The Corona Problem: Connections Between Operator Theory, Function Theory, and Geometry

Fields Institute Communications 72 The Fields Institute for Research in Mathematical Sciences Ronald G. Douglas Steven ...

Author: Ronald G Douglas | Steven G. Krantz | Eric T. Sawyer (eds.) 2014

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Fields Institute Communications 72 The Fields Institute for Research in Mathematical Sciences

Ronald G. Douglas Steven G. Krantz Eric T. Sawyer Sergei Treil Brett D. Wick Editors

The Corona Problem Connections Between Operator Theory, Function Theory, and Geometry

Fields Institute Communications VOLUME 72 The Fields Institute for Research in Mathematical Sciences Fields Institute Editorial Board: Carl R. Riehm, Managing Editor Edward Bierstone, Director of the Institute Matheus Grasselli, Deputy Director of the Institute James G. Arthur, University of Toronto Kenneth R. Davidson, University of Waterloo Lisa Jeffrey, University of Toronto Barbara Lee Keyfitz, Ohio State University Thomas S. Salisbury, York University Noriko Yui, Queen’s University

The Fields Institute is a centre for research in the mathematical sciences, located in Toronto, Canada. The Institutes mission is to advance global mathematical activity in the areas of research, education and innovation. The Fields Institute is supported by the Ontario Ministry of Training, Colleges and Universities, the Natural Sciences and Engineering Research Council of Canada, and seven Principal Sponsoring Universities in Ontario (Carleton, McMaster, Ottawa, Toronto, Waterloo, Western and York), as well as by a growing list of Affiliate Universities in Canada, the U.S. and Europe, and several commercial and industrial partners.

For further volumes: http://www.springer.com/series/10503

Ronald G. Douglas • Steven G. Krantz Eric T. Sawyer • Sergei Treil • Brett D. Wick Editors

The Corona Problem Connections Between Operator Theory, Function Theory, and Geometry

The Fields Institute for Research in the Mathematical Sciences

123

Editors Ronald G. Douglas Department of Mathematics Texas A&M University College Station, TX, USA Eric T. Sawyer Department of Mathematics and Statistics McMaster University Hamilton, ON, Canada

Steven G. Krantz Department of Mathematics Washington University St. Louis, MO, USA Sergei Treil Department of Mathematics Brown University Providence, RI, USA

Brett D. Wick Department of Mathematics Georgia Institute of Technology Atlanta, GA, USA

ISSN 1069-5265 ISSN 2194-1564 (electronic) ISBN 978-1-4939-1254-4 ISBN 978-1-4939-1255-1 (eBook) DOI 10.1007/978-1-4939-1255-1 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2014945546 Mathematics Subject Classification (2010): 30D55, 30H80, 46J15, 30H05, 47A13, 30H10, 30J99, 32A65, 32A70, 32A38, 32A35, 46J10, 46J20, 30H50, 46E25, 13M10, 26C99, 93D15, 46E22, 47B32, 32A10, 32A60 © Springer Science+Business Media New York 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Cover illustration: Drawing of J.C. Fields by Keith Yeomans Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

In June 2012, a workshop on the corona problem was held at the Fields Institute in Toronto, Ontario, Canada. The organizers were Ronald G. Douglas, Steven G. Krantz, Eric T. Sawyer, Sergei Treil, and Brett D. Wick. About forty people attended the workshop. The weeklong event was exciting, stimulating, and productive. Several new papers grew out of the interactions, and they appear in this volume. In particular, we offer a history of the corona problem—the first article of its kind. The other articles that we present describe various directions of research, and many offer new results. All of the articles were refereed to a high standard, and each represents original and incisive scholarship. We thank the Fields Institute, and particularly the Director Ed Bierstone, for providing a supportive and nurturing atmosphere, and financial support, for our mathematical work. We also thank the National Science Foundation for financial support. College Station, TX, USA St. Louis, MO, USA Hamilton, ON, Canada Providence, RI, USA Atlanta, GA, USA

Ronald G. Douglas Steven G. Krantz Eric T. Sawyer Sergei Treil Brett D. Wick

v

Contents

A History of the Corona Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ronald G. Douglas, Steven G. Krantz, Eric T. Sawyer, Sergei Treil, and Brett D. Wicks

1

Corona Problem for H1 on Riemann Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander Brudnyi

31

Connections of the Corona Problem with Operator Theory and Complex Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ronald G. Douglas

47

On the Maximal Ideal Space of a Sarason-Type Algebra on the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jörg Eschmeier

69

A Subalgebra of the Hardy Algebra Relevant in Control Theory and Its Algebraic-Analytic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marie Frentz and Amol Sasane

85

The Corona Problem in Several Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . 107 Steven G. Krantz Corona-Type Theorems and Division in Some Function Algebras on Planar Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Raymond Mortini and Rudolf Rupp The Ring of Real-Valued Multivariate Polynomials: An Analyst’s Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Raymond Mortini and Rudolf Rupp Structure in the Spectra of Some Multiplier Algebras . . . . . . . . . . . . . . . . . . . . . . . 177 Richard Rochberg

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Contents

Corona Solutions Depending Smoothly on Corona Data . . . . . . . . . . . . . . . . . . . . 201 Sergei Treil and Brett D. Wick On the Taylor Spectrum of M-Tuples of Analytic Toeplitz Operators on the Polydisk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Tavan T. Trent

A History of the Corona Problem Ronald G. Douglas, Steven G. Krantz, Eric T. Sawyer, Sergei Treil, and Brett D. Wicks

Abstract We give a history of the Corona Problem in both the one variable and the several variable setting. We also describe connections with functional analysis and operator theory. A number of open problems are described. Keywords Domain • Pseudoconvex • Corona • Maximal ideal space • Bounded analytic functions • Multiplicative linear functional

Subject Classifications: 30H80, 30H10, 30J99, 32A65, 32A70, 32A38, 32A35

R.G. Douglas () Department of Mathematics, Texas A&M University, College Station, TX 77843, USA e-mail: [email protected] S.G. Krantz