 < r. Then r is a pole of order k of Q−1 (·) and the space of all Jordan chains of Q(·) corresponding to r has a finite dimension h if and only if 1 is a pole of R(·, B+ (r)) with residue of finite rank h. 4. If r is a pole of Q−1 (·) and the space of all Jordan chains of Q(·) corresponding to r is finite-dimensional, then {λ ∈ σ(Q(·)) | | λ | = r} = r · G. here G is the union of finitely many (finite) groups of roots of unity and consists entirely of poles of Q−1 (·) 5. Let r and λ0 be poles of Q−1 (·) with | λ0 | = r. Then the following hold: (a) The order of the pole λ0 of Q−1 (·) is not greater than the order of the pole r of Q−1 (·). (b) Every nontrivial Jordan chain of Q(·) corresponding to λ0 is a linearly independent set. Proof. Let 0 < r < r0 be such that S (r) = rm . Then B(λ0 ) is invertible for all λ0 ∈ Tr by Theorem 5.1.3. Therefore Q−1 (·) has a pole of order k at λ0 if and only C −1 (·) has a pole of order k at λ0 . By [16, Lemma 22.5, p.123] the Jordan chains of Q(·) and C(·) corresponding to λ0 coincide. Now the assertions of the theorem concerning the case 0 < r < r0 follow from the corresponding results of [19]; namely, assertion 2 follows from [19, Corollary 5.5], assertion 4 follows from [19, Corollary 4.7] and assertion 5(b) follows from [19, Corollary 5.4]. Assertion 5(a) is a consequence of the Pringsheim Theorem, see [22, p.262]. Let r0 < r be such that S (r) = rm . Then C(λ0 ) is invertible for all λ0 ∈ Tr by Theorem 5.1.3. Therefore Q−1 (·) has a pole of order k at λ0 if and only if B −1 (·)

160


has a pole of order k at λ0 . Direct computation shows: xj for j = 0, 1, . . . , p form a Jordan chain of Q(·) corresponding to λ0 if and only if the vectors j j yj = C (j−i) (λ0 )xi for j = 0, 1, . . . , p i i=0 form a Jordan chain of B(·) corresponding to λ0 . Therefore Q−1 (·) has a pole of order k at λ0 if and only B −1 (·) has a pole of order k at λ0 ; recall that the order of a pole is the length of the longest nontrivial Jordan chain corresponding to it and that a Jordan chain is called nontrivial if its first vector is non-zero. Further, it follows that the space of Jordan chains of Q(·) corresponding to λ0 has a finite dimension h if and only if the space of Jordan chains of B(·) corresponding to λ0 ˜ has a finite dimension h. We define now the monic operator polynomial B(·) by l−m ˜ B(1/λ) for λ = 0. Note that (see the proof of Theorem 5.3.2) B(λ) = λ 1 1 ˜ : λ ∈ σ(B(·))} and r˜ = = (B(·)). λ r ˜ is the comonic companion operator Then the companion operator CB˜ of B(·) of B(·), see [12, p. 187]; note that the results formulated there for the matrix case hold also for bounded linear operators in Banach spaces. Note also that in [16, §12] slightly different companion operators are defined, but the two variants are cogredient, i.e., they coincide after a permutation of some operator rows and corresponding operator columns. Therefore the results of [16, §12 and §13] hold for the companion operators defined above and the following sequence of equivalences holds. Q−1 (·) has a pole of order k at λ0 (and the space of Jordan chains of Q(·) corresponding to λ0 has finite dimension h) if and only if B −1 (·) has a pole of order k at λ0 (and the space of Jordan chains of B(·) corresponding to λ0 has a finite dimension h, respectively) if and only if (I −·CB˜ )−1 has a pole of order k at λ0 (and the space of Jordan chains of (I −·CB˜ ) corresponding to λ0 has finite a dimension h, respectively), see [16, Lemma 12.5 and 12.6 and the remarks on p. 63] if and only if the resolvent R(·, CB˜ ) = (·I − CB˜ )−1 has a pole at λ˜0 = λ10 of order k (and the residuum of this resolvent at λ0 has finite rank h, respectively), see [16, Lemma 12.8] if and only if ˜ −1 (·) has a pole of order k at λ˜0 and the space of Jordan chains of B(·) ˜ correB ˜ sponding to λ0 has finite a dimension h, respectively), see [15, 2.1, Hilfssatz 1]. Now assertion 4 follows from [19, Corollary 4.7]. Assertion 5(a) is a consequence of the Pringsheim Theorem, see [22, p. 262]. From the connection between the Jordan chains of Q(·) and B(·) given above it follows directly that the nontrivial Jordan chains of Q(·) corresponding to λ0 = 0 are linearly independent if and only if the nontrivial Jordan chains of B(·) corresponding to λ0 = 0 are linearly independent if and only if the nontrivial ˜ corresponding to λ˜0 = 1 are linearly independent, see [16, Jordan chains of B(·) λ0 ˜ σ(B(·))\{0} ={


161

Lemma 12.3 and 12.7 and remarks on p.63]. Assertion 5(b) follows now from [19, Corollary 5.4]. If λ0 = r, we can continue the equivalences above by [19, Corollary 5.5], and obtain: Q−1 (·) has a pole of order k at r and the space of Jordan chains of Q(·) corresponding to r has a finite dimension h if and only if ˜+ (˜ r˜ is a pole of the resolvent R(·, r˜−l+m+1 )B r ) and the residuum of this resolvent at r˜ has finite rank h, and this assertion is equivalent to 1 is a pole of the resolvent R(·, B+ (r)) and the residuum of this resolvent at 1 has ˜+ (˜ r ) = r˜B+ (r)). finite rank h (since r˜−l+m+1 B Acknowledgement The authors wish to thank two referees for their careful work and constructive remarks.

References [1] B. Aupetit, A Primer on Spectral Theory. Springer-Verlag, New York – Berlin – Heidelberg, 1991. [2] F.F. Bonsall, Endomorphismus of a Partially Ordered Vector Space without Order Unit. J. London Math. Soc. 30 (1955), 144–153. [3] K.F. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators. Birkh¨ auser, Basel and Boston, 1981. [4] K.-H. F¨ orster and B. Nagy, Some Properties of the Spectral Radius of a Monic Operator Polynomial with Nonnegative Compact Coefficients. Integral Equations Operator Theory, 14 (1991), 794–805. [5] K.H. F¨ orster and B. Nagy, On the Linear Independence of Jordan Chains. Operator Theory: Advances and Applications, 122 (2001), 229–245. [6] H.R. Gail, S.L. Hantler and B.A. Taylor. Spectral Analysis of M/G/1 and G/M/1 Type Markov Chains. Adv. Appl. Prob. 28 (1996), 114–165. [7] H.R. Gail, S.L. Hantler and B.A. Taylor. Matrix-Geometric Invariant Measures for G/M/1 Type Markov Chains. Comm. Statist. Stochastic Models 14 (3) (1998), 537– 569. [8] I. Gohberg and I. Feldman. Convolution Equations and Projection Methods for their Solution. Amer. Math. Soc., Providence, R. I., 1974. [9] I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators. I.OT49. Birkh¨ auser Verlag, 1990. [10] I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Vol. II. Birkh¨ auser Verlag, Basel and Boston, 1993. [11] I. Gohberg, M.A. Kaashoek and I.M. Spitkovsky, An Overview of Matrix Factorization Theory and Operator Applications, in: Operator Theory: Advances and Applications, vol. 141, pp. 1–102, Birkh¨ auser Verlag, Basel and Boston, 2003. [12] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials. Academic Press, New York, 1982.

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[13] K.P. Hadeler, Eigenwerte von Operatorpolynomen. Archive Rat. Mech. Appl. 20 (1965), 72–80. [14] W.K. Hayman, P.B. Kennedy, Subharmonic Functions. Academic Press, London – New York – San Francisco, 1976. ¨ [15] G. Maibaum, Uber Scharen positiver Operatoren. Math. Ann. 184 (1970), 238–256. [16] A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils. Transl. of Math. Monographs, vol. 71. Amer. Math. Soc., Providence, 1988. [17] P. Meyer-Nieberg, Banach Lattices. Springer-Verlag, New York, 1991. [18] T.W. Palmer, Banach Algebras and The General Theory of *-Algebras. Cambridge University Press, Cambridge, 1994. [19] R.T. Rau, On the Peripheral Spectrum of a Monic Operator Polynomial with Positive Coefficients. Integral Equations Operator Theory 15 (1992), 479–495. [20] H. Raubenheimer and S. Rode, Cones in Banach Algebras. Indag. Math. (N.S.) 7 (4) (1996), 489–502. [21] L. Rodman, An Introduction to Operator Polynomials. Operator Theory: Advances and Applications, vol. 38, Birkh¨ auser: Basel-Boston-Berlin, 1989. [22] H.H. Schaefer, Banach Lattices and Positive Operators. Springer Verlag, New York, 1980. [23] H.H. Schaefer, Topological Vector Spaces. Springer Verlag, New York, 1971. Karl-Heinz F¨ orster Technische Universit¨ at Berlin Fakult¨ at II Institut f¨ ur Mathematik, MA 6-4 Straße des 17. Juni 136 D-10623 Berlin Germany e-mail: [email protected] Béla Nagy Department of Analysis Institute of Mathematics Technical University of Budapest H-1521 Budapest Hungary e-mail: [email protected]


Szeg˝ o Pairs of Orthogonal Rational Matrixvalued Functions on the Unit Circle Bernd Fritzsche, Bernd Kirstein and Andreas Lasarow Dedicated to Professor Heinz Langer

Abstract. We study distinguished pairs of orthonormal systems of rational matrix-valued functions on the unit circle, namely the so-called Szeg˝o pairs. These pairs are determined by an initial condition and a sequence of strictly contractive q × q matrices, which is called the sequence of Szeg˝o parameters. The Szeg˝ o parameters contain essential information on the underlying q × q nonnegative Hermitian Borel measure on the unit circle. Mathematics Subject Classification (2000). Primary 42C05, 47A56. Keywords. Orthogonal rational matrix-valued functions, Recurrence relations of Szeg˝ o-type, Szeg˝ o parameters, Favard-type theorem.

0. Introduction This paper continues the line of investigations started in [FKL1]–[FKL3] where we realized first steps towards a Szeg˝o theory of orthogonal rational matrix-valued functions on the unit circle T. The main feature of our conception of Szeg˝ o theory is the distinguished role of the Christoffel-Darboux formulas (see [FKL2], [FKL3]). As a cornerstone of our approach we introduced in [FKL2, Section 6] the notions of left and right Christoffel-Darboux pairs for rational matrix-valued functions. The recursion formulas for left and right Christoffel-Darboux pairs which were obtained in Section 2 of [FKL3] realize a crucial step in constructing our Szeg˝ o theory. Namely, these recursion formulas indicated that there is a one-to-one correspondence between Christoffel-Darboux pairs and sequences of matrices which are connected to particular signature matrices. The work of the third author of the present paper was supported by the German Academy of Natural Scientists Leopoldina by means of the Federal Ministry of Education and Research on badge BMBF-LPD 9901/8-88.

164

B. Fritzsche, B. Kirstein and A. Lasarow

The main topic of this paper is to discuss those pairs of orthogonal systems of rational matrix-valued functions which are direct generalizations of the Szeg˝ o pairs of orthonormal systems of matrix polynomials which were discussed by Delsarte, Genin, and Kamp in [DGK] (cf. [DFK, Section 3.6]), who developed the theory along the classical case of orthogonal polynomials of Szeg˝ o [S] (see also [GS], [G1], and [G2]). The present paper is also guided by the work of Bultheel, González-Vera, Hendriksen, and Nj˚ astad on scalar orthogonal rational functions (see, e.g., [BGHN1], [BGHN2], and the probably first work on this topic by Djrbashian [Dj]). Note that, in the case of matrix polynomials and in the scalar case of rational functions, Szeg˝ o recursions for orthogonal functions were derived in [DGK, Section V] and in [BGHN2, Chapter 4], respectively, using a different way as demonstrated below. Szeg˝o pairs of rational matrix-valued functions on the unit circle T are special pairs of orthogonal systems which are determined by an initial condition and a sequence of strictly contractive q × q matrices which are called the Szeg˝o parameters. In [FKL3] it is already shown that, in the generic case, each pair of orthonormal systems of rational matrix-valued functions is characterized by an initial condition and a sequence of jqq -unitary matrices. Roughly speaking, Szeg˝ o pairs of orthonormal systems of rational matrix-valued functions correspond to sequences of positive Hermitian jqq -unitary matrices. Since there is a one-to-one correspondence between jqq -unitary matrices and strictly contractive q × q matrices (see, e.g., [D, Theorem 1.2]), Szeg˝o pairs of rational matrix-valued functions on the unit circle T can be described by an initial condition and a sequence of strictly contractive q × q matrices. A brief synopsis is as follows. In addition to several notations we will use, we explain in Section 1 some basics on rational matrix-valued functions and, especially, we recall the Christoffel-Darboux formulas which every pair of orthonormal rational matrix-valued functions satisfies. In Section 2, we introduce the notion Szeg˝o pair of orthonormal systems in the context of rational matrix-valued functions and, by an application of the Christoffel-Darboux formulas, we will see that such pairs of orthonormal systems are determined by an initial condition and a sequence of strictly contractive q × q matrices via certain recurrence relations. Then, in Section 3, we study an inverse question about this and we prove a result which is often called Favard-type theorem. Our Favard-type theorem says that if we have a pair of rational matrix-valued functions which satisfies the initial condition and which is recursively connected by some strictly contractive q × q matrices based on the recurrence relations in Section 2, then there is a (not necessarily unique) nonnegative Hermitian q × q matrix measure on the Borelian σ-algebra on the unit circle with respect to which the sequences of rational matrix-valued functions form left and right orthonormal systems, respectively. Finally, we will see in Section 4 that the Szeg˝ o parameter En , which determines the elements Xn and Yn of a Szeg˝ o pair of rational matrix-valued functions by the recurrence relations assuming the knowledge of the previous elements Xn−1 and Yn−1 , can be computed by some integral formulas and, in particular, in terms of Xn−1 and Yn−1 .

Szeg˝o Pairs of Orthogonal Rational Matrix-valued Functions

165

1. Notation and preliminaries Throughout this paper, let p and q belong to the set N of all positive integers. Let us use C, Z, and N0 to denote the sets of all complex numbers, of all integers, and of all nonnegative integers, respectively. If m ∈ Z and if n ∈ Z or n = +∞, then we will write Nm,n for the set of all integers k which satisfy m ≤ k ≤ n. If X is a nonempty set, then let Xp×q be the set of p × q matrices each entry of which belongs to X. The notation 0p×q stands for the null matrix that belongs to the set Cp×q of all p × q complex matrices, and the identity matrix which belongs to Cq×q will be denoted by Iq . If the size of a null matrix or a identity matrix is obvious, we will omit the indices. If A ∈ Cq×q , then rank A and det A indicate the rank of A and the determinant of A, √ respectively. Furthermore, if A is a nonnegative Hermitian q × q matrix, then A stands for the (unique) nonnegative Hermitian square root of A. The symbol T stands for the unit circle, D for its interior, and E for its exterior with respect to the extended complex plane C0 := C ∪ {∞}, i.e., let T := {z ∈ C : |z| = 1}, D := {z ∈ C : |z| < 1}, and E := C0 \ (D ∪ T). We will use the notation T1 to designate the set of all sequences (αj )∞ j=1 of complex numbers which satisfy αj αk = 1 for all positive integers j and k. Obviously, if (αj )∞ j=1 ∈ T1 , then αj ∈ T for all j ∈ N. For technical reasons, throughout this paper, we set α0 := 0. Let Bp×q (respectively, B1 ) be the σ-algebra of all Borel subsets of Cp×q (respectively, C), and let BT := B1 ∩ T. The linear Lebesgue measure on (T, BT ) is designated by λ. If f is a matrix-valued function defined on a subset M of C0 with T ⊆ M, then we will write f for the restriction of f to T. If G is a nonempty ∗ subset 0 and if f is a matrix-valued function defined on G, then f (z) is short ofC ∗ for f (z) , z ∈ G. & ' If α ∈ C \ (T ∪ {0}), then bα denotes the function bα : C0 \ α1 → C given by & ' + α α−w for w ∈ C \ α1 |α| 1−αw (1.1) bα (w) := 1 for w = ∞ . |α| Further, let b0 : C → C be defined by b0 (w) := w for each w ∈ C. Clearly, if α ∈ D, then the function bα is exactly the elementary Blaschke factor corresponding to α. Now we turn our attention to left and right Cq×q -modules of rational matrixvalued functions with prescribed pole structure. Let τ ∈ N or τ = +∞, and let (αj )τj=1 be a sequence of complex numbers. Further, let πα,0 : C0 → C be the constant function with value 1 and let Rα,0 denote the set of all constant complexvalued functions defined on C0 . For each n ∈ N1,τ , let πα,n: C → C be defined by πα,n (w) :=

n A j=1

(1 − αj w)

166


and let Rα,n designate the set of all complex-valued functions f which are rational 1 P with some complex polynomial P of and which admit a representation f = πα,n degree not greater than n. Further, for each n ∈ N0,τ , let . n n $ $ 1 Pα,n := and Zα,n := {αj }, αj j=1 j=1 B & ' 1 := 0, and 0j=1 α1j := ∅. For where we use the conventions 10 := ∞, ∞ each n ∈ N0,τ , every function which belongs to Rα,n is holomorphic in C0 \ Pα,n . Obviously, if 0 ≤ n < τ , then Rα,n does not depend on the numbers αj , j ∈ Nn+1,τ . Identifying a constant complex-valued function defined on C0 with its restriction to C, one can easily see that in the case αj = 0 for all j ∈ N1,n the class Rα,n coincides with the set Pn of all complex-valued polynomials of degree not greater than n. If n1 and n2 are integers with 0 ≤ n1 ≤ n2 ≤ τ , then Rα,n1 ⊆ Rα,n2 . If a sequence (αj )∞ j=1 of complex numbers is given, then let Rα,∞ :=

∞ $

Rα,n .

n=0

Every function f ∈ Rα,∞ is holomorphic in C0 \

B ∞ ( j=1

1 αj

) .

q×q -submodule For each n ∈ N0 , the class Rp×q α,n can be considered as a right C q×q p×q of the right C -module Rα,∞ . On the other hand, for each n ∈ N0 , the class p×p -submodule of the left Cp×p -module Rp×q Rp×q α,n is also a left C α,∞ . q×q to denote the constant Now let (αj )∞ j=1 ∈ T1 . We will use Bα,0 : C0 → C (q)

(q)

matrix-valued function with value Iq and, for each n ∈ N, let the function Bα,n : C0 \ Pα,n → Cq×q be defined by ⎛ ⎞ n A (q) (w) := ⎝ bαj (w)⎠ Iq . Bα,n j=1

) ( (q) If τ ∈ N0 or τ = +∞, then Bα,k : k ∈ N0,τ is both a basis of the right q×q -module Rq×q Cq×q -module Rq×q α,τ and a basis of the left C α,τ . (For a more detailed discussion of these modules we refer to [FKL1].) In particular, if n ∈ N0 and if q×q X ∈ Rq×q such that α,n , then there are unique matrices A0 , A1 , . . . , An ∈ C #n (q) X = j=0 Aj Bα,j and the reciprocal rational (matrix-valued ) function X [α,n] of #n (q) [α,n] := j=0 A∗n−j Bβ,j where the X with respect to (αj )∞ j=1 and n is given by X sequence (βj )∞ j=1 is defined by βk := αn+1−k for each k ∈ N1,n and βj := αj for each integer j with j ≥ n + 1. In particular,

and

X [α,n] ∈ Rq×q α,n

(1.2)

X [α,n] (αn ) = A∗n .

(1.3)


167

If n ∈ N0 and if αj = 0 for each j ∈ N1,n , then every X ∈ Rq×q α,n is a q × q matrix polynomial of degree not greater than n and X [α,n] is exactly the reciprocal matrix ˜ [n] of X with respect to the unit circle T and the formal degree n polynomial X (cf. [DGK] or also [DFK, Section 1.2]). A mapping F whose domain is the σ-algebra BT of all Borelian subsets of T and whose values are nonnegative Hermitian complex q × q matrices is called nonnegative Hermitian ∞ q × q∞Borel measure on T if it is countably additive, i.e., if B # F satisfies F Ak = F (Ak ) for every infinite sequence (Ak )∞ k=1 of pairk=1

k=1

wise disjoint sets which belong to BT . We will use Mq≥ (T, BT ) to denote the set of all nonnegative Hermitian q × q Borel measures on T. Let F ∈ Mq≥ (T, BT ). For each j ∈ N1,q and each k ∈ N1,q , the entry function Fjk of F in the jth row and kth column is a complex-valued measure. One can easily #q see that F is absolutely continuous with respect to the trace measure τ F := j=1 Fjj of F . An ordered pair [Φ, Ψ] consisting of Borel measurable matrix-valued functions Φ : T → Cp×q and Ψ : T → Cs×q is called left-integrable with respect to F if each entry of the matrix-valued function ΦFτ Ψ∗ is integrable with respect to τ F and the corresponding integral is defined by ΦdF Ψ∗ := ΦFτ Ψ∗ dτ F. (1.4) T T We will also write T Φ(z)F (dz)Ψ∗ (z) for the integral given in (1.4). An ordered pair [Φ, Ψ] consisting of Borel measurable matrix-valued functions Φ : T → Cq×p and Ψ : T → Cq×s is said to be right-integrable with respect to F if [Φ∗ , Ψ∗ ] to F. It is known that the set p × q-L2l (T, BT , F ) is left-integrable with respect 2 respectively, q × p − Lr (T, BT , F ) of all matrix-valued functions Φ : T → Cp×q (respectively, Φ : T → Cq×p ) for which [Φ, Φ] is left-integrable (respectively, rightintegrable) with respect to F is a left (respectively, right) Cp×p -semi Hilbert module, where the Gramian structure is given by (Φ, Ψ)F,l := respectively, (Φ, Ψ)F,r := ΦdF Ψ∗ Φ∗ dF Ψ . (1.5) T

T

For more details on the integration theory with respect to nonnegative Hermitian q × q measures, we refer to Kats [Kt] and Rosenberg [R]. Now let a sequence (αj )∞ j=1 of numbers which belong to C \ T and a nonnegative Hermitian q × q Borel measure F on T be given. Then one can immediately 2 q×p see that {X : X ∈ Rp×q α,∞ } ⊆ p × q − Ll (T, BT , F ) and {X : X ∈ Rα,∞ } ⊆ 2 p×q q×p q × p − Lr (T, BT , F ) hold. If X and Y belong to Rα,∞ (respectively, Rα,∞ ), then (X, Y )F,l (respectively, (X, Y )F,r ) is short for (X, Y )F,l (respectively, (X, Y )F,r ). q×q For each n ∈ N0 , it is readily checked that Rq×q -semi Hilbert α,n is both a left C 2 q×q submodule of q × q − Ll (T, BT , F ) and a right C -semi Hilbert submodule of q × q − L2r (T, BT , F ). Under a certain assumption on the measure F the space q×q Rq×q -Hilbert module. This is based on a non-degeneracy concept which α,n is a C is studied in [FKL1, Section 5]. If n ∈ N0 , then a nonnegative Hermitian q × q

168


Borel measure F on T is called non-degenerate of order n if the block Toeplitz (F ) (F ) matrix Tn := (Γj−k )nj,k=0 is nonsingular where (F ) Γj := z −j F (dz), j ∈ Z, T

are the Fourier coefficients of F . For each n ∈ N0 , we will use Mq,n ≥ (T, BT ) to denote the set of all F ∈ Mq≥ (T, BT ) which are non-degenerate of order n. Clearly, (T, BT ) ⊆ Mq,n Mq,n+1 ≥ ≥ (T, BT ) holds for all n ∈ N0 . A nonnegative Hermitian q×q Borel measure F on T is said to be non-degenerate of order ∞ if F belongs to Mq,∞ ≥ (T, BT ) :=

∞ :

Mq,n ≥ (T, BT ).

n=0

Observe that in [FKL1, Section 5] several characterizations of the set Mq,n ≥ (T, BT ) q,n are given. If n ∈ N0 and if F ∈ M≥ (T, BT ), in view of [FKL1, Theorem 5.8] one can easily check that Rq×q α,n (with the Gramian structure given in (1.5)) is both a left and a right Cq×q -Hilbert module. If F belongs to Mq,∞ ≥ (T, BT ), the space q×q turns out to be both a left and a right C -pre-Hilbert module. Rq×q α,∞ q ∞ Let (αj )j=1 ∈ T1 and let F ∈ M≥ (T, BT ). Further, let τ ∈ N0 or let τ = +∞. A sequence (Xk )τk=0 of matrix-valued functions which belong to Rq×q α,∞ is called a left (respectively, right) orthonormal system corresponding to (αj )∞ j=1 and F if the following two conditions are satisfied: (i) For each k ∈ N0,τ , the function Xk belongs to Rq×q α,k . (ii) For each j ∈ N0,τ and each k ∈ N0,τ , respectively, (Xj , Xk )F,r = δjk I . (Xj , Xk )F,l = δjk I We call a pair [(Xk )τk=0 , (Yk )τk=0 ] consisting of a left orthonormal system τ corresponding to (αj )∞ j=1 and F and a right orthonormal system (Yk )k=0 corresponding to (αj )∞ j=1 and F a pair of orthonormal systems corresponding to ∞ (αj )j=1 and F . In [FKL2, Corollary 4.4] it is verified that if (αj )∞ j=1 ∈ T1 , if τ ∈ N0 or τ = +∞, and if F ∈ Mq≥ (T, BT ), then there exists a pair [(Xk )τk=0 , (Yk )τk=0 ] of orthonormal systems corresponding to (αj )∞ j=1 and F if and only if the underlying measure F is non-degenerate of order τ . Pairs of orthonormal systems satisfy the following Christoffel-Darboux formulas. (Xk )τk=0

q,τ Theorem 1.1. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Further, let [(Xk )τk=0 , (Yk )τk=0 ] be a pair of orthonormal systems corresponding to (αj )∞ j=1 and F . For all n ∈ N1,τ and every choice of v and w in C0 \ Pα,n , n−1 ∗ 1 − bαn (v)bαn (w) Xk∗ (v)Xk (w) = Yn[α,n] (v) Yn[α,n] (w) − Xn∗ (v)Xn (w), k=0

n−1 ∗ 1 − bαn (v)bαn (w) Yk (v)Yk∗ (w) = Xn[α,n] (v) Xn[α,n] (w) − Yn (v)Yn∗ (w), k=0


n Xk∗ (v)Xk (w) 1 − bαn (v)bαn (w)

= and

169

∗ Yn[α,n] (v)

k=0 Yn[α,n] (w)

− bαn (v)bαn (w)Xn∗ (v)Xn (w),

n Yk (v)Yk∗ (w) 1 − bαn (v)bαn (w) k=0

∗ = Xn[α,n] (v) Xn[α,n] (w) − bαn (v)bαn (w)Yn (v)Yn∗ (w). A proof of Theorem 1.1 is given in [FKL2, Theorem 5.4 and Corollary 5.5]. Note that, conversely, systems of rational matrix-valued functions satisfying identities of the type given in Theorem 1.1 are necessarily orthonormal systems corresponding to some nonnegative Hermitian measure (see [FKL3, Theorems 3.10 and 4.12]). q,τ Remark 1.2. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). τ τ Further, let [(Xk )k=0 , (Yk )k=0 ] be a pair of orthonormal systems corresponding to (αj )∞ j=1 and F . For each n ∈ N1,τ , the following statements are satisfied (see [FKL2, Remark 6.2, Theorems 6.7, 6.9, and 6.10]):

(a) There is a number zn ∈ T such that zn · det Xn (w) = det Yn (w)

and det Xn[α,n] (w) = zn · det Yn[α,n] (w)

are satisfied for every choice of w ∈ C0 \ Pα,n . [α,n] (b) If |αn | < 1, then det Xn vanishes nowhere in E \ Pα,n and det Xn vanishes nowhere in D \ Pα,n−1 . [α,n] (c) If |αn | > 1, then det Xn vanishes nowhere in D \ Pα,n and det Xn vanishes nowhere in E \ Pα,n−1 .

2. Szeg˝ o pairs of orthonormal systems Inspired by the case of matrix polynomials we turn our attention to distinguished pairs of orthonormal systems of rational matrix-valued functions. q,τ Definition 2.1. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). For each k ∈ N0,τ , let + α k if αk = 0 |αk | (2.1) ηk := −1 if αk = 0 .

Let [(Xk )τk=0 , (Yk )τk=0 ] be a pair of orthonormal systems corresponding to (αj )∞ j=1 and F . Then [(Xk )τk=0 , (Yk )τk=0 ] is called a Szeg˝ o pair of orthonormal systems corresponding to (αj )∞ j=1 and F if for each n ∈ N1,τ the following statements holds: (i) If (1 − |αn |)(1 − |αn−1 |) > 0, then the matrices −1 ηn ηn−1 (1 − |αn−1 |2 ) [α,n−1] Yn−1 (αn−1 ) Yn[α,n] (αn−1 ) 1 − αn αn−1

(2.2)

170

B. Fritzsche, B. Kirstein and A. Lasarow and

−1 [α,n−1] ηn ηn−1 (1 − |αn−1 |2 ) [α,n] Xn (αn−1 ) Xn−1 (αn−1 ) 1 − αn αn−1

(2.3)

are both positive Hermitian. (ii) If (1 − |αn |)(1 − |αn−1 |) < 0, then the matrices

and

−1 [α,n−1] |αn−1 |2 − 1 Yn (αn−1 ) Xn−1 (αn−1 ) 1 − αn αn−1

(2.4)

−1 |αn−1 |2 − 1 [α,n−1] Yn−1 (αn−1 ) Xn (αn−1 ) 1 − αn αn−1

(2.5)

are both positive Hermitian. Observe that, in view of Remark 1.2, the inverse matrices occurring in the formulae (2.2)–(2.5) are well defined. The notion introduced in Definition 2.1 is a generalization of the corresponding notion in the matrix polynomial case, i.e., in the case that αj = 0 for all j ∈ N (see [DGK] and [DFK, Definition 3.6.5]). q,τ Remark 2.2. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). From [FKL2, Proposition 3.7 and Corollary 4.4] and the polar decomposition of matrices one can immediately see that there exists a Szeg˝o pair of orthonormal systems corresponding to (αj )∞ j=1 and F . q,τ Remark 2.3. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). From [FKL2, Proposition 3.7] and the polar decomposition of matrices one can also see that the following statements hold:

(a) If [(Xk )τk=0 , (Yk )τk=0 ] is a Szeg˝o pair of orthonormal systems corresponding to (αj )∞ j=1 and F , then, for all unitary complex q × q matrices U and V, the ˜ pair [(Xk )τk=0 , (Y˜k )τk=0 ] given for all k ∈ N0,τ by + + UXk if αk ∈ D Yk V if αk ∈ D ˜ ˜ and Yk := Xk := ∗ V Xk if αk ∈ E Yk U∗ if αk ∈ E is also a Szeg˝o pair of orthonormal systems corresponding to (αj )∞ j=1 and F . ˜ k )τ , (Y˜k )τ ] are Szeg˝ (b) If [(Xk )τk=0 , (Yk )τk=0 ] and [(X o pairs of orthonormal k=0 k=0 ˜ 0 (0) −1 and systems corresponding to (αj )∞ and F , then U := X0 (0) X j=1 −1 Y0 (0) are unitary complex q × q matrices and for all k ∈ N0,τ V := Y˜0 (0) the identities + + UXk if αk ∈ D Yk V if αk ∈ D ˜k = and Y˜k = X ∗ V Xk if αk ∈ E Yk U∗ if αk ∈ E are satisfied.


171

q,τ Remark 2.4. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Furτ τ ther, let [(Xk )k=0 , (Yk )k=0 ] be a Szeg˝o pair of orthonormal systems corresponding to (αj )∞ j=1 and F . In view of Remark 1.2 and Definition 2.1, one can get that there exists a z ∈ T such that for all n ∈ N1,τ and all w ∈ C0 \ Pα,n the equalities

z · det Xn (w) = det Yn (w)

and

det Xn[α,n] (w) = z · det Yn[α,n] (w)

are satisfied. q,τ Definition 2.5. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). For each k ∈ N0,τ , let ηk be given by (2.1). Further, let [(Xk )τk=0 , (Yk )τk=0 ] be a τ Szeg˝o pair of orthonormal systems corresponding to (αj )∞ j=1 and F . Then (En )n=1 given by ⎧ [α,n] −1 ⎨ ηn ηn−1 Xn (αn−1 ) Yn (αn−1 ) if (1−|αn |)(1−|αn−1 |) > 0 ∗ (2.6) En:= −1 [α,n] ⎩η η (αn−1 ) Xn (αn−1 ) if (1−|αn |)(1−|αn−1 |) < 0 . n n−1 Yn

is said to be the sequence of Szeg˝ o parameters corresponding to [(Xk )τk=0 , (Yk )τk=0 ]. Note that, in view of Remark 1.2, the inverses occurring in (2.6) are well defined. The notion introduced in Definition 2.5 is a generalization of the corresponding notion in the matrix polynomials case (see [DGK] and [DFK, Definition 3.6.6]). This can be seen from [DFK, Lemma 3.6.14 and Proposition 3.6.5]. Moreover, under the assumptions of Definition 2.5 from [FKL2, Remark 6.2 and Lemma 6.5] and Remark 1.2, for each n ∈ N1,τ , it follows that ⎧ −1 ⎨ ηn ηn−1 Xn[α,n] (αn−1 ) Yn (αn−1 ) if (1−|αn |)(1−|αn−1 |) > 0 ∗ En = (2.7) −1 [α,n] ⎩η η Y (α ) X (α ) if (1−|αn |)(1−|αn−1 |) < 0 . n n n−1 n n−1 n−1 In the matrix polynomial case the Szeg˝ o parameters can be used to obtain recursion formulas for Szeg˝o pairs of orthonormal systems (of matrix polynomials). We will see that these formulas can be generalized to the rational case. q,τ Remark 2.6. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝o pair of orthonormal systems corresponding to τ (αj )∞ o parameters. Then j=1 and F , and let (En )n=1 be the sequence of its Szeg˝ q,τ T F belongs to M≥ (T, BT ). Moreover, [(YkT )τk=0 , (XkT )τk=0 ] is a Szeg˝o pair of orT T τ thonormal systems corresponding to (αj )∞ j=1 and F , and (En )n=1 is the sequence of its Szeg˝ o parameters (see [FKL1, Remark 5.11], [FKL2, Remark 2.10 and Remark 3.5], and (2.7)). q,τ Remark 2.7. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). If A is a positive Hermitian q×q matrix, then [FKL1, Remarks 5.3, 3.10, and 5.13] yield √ −1 −1 √ A belongs that H : BT → Cq×q given by H(B) := A F (T) F (B) F (T) τ τ also to Mq,τ (T, B ) and satisfies H(T) = A. Moreover, if [(X ) , (Y T k k=0 k )k=0 ] is a ≥ ∞ Szeg˝o pair of orthonormal systems corresponding to (αj )j=1 and F , and if (En )τn=1

172


is the sequence of its Szeg˝o parameters, then, in view of [FKL2, Remark 2.8], it √ −1 τ τ √ −1 is not hard to see that Xk F (T) A , A F (T)Yk k=0 is a Szeg˝o k=0 τ pair of orthonormal systems corresponding to (αj )∞ j=1 and H, and (En )n=1 is also the sequence of the Szeg˝o parameters corresponding to this Szeg˝ o pair. q,τ Lemma 2.8. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Let τ τ [(Xk )k=0 , (Yk )k=0 ] be a pair of orthonormal systems corresponding to (αj )∞ j=1 and F . For each n ∈ N1,τ and each w ∈ C \ Pα,n , then ∗ ∗ [α,n] Yn (αn−1 ) Yn[α,n] (w) − Xn (αn−1 ) Xn (w) [α,n−1] ∗ [α,n−1] = 1 − bαn (αn−1 )bαn (w) Yn−1 (αn−1 ) Yn−1 (w)

and

∗ ∗ Xn[α,n] (w) Xn[α,n] (αn−1 ) − Yn (w) Yn (αn−1 ) [α,n−1] [α,n−1] ∗ = 1 − bαn (w)bαn (αn−1 ) Xn−1 (w) Xn−1 (αn−1 ) .

Proof. Apply Theorem 1.1 and use (1.1). Proposition 2.9. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ For each k ∈ N0,τ , let ηk be given by (2.1). For each n ∈ N1,τ , let ⎧ C 1−|αn |2 ⎨ if (1 − |αn |)(1 − |αn−1 |) > 0 1−|αn−1 |2 C ρn := ⎩ − |αn |2 −1 if (1 − |αn |)(1 − |αn−1 |) < 0 . 1−|αn−1 |2

Mq,τ ≥ (T, BT ).

(2.8)

o pair of orthonormal systems corresponding to Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝ τ (αj )∞ and F , and let (E ) be the sequence of its Szeg˝ o parameters. For each n n=1 j=1 n ∈ N1,τ , the following statements hold: (a) If (1 − |αn |)(1 − |αn−1 |) > 0, then −1 2 ηn ηn−1 (1 − |αn−1 |2 ) [α,n−1] I − E∗n En = ρ2n Yn−1 (αn−1 ) Yn[α,n] (αn−1 ) 1 − αn αn−1 and I−

En E∗n

=

ρ2n

2 −1 [α,n−1] ηn ηn−1 (1 − |αn−1 |2 ) [α,n] Xn (αn−1 ) Xn−1 (αn−1 ) . 1 − αn αn−1

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then 2 −1 [α,n−1] |αn−1 |2 − 1 ∗ 2 I − E n E n = ρn Yn (αn−1 ) Xn−1 (αn−1 ) 1 − αn αn−1 and I − En E∗n = ρ2n

−1 |αn−1 |2 − 1 [α,n−1] Yn−1 (αn−1 ) Xn (αn−1 ) 1 − αn αn−1

(c) The Szeg˝ o parameter En is a strictly contractive q × q matrix.

2 .


173

Proof. Let n ∈ N1,τ . First we consider the case (1 − |αn |)(1 − |αn−1 |) > 0. Hence, Remark 1.2 implies det Yn[α,n] (αn−1 ) = 0. (2.9) Using (2.9), (2.6), and Lemma 2.8 we obtain [α,n] ∗ Yn (αn−1 ) I − E∗n En Yn[α,n] (αn−1 ) ∗ ∗ = Yn[α,n] (αn−1 ) Yn[α,n] (αn−1 ) − Xn (αn−1 ) Xn (αn−1 ) [α,n−1] ∗ [α,n−1] (2.10) = 1 − |bαn (αn−1 )|2 Yn−1 (αn−1 ) Yn−1 (αn−1 ). From (1.1) we see that 1 − |bαn (αn−1 )|2 =

(1 − |αn |2 )(1 − |αn−1 |2 ) . (1 − αn αn−1 )(1 − αn αn−1 )

(2.11)

Since ηn and ηn−1 are unimodular complex numbers from (2.8), (2.10), (2.9), and (2.11), we can conclude that, in the considered case, the first equation in (a) is fulfilled. In view of (2.7), in the case (1 − |αn |)(1 − |αn−1 |) < 0, the first formula in (b) can be verified similarly. Using Remark 2.6 the second equation in (a) and (b) follows from the first one, respectively. Thus, parts (a) and (b) are proved. In view of Definition 2.1 and (2.8), part (c) is a consequence of parts (a) and (b). In the following, if a sequence (αj )∞ j=1 ∈ T1 is given, we will use the notations ηk and ρn which are defined by (2.1) for all k ∈ N0 and by (2.8) for all n ∈ N, respectively. q,τ Proposition 2.10. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). τ τ o pair of orthonormal systems corresponding to Let [(Xk )k=0 , (Yk )k=0 ] be a Szeg˝ τ (αj )∞ and F , and let (E ) be the sequence of its Szeg˝ o parameters. For each n n=1 j=1 n ∈ N1,τ and each w ∈ C \ Pα,n , then the following statements hold:

(a) If (1 − |αn |)(1 − |αn−1 |) > 0, then Yn (w) − ηn ηn−1 Xn[α,n] (w)En = ρn

1 − αn−1 w bαn−1 (w)Yn−1 (w) I − E∗n En 1 − αn w

and Xn (w) − ηn ηn−1 En Yn[α,n] (w) = ρn

1 − αn−1 w bαn−1 (w) I − En E∗n Xn−1 (w). 1 − αn w

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then Yn (w) − ηn ηn−1 Xn[α,n] (w)En = ρn

1 − αn−1 w [α,n−1] Xn−1 (w) I − E∗n En 1 − αn w

and Xn (w) − ηn ηn−1 En Yn[α,n] (w) = ρn

1 − αn−1 w [α,n−1] I − En E∗n Yn−1 (w). 1 − αn w

174


Proof. Let n ∈ N1,τ and w ∈ C \ Pα,n . At first we shall prove the first equation in (a) and hence we consider in the following the case (1 − |αn |)(1 − |αn−1 |) > 0. Using (2.6) and Lemma 2.8 we get Yn[α,n] (w) − ηn ηn−1 E∗n Xn (w) −∗ [α,n] ∗ ∗ Yn (αn−1 ) Yn[α,n] (w) − Xn (αn−1 ) Xn (w) = Yn[α,n] (αn−1 ) −∗ [α,n−1] ∗ [α,n−1] Yn−1 (αn−1 ) Yn−1 (w) = 1 − bαn (αn−1 )bαn (w) Yn[α,n] (αn−1 ) and therefore, in view of [FKL2, Remarks 2.4 and 2.8] and Proposition 2.9, Yn (w) − ηn ηn−1 Xn[α,n] (w)En −1 = bαn (w) − bαn (αn−1 ) Yn−1 (w)Yn[α,n−1] (αn−1 ) Yn[α,n] (αn−1 ) bαn (w) − bαn (αn−1 ) ηn ηn−1 (1 − αn αn−1 ) = Yn−1 (w) I − E∗n En . 2 (1 − |αn−1 | )ρn Taking into account the identity bαn (w) − bαn (αn−1 ) ηn ηn−1 (1 − αn αn−1 ) 1 − αn−1 w bαn−1 (w) = 1 − αn w 1 − |αn |2 it follows the first identity in (a). Now let (1 − |αn |)(1 − |αn−1 |) < 0. Using (2.7), Lemma 2.8, (1.1), (2.8), and Proposition 2.9 we can conclude Yn (w) − ηn ηn−1 Xn[α,n] (w)En ∗ ∗ −∗ Yn (αn−1 ) = Yn (w) Yn (αn−1 ) − Xn[α,n] (w) Xn[α,n] (αn−1 ) [α,n−1] [α,n−1] ∗ −∗ = bαn (w)bαn (αn−1 ) − 1 Xn−1 (w) Xn−1 (αn−1 ) Yn (αn−1 ) ∗ −1 [α,n−1] (|αn |2 − 1)(1 − αn−1 w) [α,n−1] Xn−1 (w) Yn (αn−1 ) = Xn−1 (αn−1 ) (1 − αn αn−1 )(1 − αn w) 1 − αn−1 w [α,n−1] Xn−1 (w) I − E∗n En . = ρn 1 − αn w Hence, the first identity in (b) is also proved. The second equation in (a) and (b) can be obtained using Remark 2.6, [FKL2, Remark 2.11], and the first one, respectively. For each strictly contractive complex q × q matrix E, let H(E) be the so-called Halmos extension of E, i.e., let √ √ −1 −1 I − EE∗ E I − E∗ E H(E) := . √ √ −1 −1 E∗ I − EE∗ I − E∗ E Furthermore, in view of Definition 2.1, Definition 2.5, and part (c) of Proposition τ 2.9, if (αj )∞ j=1 ∈ T1 and if (En )n=1 is a sequence of strictly contractive complex q × q matrices where τ ∈ N or τ = ∞, then for n ∈ N1,τ and w ∈ C \ Pα,n we set 0 (w)Iq 0 I b H(En ) αn−1 Hα,E;n (w) := q 0 ηn ηn−1 Iq 0 Iq


175

0 bαn−1 (w)Iq 0 I H(E∗n ) q 0 Iq 0 ηn ηn−1 Iq if (1 − |αn |)(1 − |αn−1 |) > 0 and in the case (1 − |αn |)(1 − |αn−1 |) < 0 we put bαn−1 (w)Iq 0 0 Iq 0 Iq Hα,E;n (w) := H(En ) 0 ηn ηn−1 Iq 0 Iq Iq 0 and 0 Iq 0 bαn−1 (w)Iq 0 Iq ∗ H(En ) Gα,E;n (w) := . 0 Iq 0 ηn ηn−1 Iq Iq 0

and

Gα,E;n (w) :=

q,τ Theorem 2.11. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). τ τ o pair of orthonormal systems corresponding to Let [(Xk )k=0 , (Yk )k=0 ] be a Szeg˝ τ (αj )∞ and F , and let (E ) o parameters. For all n n=1 be the sequence of its Szeg˝ j=1 n ∈ N1,τ and all w ∈ C \ Pα,n , the following recurrence relations are satisfied: Xn−1 (w) Xn (w) 1 − αn−1 w Hα,E;n (w) = ρn [α,n] [α,n−1] 1 − αn w Yn (w) Yn−1 (w) and 1 − αn−1 w [α,n−1] Yn−1 (w), Xn−1 (w) Gα,E;n (w). Yn (w), Xn[α,n] (w) = ρn 1 − αn w

Proof. Let n ∈ N1,τ and w ∈ C \ Pα,n . Suppose (1−|αn |)(1−|αn−1 |) > 0. We have Xn (w) 0 I H(−En ) q [α,n] 0 ηn ηn−1 Iq Yn (w) −1 [α,n] −1 I − En E∗n Xn (w) − ηn ηn−1 En I − E∗n En Yn (w) = −1 [α,n] −1 −E∗n I − En E∗n Xn (w) + ηn ηn−1 I − E∗n En Yn (w) ⎛ ⎞ −1 [α,n] Xn (w) − ηn ηn−1 En Yn (w) I − En E∗n ⎠ . = ⎝ (2.12) −1 [α,n] −E∗n Xn (w) + ηn ηn−1 Yn (w) I − E∗n En From Proposition 2.10, [FKL2, Remark 2.8 and Proposition 2.13], and the identity ηn−1 (αn−1 − w) 1 − αn−1 w bαn−1 (w) = 1 − αn w 1 − αn w we get Yn[α,n] (w) − ηn ηn−1 E∗n Xn (w) −ηn (ηn−1 αn−1 w − ηn−1 ) [α,n−1] I − E∗n En Yn−1 (w) = ρn 1 − αn w 1 − αn−1 w [α,n−1] I − E∗n En Yn−1 (w). (2.13) = ρn ηn ηn−1 1 − αn w Using (2.13) and again Proposition 2.10 we can see that the right-hand side of (2.12) is equal to 1 − αn−1 w bαn−1 (w)Xn−1 (w) ρn . [α,n−1] 1 − αn w Yn−1 (w)

176


Taking into account H(En )H(−En ) = I thus the first identity for the considered case (1−|αn |)(1−|αn−1 |) > 0 follows. Analogously, one can prove that this identity is satisfied if (1 − |αn |)(1 − |αn−1 |) < 0. In view of [FKL2, Remark 2.8 and Proposition 2.13], the second recurrence formula is a consequence of the first one. q,τ Corollary 2.12. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). τ τ o pair of orthonormal systems corresponding to Let [(Xk )k=0 , (Yk )k=0 ] be a Szeg˝ τ (αj )∞ and F , and let (E ) be the sequence of its Szeg˝ o parameters. For every n n=1 j=1 choice of n ∈ N1,τ and w ∈ C \ Pα,n , the following statements hold: (a) If (1 − |αn |)(1 − |αn−1 |) > 0, then −1 1 − αn−1 w [α,n−1] bαn−1 (w)Xn−1 (w) + En Yn−1 (w) I − En E∗n Xn (w) = ρn 1 − αn w

and Yn (w) = ρn

−1 1 − αn−1 w [α,n−1] bαn−1 (w)Yn−1 (w) + Xn−1 (w)En I − E∗n En . 1 − αn w

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then −1 1 − αn−1 w [α,n−1] bαn−1 (w)En Xn−1 (w) + Yn−1 (w) I − En E∗n Xn (w) = ρn 1 − αn w and Yn (w) = ρn

−1 1 − αn−1 w [α,n−1] bαn−1 (w)Yn−1 (w)En + Xn−1 (w) I − E∗n En . 1 − αn w

Proof. Apply Theorem 2.11.

At the end of this section we note again that the particular case of q × q matrix polynomials is considered in [DGK] and [DFK, Section 3.6]. In this special case the recurrence relations stated in Corollary 2.12 coincide with those given in [DFK, Lemma 3.6.12].

3. Szeg˝ o pairs of rational matrix-valued functions Motivated by the matrix polynomial case and Corollary 2.12 we introduce the following notion. τ Definition 3.1. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let (En )n=1 be a sequence of strictly contractive q × q matrices. Further, let X0 and Y0 be nonsingular complex q×q matrices such that X∗0 X0 = Y0 Y0∗ , and let X0 and Y0 be the constant matrix-valued functions (defined on C0 ) with values X0 and Y0 , respectively. For all n ∈ N1,τ such that (1−|αn |)(1−|αn−1 |) > 0, let Xn and Yn be the matrix-valued functions which are given for all w ∈ C \ Pα,n by −1 1 − αn−1 w [α,n−1] bαn−1 (w)Xn−1 (w) + En Yn−1 (w) I − En E∗n Xn (w) := ρn 1 − αn w

Szeg˝o Pairs of Orthogonal Rational Matrix-valued Functions and Yn (w) := ρn

177

−1 1 − αn−1 w [α,n−1] bαn−1 (w)Yn−1 (w) + Xn−1 (w)En I − E∗n En . 1 − αn w

For all n ∈ N1,τ such that (1 − |αn |)(1 − |αn−1 |) < 0, let Xn and Yn be the matrix-valued functions which are defined for all w ∈ C \ Pα,n by −1 1 − αn−1 w [α,n−1] bαn−1 (w)En Xn−1 (w) + Yn−1 (w) I − En E∗n Xn (w) := ρn 1 − αn w and Yn (w) := ρn

−1 1 − αn−1 w [α,n−1] bαn−1 (w)Yn−1 (w)En + Xn−1 (w) I − E∗n En . 1 − αn w

o pair of rational matrix-valued Then the pair [(Xk )τk=0 , (Yk )τk=0 ] is called the Szeg˝ τ functions generated by [(αj )∞ j=1 ; (En )n=1 ; X0 , Y0 ]. q,τ Remark 3.2. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). τ τ Let [(Xk )k=0 , (Yk )k=0 ] be a Szeg˝o pair of orthonormal systems corresponding to τ (αj )∞ o parameters. Further, let j=1 and F , and let (En )n=1 be a sequence of Szeg˝ X0 := X0 (0) and Y0 := Y0 (0). From [FKL2, Remark 5.3] we know that X0 and Y0 are nonsingular matrices such that X∗0 X0 = Y0 Y0∗ holds. Hence, Corollary 2.12 shows that [(Xk )τk=0 , (Yk )τk=0 ] is the Szeg˝o pair of rational matrix-valued functions τ generated by [(αj )∞ j=1 ; (En )n=1 ; X0 , Y0 ]. τ Remark 3.3. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let (En )n=1 be a sequence of strictly contractive q × q matrices. Further, let X0 and Y0 be nonsingular complex q × q matrices such that X∗0 X0 = Y0 Y0∗ . Let [(Xk )τk=0 , (Yk )τk=0 ] be the Szeg˝ o τ pair of rational matrix-valued functions generated by [(αj )∞ ; (E ) ; X , Y ]. n 0 0 n=1 j=1 For each k ∈ N0,τ , from Definition 3.1, (1.1), and (1.2) one can easily see that the matrix-valued functions Xk and Yk both belong to Rq×q α,k . τ Remark 3.4. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let (En )n=1 be a sequence of strictly contractive complex q × q matrices. Furthermore, let X0 and Y0 be nonsingular complex q × q matrices such that X∗0 X0 = Y0 Y0∗ , let X0 and Y0 be the constant matrix-valued functions with values X0 and Y0 respectively, and for each n ∈ N1,τ let Xn and Yn be a matrix-valued function belonging to Rq×q α,n . By using the same arguments as in the proof of Theorem 2.11, one can verify that [(Xk )τk=0 , (Yk )τk=0 ] is the Szeg˝o pair of rational matrix-valued functions generated τ by [(αj )∞ j=1 ; (En )n=1 ; X0 , Y0 ] if and only if for each n ∈ N1,τ and each w ∈ C\Pα,n the recurrence relations stated in Proposition 2.10 are fulfilled.

As already in the introduction announced, we are going now to prove some results with respect to the pairs of sequences of rational matrix-valued functions defined by Definition 3.1, which are often called Favard-type theorems (cf. [BGHN2, Chapter 8]). At first, we study the case that a finite sequence of strictly contractive complex q × q matrices forms the basis of the pairs.

178


m Theorem 3.5. Let (αj )∞ j=1 ∈ T1 and let m ∈ N. Further, let (En )n=1 be a sequence of strictly contractive complex q×q matrices, let X0 and Y0 be nonsingular complex m q × q matrices such that X∗0 X0 = Y0 Y0∗ , and let [(Xk )m o k=0 , (Yk )k=0 ] be the Szeg˝ ∞ pair of rational matrix-valued functions generated by [(αj )j=1 ; (En )m n=1 ; X0 , Y0 ]. Then Fm : BT → Cq×q defined by % %% −1 −∗ 1 − |αm |2 % 1 Fm (B) := Xm (z) Xm (z) λ(dz) (3.1) 2 2π B |z − αm |

is a (well-defined ) nonnegative Hermitian q × q measure which belongs to the set m Mq,∞ (T, BT ) and [(Xk )m o pair of orthonormal systems cork=0 , (Yk )k=0 ] is a Szeg˝ ≥ ∞ o responding to (αj )j=1 and Fm . Moreover, (En )m n=1 is exactly the sequence of Szeg˝ m , (Y ) ]. parameters corresponding to [(Xk )m k k=0 k=0 Proof. From Remark 3.3 we know that, for each k ∈ N0,m , the matrix-valued functions Xk and Yk both belong to Rq×q α,k . The matrix 0 I jqq := q 0 −Iq is obviously a 2q × 2q signature matrix, i.e., j2qq = jqq and j∗qq = jqq hold. We are m q×q going to verify that [(Xk )m k=0 , (Yk )k=0 ] is a pair of Rα,∞ -sequences which is jqq recursively connected, i.e., that X0 and Y0 are constant matrix-valued functions which satisfy X0∗ X0 = Y0 Y0∗ , that there is a sequence (Un )m n=1 of complex 2q × 2q matrices such that + jqq if (1 − |αn |)(1 − |αn−1 |) > 0 U∗n jqq Un = (3.2) −jqq if (1 − |αn |)(1 − |αn−1 |) < 0 for all n ∈ N1,m and that Xn (w) 1 − αn−1 w (w)Iq b = ρn Un αn−1 [α,n] 0 Yn (w) 1 − αn

0 Iq

Xn−1 (w) [α,n−1] Yn−1 (w)

(3.3)

for all n ∈ N1,m and all w ∈ C \ Pα,n . Obviously, X0 and Y0 are constant matrixvalued functions which satisfy X0∗ X0 = Y0 Y0∗ . For each n ∈ N1,m , the matrix ⎧ Iq 0 ⎪ ⎪ H(En ) if (1 − |αn |)(1 − |αn−1 |) > 0 ⎪ ⎨ 0 ηn ηn−1 Iq Un := ⎪ 0 Iq 0 Iq ⎪ ⎪ if (1 − |αn |)(1 − |αn−1 |) < 0 H(E ) ⎩ n 0 ηn ηn−1 Iq Iq 0 fulfills (3.2). Let n ∈ N1,m and let w ∈ C \ Pα,n . In view of Definition 3.1 and [FKL2, Remark 2.8 and Proposition 2.13], if (1 − |αn |)(1 − |αn−1 |) > 0 then Yn[α,n] (w) = ρn ηn ηn−1

−1 1−αn−1w [α,n−1] Yn−1 (w)+bαn−1 (w)E∗n Xn−1 (w) . (3.4) I−E∗n En 1−αn w


179

Analogously, in the case (1 − |αn |)(1 − |αn−1 |) < 0, we have Yn[α,n] (w) = ρn ηn ηn−1

−1 1−αn−1w [α,n−1] E∗n Yn−1 (w)+bαn−1 (w)Xn−1 (w) . (3.5) I−E∗n En 1−αn w

Consequently, (3.3) is valid. By virtue of [FKL3, Lemma 3.11], for all z ∈ T, the matrix Xm (z) is nonsingular. From [FKL1, Example 5.4] we know then that Fm is a well-defined nonnegative Hermitian measure which belongs to Mq,∞ ≥ (T, BT ). m , (Y ) ] is an orthonorApplying [FKL3, Theorem 4.4] we obtain that [(Xk )m k k=0 k=0 mal system corresponding to (αj )∞ and F . Let n ∈ N . We consider the case m 1,m j=1 that the inequality (1 − |αn |)(1 − |αn−1 |) > 0 holds. From (3.4) we see that

I − E∗n En Yn[α,n] (αn−1 ) = ρn ηn ηn−1

1 − |αn−1 |2 [α,n−1] Y (αn−1 ) 1 − αn αn−1 n−1

and, in view of [FKL3, Lemma 3.11], −1 1 − |αn−1 |2 [α,n−1] I − E∗n En = ρn ηn ηn−1 Yn−1 (αn−1 ) Yn[α,n] (αn−1 ) . 1 − αn αn−1

(3.6)

In particular, the matrix stated in (2.2) is positive Hermitian. For each w ∈ C\Pα,n , from Definition 3.1 we get Xn[α,n] (w) I − En E∗n 1 − αn−1 w [α,n−1] Xn−1 (w) + bαn−1 (w)Yn−1 (w)E∗n = ρn ηn ηn−1 1 − αn w and consequently, in view of [FKL3, Lemma 3.12], −1 [α,n−1] 1 − |αn−1 |2 [α,n] X I − En E∗n = ρn ηn ηn−1 (αn−1 ) Xn−1 (αn−1 ). 1 − αn αn−1 n Therefore the matrix stated in (2.3) is also positive Hermitian. In the case that (1 − |αn |)(1 − |αn−1 |) < 0 holds one can analogously check that the matrices stated in (2.4) and (2.5) are both positive Hermitian. If (1 − |αn |)(1 − |αn−1 |) > 0, then Definition 3.1 and (3.6) provide us −1 ηn ηn−1 Xn (αn−1 ) Yn[α,n] (αn−1 ) −1 −1 1−|αn−1 |2 [α,n−1] = ηn ηn−1 ρn I−En E∗n En Yn−1 (αn−1 ) Yn[α,n] (αn−1 ) = En . 1−αn αn−1 If (1 − |αn |)(1 − |αn−1 |) < 0, from (3.5) and Definition 3.1 we obtain similarly −1 ∗ ηn ηn−1 Yn[α,n] (αn−1 ) Xn (αn−1 ) −1 ∗ −1 ∗ [α,n−1] 1 − |αn−1 |2 ∗ = ρn I − En En En Yn−1 (αn−1 ) Xn (αn−1 ) = En . 1 − αn αn−1 The proof is complete.

180


τ Corollary 3.6. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N, let (En )n=1 be a sequence of strictly contractive complex q × q matrices, let X0 and Y0 be nonsingular complex q × q matrices such that X∗0 X0 = Y0 Y0∗ , and let [(Xk )τk=0 , (Yk )τk=0 ] be the Szeg˝ o pair of τ rational matrix-valued functions generated by [(αj )∞ j=1 ; (En )n=1 ; X0 , Y0 ]. Further, ∞ let the sequences (Xm )∞ m=τ +1 and (Ym )m=τ +1 of rational matrix-valued functions be defined for all m ∈ Nτ +1,∞ and w ∈ C0 \ Pα,m by C ⎧ 1−|αm |2 αm−1 −w ⎨ ηm−1 1−|α if (1−|αm |)(1−|αm−1 |) > 0 2 1−α w Xm−1 (w) m−1 | m C Xm (w) := ⎩ − |αm |2 −1 1−αm−1 w [α,m−1] (w) if (1−|αm |)(1−|αm−1 |) < 0 1−|αm−1 |2 1−αm w Ym−1

and

C ⎧ 1−|αm |2 αm−1 −w ⎨ ηm−1 1−|α 2 1−α w Ym−1 (w) m−1 | m C Ym (w) := ⎩ − |αm |2 −1 1−αm−1 w [α,m−1] (w) 1−|αm−1 |2 1−αm w Xm−1

if (1−|αm |)(1−|αm−1 |) > 0 if (1−|αm |)(1−|αm−1 |) < 0 ,

∞ o and let Em := 0q×q for all m ∈ Nτ +1,∞ . Then [(Xk )∞ k=0 , (Yk )k=0 ] is the Szeg˝ ∞ pair of rational matrix-valued functions generated by [(αj )j=1 ; (En )∞ ; X , Y ]. 0 0 n=1 Moreover, for each m ∈ Nτ,∞ , if the nonnegative Hermitian q × q Borel measure ∞ Fm on T is given as in (3.1) then Fm = Fτ , [(Xk )∞ o pair k=0 , (Yk )k=0 ] is the Szeg˝ ∞ of orthonormal systems corresponding to (αj )j=1 and Fτ , and (En )∞ is exactly n=1 ∞ the sequence of Szeg˝ o parameters corresponding to [(Xk )∞ k=0 , (Yk )k=0 ].

Proof. Since 0q×q is a strictly contractive q × q matrix, from Definition 3.1, (2.8), ∞ (1.1), and (2.1) it follows immediately that [(Xk )∞ o pair k=0 , (Yk )k=0 ] is the Szeg˝ ∞ of rational matrix-valued functions generated by [(αj )j=1 ; (En )∞ n=1 ; X0 , Y0 ]. Furthermore, an application of Theorem 3.5 and [FKL2, Remark 2.5, Remark 6.2, and Lemma 6.5] implies for all m ∈ Nτ +1,∞ and z ∈ T the identity % % % 1 − |αm |2 % |z − ατ |2 ∗ ∗ % Xm (z) Xm (z) = %% Xτ (z) Xτ (z). % 2 2 1 − |ατ | |z − αm | Therefore, we obtain for each m ∈ Nτ +1,∞ the equality Fm = Fτ . Thus, we ∞ can finally conclude from Theorem 3.5 that [(Xk )∞ o pair of k=0 , (Yk )k=0 ] is the Szeg˝ ∞ orthonormal systems corresponding to (αj )j=1 and Fτ , and that (En )∞ n=1 is exactly ∞ the sequence of Szeg˝o parameters corresponding to [(Xk )∞ k=0 , (Yk )k=0 ]. ∞ Theorem 3.7. Let (αj )∞ j=1 ∈ T1 , let (En )n=1 be a sequence of strictly contractive complex q × q matrices, and let X0 and Y0 be nonsingular complex q × q matri∞ ces such that X∗0 X0 = Y0 Y0∗ . Let [(Xk )∞ o pair of rational k=0 , (Yk )k=0 ] be a Szeg˝ ∞ matrix-valued functions generated by [(αj )j=1 ; (En )∞ ; X , Y ]. 0 0 Then there is an n=1 ∞ ∞ (T, B ) such that [(X ) , (Y ) ] is a Szeg˝ o pair of orthonormal F ∈ Mq,∞ T k k=0 k k=0 ≥ ∞ ∞ systems corresponding to (αj )j=1 and F . Moreover, (En )n=1 is the sequence of ∞ Szeg˝ o parameters corresponding to [(Xk )∞ k=0 , (Yk )k=0 ].


181

m Proof. According to Theorem 3.5, for each m ∈ N, [(Xk )m o k=0 , (Yk )k=0 ] is a Szeg˝ ∞ pair of orthonormal systems corresponding to (αj )j=1 and the nonnegative Hermitian q × q measure Fm which is given by (3.1). From [FKL2, Remark 5.3] we know that Fm (T) = (X∗0 X0 )−1 for all m ∈ N. By virtue of [FKL3, Lemma 4.7], ∞ there are an F ∈ Mq≥ (T, BT ) and a subsequence (Fmn )∞ n=1 of (Fm )m=1 which converges weakly to F (cf. [FK] or [FKL3, Section 4]). Applying [FKL3, Lemma 4.6] (see also [FK, Satz 3]), for all nonnegative integers j and k, we obtain ∗ Xj dF Xk = lim Xj dFmn Xk∗ (Xj , Xk )F,l = T

n→∞

T

= lim (Xj , Xk )Fmn ,l = δj,k Iq n→∞

and analogously (Yj , Yk )F,r = δjk Iq . ∞ Hence, in view of Remark 3.3, [(Xk )∞ k=0 , (Yk )k=0 ] is a pair of orthonormal systems ∞ corresponding to (αj )j=1 and F . The rest of the assertion follows immediately from [FKL2, Corollary 4.4] and Theorem 3.5.

4. Integral representations of Szeg˝ o parameters In the following, for each w ∈ C, let fw : C → C and gw : C → C be defined by fw (z) := z − w

and gw (z) := 1 − wz,

(4.1)

respectively. If a sequence (αj )∞ j=1 ∈ T1 is given, we will again use the notation ηk given by (2.2). q Remark 4.1. Let (αj )∞ j=1 ∈ T1 , let n ∈ N, and let F ∈ M≥ (T, BT ). Let X and Y belong to Rq×q α,n−1 . Further, let P be a polynomial over C of degree one. In view of [FKL2, Remark 2.5], one obtains that gα1 P X [α,n−1] and gα1 P Y belong to Rq×q α,n n n and that 1 1 P X [α,n−1], Y [α,n−1] = X, PY . gαn gαn F,l F,r q,τ Lemma 4.2. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). o pair of orthonormal systems corresponding to Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝ τ (αj )∞ o parameters. For each j=1 and F , and let (En )n=1 be the sequence of its Szeg˝ n ∈ N1,τ , the following statements hold:

(a) If (1 − |αn |)(1 − |αn−1 |) > 0, then for all Z ∈ Rq×q α,n−1 , we have gαn−1 [α,n−1] fαn−1 En Yn−1 , Z = ηn−1 Xn−1 , Z , gαn gαn F,l F,l gα fα [α,n−1] Z, n−1 Xn−1 En = ηn−1 Z, n−1 Yn−1 , gαn gαn F,r F,r

182


and

gαn−1 Z, Xn−1 gαn

En = ηn−1

F,l

fαn−1 [α,n−1] Z, Yn−1 gαn

, F,l

gα [α,n−1] fα = ηn−1 Xn−1 , n−1 Z . En Yn−1 , n−1 Z gαn gαn F,r F,r

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then fαn−1 gαn−1 [α,n−1] En Xn−1 , Z = ηn−1 Y ,Z , gαn gαn n−1 F,l F,l gα fα [α,n−1] Z, n−1 Yn−1 En = ηn−1 Z, n−1 Xn−1 , gαn gαn F,r F,r fαn−1 gαn−1 [α,n−1] Z, Yn−1 En = ηn−1 Z, Xn−1 , gαn gαn F,l F,l and

gαn−1 [α,n−1] fαn−1 Z = ηn−1 Yn−1 , Z . En Xn−1 , gαn gαn F,r F,r

Proof. Let n ∈ N1,τ and Z ∈ Rq×q α,n−1 . From [FKL2, Remark 2.1 and Lemma 3.6] we get that (Xn , Z)F,l = 0 and (Z, Yn )F,r = 0. Application of Corollary 2.12 yields the first and the second identity of (a) if (1 − |αn |)(1 − |αn−1 |) > 0 and in the case (1 − |αn |)(1 − |αn−1 |) < 0 the first and the second identity of (b). The other identities follow then from Remark 4.1. q,τ Lemma 4.3. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Let [(Xk )τk=0 , (Yk )τk=0 ] be a pair of orthonormal systems corresponding to (αj )∞ j=1 and F . Further, let n ∈ N1,τ and let Z ∈ Rq×q . α,n−1

(a) If (1 − |αn |)(1 − |αn−1 |) > 0, then the following statements are equivalent: g α = 0. (i) det gαn−1 Z, Xn−1 n F,l gα = 0. (ii) det Yn−1 , gαn−1 Z n

(iii) det Z

[α,n−1]

F,r

(αn ) = 0.

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then the following statements are equivalent: f α [α,n−1] = 0. (iv) det gαn−1 Z, Yn−1 n F,l [α,n−1] fα = 0. (v) det Xn−1 , gαn−1 Z n

(vi) det Z

[α,n−1]

(αn ) = 0.

F,r


183

gα

Proof. (a) Suppose (1 − |αn |)(1 − |αn−1 |) > 0. Obviously, W := gαn−1 Z belongs n to Rq×q α,n . Using [FKL2, Remark 2.8 and Proposition 2.13] we obtain W [α,n] = −ηn

fαn−1 [α,n−1] Z . gαn

(4.2)

From fαn−1 (αn−1 ) = 0 it follows W [α,n] (αn−1 ) = 0. In view of [FKL2, Remark 2.1], let q × q matrices such that

(Wk )nk=0

W =

n

(4.3)

be the unique sequence of complex (q)

Wk Bα,k

k=0

and for each j ∈ N0,n let (Xjk )nk=0 be the sequence of complex q × q matrices such that n (q) Xj = Xjk Bα,k . (4.4) k=0

We have Xn[α,n] (αn−1 ) = X∗nn + bαn (αn−1 )X∗n,n−1

(4.5)

and ∗ . W [α,n] (αn−1 ) = Wn∗ + bαn (αn−1 )Wn−1

From (4.3) it follows Wn = −bα,n (αn−1 )Wn−1 .

(4.6)

Using (1.3) and (4.2) we get Wn∗ = W [α,n] (αn ) = −ηn = −

fαn−1 (αn ) [α,n−1] Z (αn ) gαn (αn )

1 − αn αn−1 bαn (αn−1 )Z [α,n−1] (αn ). 1 − |αn |2

If αn = αn−1 , then comparing this with (4.6) yields 1 − αn αn−1 [α,n−1] ∗ Z (αn ) = Wn−1 . 1 − |αn |2

(4.7)

If αn = αn−1 we have W = Z and hence, in view of (1.3), equation (4.7) also holds in this case. From (4.4) and [FKL2, Lemma 3.6] we obtain n−1 (q) (q) 0 = (Xn , Xn−1 )F,l = Xnn (Bα,n , Xn−1 )F,l + Xnk Bα,k , Xn−1 k=0

=

(q) Xnn (Bα,n , Xn−1 )F,l

+

Xn,n−1 X−1 n−1,n−1 .

Analogously, we get (q) (W, Xn−1 )F,l = Wn (Bα,n , Xn−1 )F,l + Wn−1 X−1 n−1,n−1 .

F,l

(4.8)

184


Hence from (4.8), (4.6), (4.5), and (4.7) we can conclude −1 (W, Xn−1 )F,l = −Wn X−1 nn Xn,n−1 + Wn−1 Xn−1,n−1 −1 = Wn−1 X−1 nn bαn (αn−1 )Xn,n−1 + Xnn Xn−1,n−1 [α,n] ∗ = Wn−1 X−1 (αn−1 ) X−1 nn Xn n−1,n−1 ∗ −1 [α,n] ∗ 1 − αn−1 αn [α,n−1] Z = (αn ) Xnn Xn (αn−1 ) X−1 n−1,n−1 . 2 1 − |αn |

(4.9)

[α,n]

Since we know from Remark 1.2 that det Xn (αn−1 ) = 0 holds we see then from (4.9) that (i) and (iii) are equivalent. In view of [FKL2, Remark 2.11 and Remark 3.5], we also obtain that (ii) and (iii) are equivalent. fα (b) Now let (1−|αn |)(1−|αn−1 |) < 0. Let V := gαn−1 Z, let (Vk )nk=0 be the unique n sequence of complex q × q matrices such that n (q) V = Vk Bα,k k=0

and let

(Ynk )nk=0

be the sequence of complex q × q matrices such that Yn =

n

(q)

Ynk Bα,k .

k=0

Obviously, R :=

n−1 # k=0

(q)

Ynk Bα,k belongs to Rq×q α,n−1 . From [FKL2, Remark 2.8 and

Proposition 2.13] we get V [α,n] = −ηn

gαn−1 [α,n−1] Z gαn

and (1.3) yields then ∗ ∗ 1 − αn αn−1 [α,n−1] Vn = V [α,n] (αn ) = −ηn Z (αn ) . 2 1 − |αn |

(4.10)

[α,n−1]

belongs to Rq×q Since Xn−1 α,n−1 , from [FKL2, Lemma 3.6 and Remark 4.2] and (1.3) we obtain ∗ [α,n−1] [α,n−1] (q) = Xn−1 , Bα,n Ynn + R[α,n−1] , Xn−1 0 = Xn−1 , Yn F,r F,r F,l ∗ ∗ −1 [α,n−1] (q) = Xn−1 , Bα,n Ynn + R(αn−1 ) Xn−1,n−1 . (4.11) F,r

Furthermore, we have (q) R(αn−1 ) = Yn (αn−1 ) − Ynn Bα,n (αn−1 ) = Yn (αn−1 ).

The matrix-valued function H :=

n−1 k=0

(q)

Vk Bα,k

(4.12)


185

belongs to Rq×q α,n−1 and satisfies (q) H(αn−1 ) = V (αn−1 )−Vn Bα,n (αn−1 ) = V (αn−1 ) =

fαn−1(αn−1 ) Z(αn−1 ) = 0. gαn(αn−1 )

By virtue of [FKL2, Lemma 3.6 and Remark 4.2], (1.3), (4.10), (4.11), and (4.12) it follows ∗ ∗ [α,n−1] [α,n−1] (q) = Xn−1 , Bα,n Vn + H(αn−1 ) X−1 Xn−1 , V n−1,n−1 F,r F,r ∗ ∗ [α,n−1] (q) −1 = Xn−1 , Bα,n Vn = − R(αn−1 ) X−1 Ynn Vn n−1,n−1 F,r ∗ [α,n−1] ∗ −1 ∗ 1−αn αn−1 −1 Y Z = −ηn (α ) X Ynn (αn ) . n n−1 n−1,n−1 1−|αn |2 Since Remark 1.2 yields that det Yn (αn−1 ) = 0 is fulfilled we see that (v) and (vi) are equivalent. The equivalence of (iv) and (vi) follows then by application of [FKL2, Remark 2.11 and Remark 3.5]. q,τ Theorem 4.4. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). o pair of orthonormal systems corresponding to Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝ τ (αj )∞ and F , and let (E ) be the sequence of its Szeg˝ o parameters. For each n n=1 j=1 n ∈ N1,τ , the following statements hold: (a) If (1 − |αn |)(1 − |αn−1 |) > 0, then −1 [α,n−1] ∗ gαn−1 fαn−1 ∗ En = ηn−1 ZdF Xn−1 ZdF Yn−1 T gαn T gαn

and

En = ηn−1

T

[α,n−1] ∗ Xn−1 dF

f

αn−1

gαn

Z T

∗ Yn−1 dF

g

αn−1

gαn

−1 Z

for all Z ∈ Rq×q α,n−1 which satisfy det Z [α,n−1] (αn ) = 0.

(4.13)

(b) If (1 − |αn |)(1 − |αn−1 |) < 0, then [α,n−1] ∗ −1 fαn−1 gαn−1 ∗ En = ηn−1 ZdF Yn−1 ZdF Xn−1 T gαn T gαn and

En = ηn−1

T

∗ Yn−1 dF

Rq×q α,n−1

g

αn−1

gαn

Z

T

[α,n−1] ∗

Xn−1

dF

f

αn−1

gαn

−1 Z

for all Z ∈ which satisfy (4.13). (c) If (1 − |αn |)(1 − |αn−1 |) > 0, then −1 fαn−1 gαn−1 [α,n−1] ∗ ∗ En = ηn−1 Xn−1 dF Z Yn−1 dF Z T gαn T gαn

186

B. Fritzsche, B. Kirstein and A. Lasarow and

En = ηn−1 for all Z ∈

Z ∗ dF T

Rq×q α,n−1

g

αn−1

gαn

[α,n−1]

−1

Xn−1

Z ∗ dF

f

αn−1

gαn

T

Yn−1

which satisfy det Z(αn ) = 0.

(4.14)

(d) If (1 − |αn |)(1 − |αn−1 |) < 0, then −1 gαn−1 [α,n−1] fαn−1 En = ηn−1 Yn−1 dF Z ∗ Xn−1 dF Z ∗ T gαn T gαn and

En = ηn−1

∗

Z dF T

f

αn−1

gαn

−1 Yn−1

T

Z ∗ dF

g

αn−1

gαn

[α,n−1]

Xn−1

for all Z ∈ Rq×q α,n−1 which satisfy (4.14). Proof. Application of Lemma 4.2 and Lemma 4.3 yields the proof of parts (a) and (b). Let Z ∈ Rq×q α,n−1 . In view of Remark 4.1 and [FKL2, Remark 2.4], we have gαn−1 [α,n−1] gαn−1 [α,n−1] Z , Xn−1 = Z, X , gαn gαn n−1 F,l F,r gα gαn−1 [α,n−1] Yn−1 , n−1 Z [α,n−1] = Yn−1 , Z , gαn gαn F,r F,l fαn−1 fαn−1 [α,n−1] [α,n−1] Z , Yn−1 = Z, Yn−1 , gαn gαn F,l F,r and

fαn−1 [α,n−1] fαn−1 [α,n−1] Xn−1 , Z = Xn−1 , Z . gαn gαn F,r F,l

All the matrices stated in the left-hand sides of these equalities are in view of (4.14), [FKL2, Remark 2.4], and Lemma 4.3 nonsingular. Application of Lemma 4.2 completes the proof of parts (c) and (d). The following remark presents particular matrix-valued functions Z which satisfy (4.13) and (4.14), respectively. q,τ Remark 4.5. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝o pair of orthonormal systems corresponding to τ (αj )∞ o parameters. From j=1 and F , and let (En )n=1 be the sequence of its Szeg˝ Remark 1.2 one can see that for all n ∈ N1,τ , the following statements hold: [α,n−1]

(a) If (1 − |αn |)(1 − |αn−1 |) > 0, then the relations det Xn−1 (αn ) = 0 and [α,n−1] det Yn−1 (αn ) = 0 are satisfied. (b) If (1 − |αn |)(1 − |αn−1 |) < 0, then det Xn−1 (αn ) = 0 and det Yn−1 (αn ) = 0.


187

We want to draw the attention of the reader to the special case αn−1 = αn where the integral representations of the Szeg˝o parameters stated in Theorem 4.4 can be simplified. In particular, this situation will be met if one studies orthogonal q × q matrix polynomials. q,τ Corollary 4.6. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). Let o pair of orthonormal systems corresponding to (αj )∞ [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝ j=1 τ and F , and let (En )n=1 be the sequence of its Szeg˝ o parameters. Let n ∈ N1,τ and let αn−1 = αn . For all Z ∈ Rq×q α,n−1 which satisfy (4.13), ∗ −1 [α,n−1] [α,n−1] ∗ [α,n−1] (αn−1 ) Xn−1 (αn−1 ) bαn ZdF Yn−1 (4.15) En = − Z T

and En = −

T

[α,n−1] ∗

Xn−1

[α,n−1] −1 ∗ dF bαn Z Yn−1 (αn−1 ) Z [α,n−1] (αn−1 ) . (4.16)

In particular,

En = −

and En = −

T

[α,n−1] ∗ bαn Xn−1 dF Yn−1

T

[α,n−1] ∗

Xn−1

dF bαn Yn−1 .

(4.17)

(4.18)

Proof. In view of (1.1), (2.1), and (4.1), we have ηn−1

fαn−1 fα = ηn−1 n−1 = −bαn−1 = −bαn . gαn gαn−1

Further, using [FKL2, Remark 2.1 and Lemma 3.6] and (1.3), we obtain ∗ [α,n−1] −∗ gαn−1 ∗ ∗ ZdF Xn−1 = ZdF Xn−1 = Z [α,n−1] (αn−1 ) Xn−1 (αn−1 ) . g αn T T Consequently, part (a) of Theorem 4.4 yields (4.15). Equation (4.16) follows analogously. By virtue of part (a) of Remark 4.5, choosing Z = Xn−1 (respectively, Z = Yn−1 ) from (4.15) and (4.16) we get (4.17) and (4.18). Note that, similar as in Corollary 4.6, starting from part (c) of Theorem 4.4 one can also obtain simpler expressions for the Szeg˝ o parameter En in the special case αn−1 = αn . In view of Corollary 3.6, we consider finally the situation that the Szeg˝ o parameter En is equal to zero for some n ∈ N1,τ . q,τ Corollary 4.7. Let (αj )∞ j=1 ∈ T1 , let τ ∈ N or τ = ∞, and let F ∈ M≥ (T, BT ). o pair of orthonormal systems corresponding to Let [(Xk )τk=0 , (Yk )τk=0 ] be a Szeg˝ τ (αj )∞ o parameters. Further, j=1 and F , and let (En )n=1 be the sequence of its Szeg˝ let n ∈ N1,τ and let (1−|αn |)(1−|αn−1 |) > 0. The following statements are equivalent: (i) En = 0. (ii) Xn (αn−1 ) = 0.

188


(iv) For all Z ∈

C

1−|αn |2 fαn−1 1−|αn−1 |2 gαn Rq×q α,n−1 ,

(iii) Xn = −ηn−1

(v) There exists a Z ∈ 0. (vi) Yn (αn−1 ) = C

(vii) Yn = −ηn−1 (viii) For all Z ∈

fαn−1 Xn−1 dF Z ∗ = 0. gαn

T q×q Rα,n−1

(ix) There exists a Z ∈

Yn−1 .

Z ∗ dF

T q×q Rα,n−1

(4.19)

such that (4.14) and (4.19) hold.

1−|αn |2 fαn−1 1−|αn−1 |2 gαn Rq×q α,n−1 ,

Xn−1 .

f

αn−1

gαn

Yn−1 = 0.

(4.20)

such that (4.14) and (4.20) hold.

Proof. (i) ⇔ (ii): This equivalence follows immediately from (2.6) and (2.1). (i) ⇒ (iii): Use part (a) of Corollary 2.12, (1.1), (2.1), (2.8), and (4.1). (iii) ⇒ (iv): This implication is an easy consequence of [FKL2, Remark 2.1 and Lemma 3.6]. (iv) ⇒ (v): From [FKL2, Remark 2.4] and part (a) of Remark 4.5 it follows that (v) is necessary for (iv). (v) ⇒ (i): Apply part (c) of Theorem 4.4. Analogously, in view of (2.7), one can verify that (i), (vi), (vii), (viii), and (ix) are equivalent. Observe that if (1 − |αn |)(1 − |αn−1 |) < 0 is fulfilled, the case En = 0 can be similarly characterized as in Corollary 4.7. We omit the details.

References [BGHN1] A. Bultheel, P. Gonz´ alez-Vera, E. Hendriksen, O. Nj˚ astad, A Szeg˝ o Theory for Rational Functions. Report TW 131, Department of Computer Science, K.U. Leuven 1990. [BGHN2] A. Bultheel, P. Gonz´ alez-Vera, E. Hendriksen, O. Nj˚ astad, Orthogonal Rational Functions. Cambridge Monographs on Applied and Comput. Math. 5, Cambridge University Press, Cambridge 1999. [DGK] P. Delsarte, Y. Genin, Y. Kamp, Orthogonal polynomial matrices on the unit circle. IEEE Trans. Circuits and Systems CAS 25 (1978), 145–160. [Dj] M. M. Djrbashian, Orthogonal systems of rational functions on the circle with given set of poles (in Russian). Dokl. Akad. Nauk SSSR 147 (1962), 1278–1281. [DFK] V. K. Dubovoj, B. Fritzsche, B. Kirstein, Matricial Version of the Classical Schur Problem. Teubner-Texte zur Mathematik 129, Teubner, Leipzig 1992. [D] H. Dym, J-contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation. CBMS Regional Conf. Ser. Math. 71, Amer. Math. Soc., Providence, R. I. 1989.

Szeg˝o Pairs of Orthogonal Rational Matrix-valued Functions [FK]

[FKL1]

[FKL2] [FKL3]

[G1]

[G2] [GS] [Kt]

[R] [S]

189

B. Fritzsche, B. Kirstein, Schwache Konvergenz nichtnegativ hermitescher Borelmaße. Wiss. Z. Karl-Marx-Univ. Leipzig, Math.-Naturwiss. R. 37 (1988), 375–398. B. Fritzsche, B. Kirstein, A. Lasarow, On rank invariance of moment matrices of nonnegative Hermitian-valued Borel measures on the unit circle. Math. Nachr. 263–264 (2004), 103–132. B. Fritzsche, B. Kirstein, A. Lasarow, Orthogonal rational matrix-valued functions on the unit circle. Math. Nachr. 278 (2005), 525–553. B. Fritzsche, B. Kirstein, A. Lasarow, Orthogonal rational matrix-valued functions on the unit circle: Recurrence relations and a Favard-type theorem. to appear in: Math. Nachr. Ja. L. Geronimus, On polynomials orthogonal on the unit circle, on the trigonometric moment problem and on associated functions of classes of Carath´ eodory-Schur (in Russian). Mat. USSR-Sb. 15 (1944), 99–130. Ja. L. Geronimus, Orthogonal Polynomials (in Russian). Fizmatgiz, Moskva 1958. U. Grenander, G. Szeg˝ o, Toeplitz Forms and Their Applications. University of California Press, Berkeley-Los Angeles 1958. I. S. Kats, On Hilbert spaces generated by Hermitian monotone matrix functions (in Russian). Zupiski Nauc.-issled. Inst. Mat. i Mech. i Kharkov. Mat. Obsh. 22 (1950), 95–113. M. Rosenberg, The square integrability of matrix-valued functions with respect to a non-negative Hermitian measure. Duke Math. J. 31 (1964), 291–298. G. Szeg˝ o, Orthogonal Polynomials. Amer. Math. Soc. Coll. Publ. 23, Providence, R. I. 1939.

Bernd Fritzsche, Bernd Kirstein Fakult¨ at f¨ ur Mathematik und Informatik Universit¨ at Leipzig Augustusplatz 10 D-04109 Leipzig Germany e-mail: {fritzsche,kirstein}@mathematik.uni-leipzig.de Andreas Lasarow Katholieke Universiteit Leuven Departement Computerwetenschappen Celestijnenlaan 200A B-3001 Heverlee (Leuven) Belgium e-mail: [email protected]


Singularities of Generalized Strings Michael Kaltenbäck, Henrik Winkler and Harald Woracek Abstract. We investigate the structure of a maximal chain of matrix functions whose Weyl coefficient belongs to Nκ+ . It is shown that its singularities must be of a very particular type. As an application we obtain detailed results on the structure of the singularities of a generalized string which are explicitly stated in terms of the mass function and the dipole function. The main tool is a transformation of matrices, the construction of which is based on the theory of symmetric and semibounded de Branges spaces of entire functions. As byproducts we obtain inverse spectral results for the classes of symmetric and essentially positive generalized Nevanlinna functions. Mathematics Subject Classification (2000). 46C20, 34A55, 30H05. Keywords. generalized string, chain of matrices, de Branges space.

1. Introduction A vibrating string S[L, m] with inhomogeneous mass distribution is given by its length L > 0 and a function m : [0, L) → [0, ∞) which is nondecreasing and continuous from the left. The function m measures the total mass of the part of the string between 0 and x. In the description of the motion of the string the following boundary value problem appears: x y (x) + z y(t)dm(t) = 0 , 0

y (0) = 0, y(L) = 0 if L + m(L) < ∞ . Thereby z is a complex parameter. The concept of the principal Titchmarsh-Weyl coefficient qS of a string S was introduced by I.S.Kac and M.G.Krein, cf. [KaK1]. It turned out to be of fundamental importance. The principal Titchmarsh-Weyl coefficient belongs to the Stieltjes class S, i.e., it is analytic in the open upper half-plane C+ and satisfies Im qS (z) ≥ 0, Im zqS (z) ≥ 0, z ∈ C+ .

192

M. Kaltenbäck, H. Winkler and H. Woracek

A basic inverse result states that to each function q ∈ S there exists a unique string S[L, m], such that q is the principal Titchmarsh-Weyl coefficient of S[L, m]. A canonical system of differential equations, or one-dimensional Hamiltonian system, is a 2 × 2-system of differential equations of the form y (x) = zJH(x)y(x), x ∈ [0, lH ) ,

(1.1)

where H is a locally integrable 2×2-matrix-valued function on [0, lH ) whose values are real and nonnegative matrices. Moreover, J is the matrix 0 −1 . J := 1 0 If x is interpreted as time parameter, it models the motion of a particle under the influence of a time-dependent potential. The function H is called the Hamiltonian of the system under consideration and describes its total energy. To a canonical system there is associated its Weyl coefficient qH , which is a function belonging to the Nevanlinna class N , i.e., is analytic in C+ and satisfies Im qH (z) ≥ 0, z ∈ C+ . A basic inverse result of L. de Branges, cf. [dB1] (a proper formulation can be found, e.g., in [W1]), states that to each function q ∈ N there exists an essentially unique Hamiltonian which has q as its Weyl coefficient. The notions of strings and canonical systems are closely related. In fact, in view of the above inverse results, we know that to each string S[L, m] there exists a unique Hamiltonian Hs such that qS = qHs , and that the behavior of the string is completely determined by the behavior of the canonical system with Hamiltonian Hs . There are also other ways to relate strings and canonical systems. Since qS ∈ S we know that also zqS (z) ∈ N , and therefore that there exists a Hamiltonian H0 such that qH0 = zqS (z). Moreover, it is known that, if q ∈ S, then also zq(z 2 ) ∈ N . Thus we have naturally associated yet another Hamiltonian Hd , namely such that qHd = zqS (z 2 ). Each of the Hamiltonians Hs , H0 and Hd fully describes the string, the most natural choice is Hs . Each of Hs , H0 and Hd can be determined explicitly in terms of the string S[L, m]. A detailed exposition of these topics is given in [KWW3]. During the last decades a theory was developed which deals with generalizations of the notions and theorems mentioned above to an indefinite setting. The class N0 = ind− P(EWt )e + ind− P(EWt )o + . = ind− P(EWt ) + 1 , l t+ , t+ ∈ I: In this case we must have t− ∈ I. To shorten notation put ˆt , At , etc. be defined correspondingly. Et := EWt and let E The interval [t− , t] is in ω indivisible of type π2 and positive length l > 0. Hence At− = At ∈ P(Et ). Since t+ ∈ I, we have [At , At ]P(Es ) = 0. It follows that P(Et )e P(Et− )e ˆt ) P(E ˆt− ) with codimension 1. The with codimension 1 and hence also P(E ˆ ˆs ) is a degenerated set P(Et ) endowed with the inner product inherited from P(E ˆ dB-subspace P of P(Es ), and we have & & √ ' √ ' ˆt− )[+] ˙ span At− ( z) , P = P(E P◦ = span At− ( z) . ˆ t− s ). Consider the space K(W By [KW2, Lemma 7.6] it contains a constant (cos ψ, sin ψ)T . This constant has the property that the space & ' ˆt− cos ψ + Aˆt− sin ψ ˙ span − B P(Eˆt− )[+] where cos ψ cos ψ ˆt− cos ψ + Aˆt− sin ψ, −B ˆ t− cos ψ + Aˆt− sin ψ = , −B ˆ t s) sin ψ K(W sin ψ − ˆs ). It follows that is a dB-subspace of P(E & ' ˆt− )[+] ˆt− cos ψ + Aˆt− sin ψ = P , ˙ span − B P(E and hence that

& ' & √ ' ˆ t− cos ψ + Aˆt− sin ψ = span At− ( z) , span − B

i.e.,

√ ˆt− (z) cos ψ + Aˆt− (z) sin ψ At− ( z) = λ − B for some λ ∈ C, and that cos ψ cos ψ , = 0. ˆ t s) sin ψ K(W sin ψ − From the definition of the transformation T√ we obtain ˆt− (z 2 ) = zBt− (z), Aˆt− (z 2 ) = At− (z) + wt ,12 (0)zBt− (z) . B − It follows that

√ ˆt− (z) . At− ( z) = Aˆt− (z) − wt − ,12 (0)B

Comparing this with (5.4) we obtain ψ = Arccot wt − ,12 (0).

(5.4)

226


ˆ t− t = W(l ,ψ) with some l > 0. Hence the Proposition 3.6 shows that W T ˆ t− s ) as well as to K(W ˆ t−1t ). In the first constant (cos ψ, sin ψ) belongs to K(W − space it is neutral, in the second (one-dimensional) space it is negative. It follows (cf. [ADRS]) that ˆ −1 W ˆ ˆ ind− W t− t t− s = ind− Wt− s . ˆ ts = Wt−1 Ws = Wt−1t Wt− s ) ˆ t = ind− W ˆ t− and hence (W We have ind− W −

ˆ s = ind− W ˆ t− + ind− W ˆ t− s = ind− W ˆ t + ind− W ˆ ts . ind− W We have seen that in any case in the relation Ws = Wt Wts negative indices add. Step 3: We show that T√ ◦ ω satisfies (C2). Proof of Step 3. Also this proof is divided into several cases. We use again the ˆ t. notation ω(t) = Wt and (T√ ◦ ω)(t) = W Case 1: ω ends with an indivisible interval (m, sup I) of type 0. ˆt = W ˆ m , t ∈ (m, sup I), and hence Then W ˆ t ) = t(W ˆ m ) < +∞ lim sup t(W tsup I

and ˆm . ˆt = W lim W

tsup I

We are therefore in case (a) of (C2). By the already proved property (W3) we have ˆ t ≤ ind− W ˆ m , t ∈ I, t ≤ m . ind− W ˆ t = ind− W ˆ m . Hence the implication (a) of (C2) For t ∈ (m, sup I) trivially ind− W holds true. Case 2: ω ends neither with an indivisible interval of type zero nor is the last component I∞ of ω of the form (m− , m] ∪ [m, sup I) with an indivisible interval (m− , m) of type 0 and an indivisible interval (m, sup I) of type π2 . ˆ u = maxv∈I ind− W ˆ v} We claim that there exist s, t ∈ Iˆ∞ := {u ∈ I : ind− W −1 ˆs W ˆ t ) is not a linear polynomial. This excludes the occurrence ˆ st (:= W such that W of case (b) of (C2). In case (a) we appeal to [KW3, Lemma 3.4] to obtain the validity of the desired implication. In order to establish our claim assume on the contrary that for some φ ∈ [0, π) ˆ st = W(l(s,t),φ) whenever s, t ∈ Iˆ∞ . By Lemma 5.4 this relation holds we have W especially for s, t ∈ I∞ = {u ∈ I : ind− Wu = maxv∈I ind− Wv }. φ = 0: Choose s, t ∈ I∞ , s < t, such that Wst is not a linear polynomial. By Remark 3.7 we have for some α ∈ R 1 αz + l(s, t)z 3 . Wst = 0 1 Since Wst is not a linear polynomial we must have l(s, t) = 0. Thus ind− Wst > 0, a contradiction.

Singularities of Generalized Strings

227

φ = 0: We conclude from Remark 3.7 that for all s, t ∈ I∞ , s < t, Wst = W(l1 ,0) W(l2 , π2 ) W(l3 ,0) with some li ≥ 0. By Lemma 5.2 we must be in one ⎧ ⎪ ⎨(m0 , m1 ] ∪ [m1 , m2 ] ∪ [m2 , m3 ) I∞ = (m0 , m1 ] ∪ [m1 , m2 ) ⎪ ⎩ (m0 , m1 ] ∪ [m1 , m2 )

of the following situations: of types 0, π2 , 0 of types π2 , 0 of types 0, π2

Again a contradiction since these are exactly those cases we do not consider in the present step of the proof. Case 3: The last component I∞ of ω is equal to (m0 , m1 ] ∪ [m1 , m2 ) with an indivisible interval (m0 , m1 ] of type 0 and an indivisible interval [m1 , m2 ) of type π2 . ˆ t )t∈I of some type ψ ∈ In this case the interval (m0 , m2 ) is indivisible in (W ˆ t ) = +∞, which rules out the occurrence of (0, π). It follows that limtsup I t(W case (a) in (C2). Fix t0 ∈ (m1 , m2 ). Since [m1 , t0 ] is indivisible of type π2 in (Wt )t∈I and has positive length, we obtain that Am1 ∈ P(Et0 ), [Am1 , Am1 ]P(Et0 ) > 0 . ˆt0 which As we saw in the proof of (W3), the linear combination Sˆψ of Aˆt0 and B ˆt0 ) is linearly dependent with Am1 (√z) and hence belongs to the space P(E [Sˆψ , Sˆψ ] ˆ > 0 . P(Et0 )

It follows from [KW2, Lemma 5.12] that ˆ t0 cot ψ = ind− W ˆ t0 . ind− W We conclude that the implication (b) of (C2) holds true.

We have established that T√ ◦ ω can be extended to a maximal chain by means of Lemma 4.2. If limtM t((T√ ◦ ω)(t)) = +∞, this extension is unique and satisfies q∞ () = lim (T√ ◦ ω(t)) ∞ . tM

By (3.4) this yields (5.5) zq∞ ()(z 2 ) = q∞ (ω)(z) . If limtM t((T√ ◦ ω)(t)) < +∞, so that the extension of T√ ◦ ω is not unique, we should choose the parameter τ = ∞ in Lemma 4.2 in order to achieve the relation (5.5). The fact that this choice is permitted needs justification. However, ˆ st is not of we saw in the proof of Step 3 that there exist s, t ∈ I∞ such that W the form W(l,0) . As indicated in the proof of [KW2, Lemma 8.5] this implies that ind− [limtM T√ ◦ ω(t)] ∞ = ind− limtM T√ ◦ ω(t). Let be the maximal chain with zq∞ ()(z 2 ) = q∞ (ω)(z). Due to the previous steps there exists a nondecreasing function µ ˆ : I → dom such that T√ ◦ ω = ◦ µ ˆ.

228


Step 4: There exists a surjective function µ : dom ω√ → dom such that ω√ = ◦ µ. Proof of Step 4. Write I = (σ0 , σ1 ) ∪ · · · ∪ (σn , σn+1 ). By Lemma 5.4 for each i the set µ ˆ ((σi , σi+1 )) is contained in one connected component of dom . As it is seen from (3.3), the function (t ◦ )(ˆ µ(t)) depends continuously on t ∈ (σi , σi+1 ). Thus µ ˆ is continuous, and it follows that µ ˆ((σi , σi+1 )) is an interval. Put ξi,− := lim µ ˆ(t), ξi,+ := lim µ ˆ(t) . t→σi −

t→σi +

Since T (W ) depends continuously on W and T (I) = I, we have ξ0,+ = 0. We have ξn+1,− = σn+1 if and only if limtσn+1 t((T√ ◦ ω)(t)) = +∞. Otherwise, by (5.5) and the definition of , we have √

√

q∞ () =

lim (T√ ◦ ω)(t) ∞ = (ξn+1,− ) ∞ .

tσn+1

This implies that the interval (ξn+1,− , sup dom ) is indivisible of type 0 in . Consider a point σi , i ∈ {1, . . . , n}. The essential observation for the following proof is that, by (3.4) and the existence of intermediate Weyl-coefficients (see [KW3, §5]), the function (T√ ◦ ω) ∞ has a continuous extension to [σ0 , σn+1 ]. We divide cases similar as in the definition of ςi . ξi,− , ξi,+ ∈ dom : Denote by qξi,± the intermediate Weyl coefficient of at the singularity ξi,± . We have qξi,− =

lim

s→ξi,− −

(s) ∞ = lim (T√ ◦ ω)(t) ∞ = lim (T√ ◦ ω)(t) ∞ = t→σi −

=

lim

s→ξi,+ +

t→σi +

(s) ∞ = qξi,+ ,

and hence ξi,− = ξi,+ . ξi,− ∈ dom , ξi,+ ∈ dom : We have qξi,+ = lim (T√ ◦ ω)(t) ∞ = lim (T√ ◦ ω)(t) ∞ = (ξi,− ) ∞ . t→σi −

t→σi +

Since qξi,+ is the Weyl-coefficient of the maximal chain |dom ∩(0,ξi,+ ) , it follows that (ξi,− , ξi,+ ) is indivisible of type 0 and infinite length. ξi,− ∈ dom , ξi,+ ∈ dom : The same argument as above yields qξi,− = (ξi,+ ) ∞ and hence that ind− (ξi,+ ) ∞ < ind− (ξi,+ ). This implies that the interval (ξi,− , ξi,+ ) is indivisible of type 0 and infinite length. ξi,− ∈ dom , ξi,+ ∈ dom : In this case we find (ξi,− ) ∞ = (ξi,+ ) ∞ and hence, by Lemma 5.3, 1 p(z) (ξi,− )−1 (ξi,+ ) = 0 1 If ind− (ξi,− ) = ind− (ξi,+ ), the interval (ξi,− , ξi,+ ) is indivisible of type 0 and positive length t( lim T√ (ω(t))) − t( lim T√ (ω(t))) . tσi

tσi

Singularities of Generalized Strings

229

If ind− (ξi,− ) < ind− (ξi,+ ), there exists a singularity ξ of in the interval (ξi,− , ξi,+ ). By Lemma 5.3 the intervals (ξi,− , ξ) and (ξ, ξi,+ ) are indivisible of type 0 and infinite length. The case ind− (ξi,− ) > ind− (ξi,+ ) can not occur. We saw that in any case |(ξi,− ,ξi,+ ) ∼ ςi . The required function µ is now defined in the obvious way. We have constructed the function µ required in Theorem 5.1. To complete the proof of Theorem 5.1 choose for λ a right inverse of µ. ✌ 5.5. Corollary. Let ω ∈ Msym and assume that ω satisfies (K− ). Then the maximal κ chain with zq∞ ()(z 2 ) = q∞ (ω)(z) belongs to Mep ≤κ and also satisfies (K− ). Proof. We know from Theorem 5.1 that = ω√ ◦ λ. By (3.4) the number of zeros of T√ (ω(t))21 in C \ [0, ∞) equals the number of non-real zeros of ω(t)21 and is therefore, by Corollary 2.10, bounded uniformly by ind− ω. On indivisible intervals of type 0 the entry (s)21 is constant. Hence the number of zeros of (s)21 in C \ [0, ∞) is bounded independently of s. By Corollary 2.10, (ii), the maximum number of zeros that any entry of (s) can have in C\[0, ∞) is bounded independently of s. The fact that satisfies (K− ) readily follows from the relation of Weyl coefficients and [KW2, Theorem 5.7]. We deduce an inverse result for the class Mep

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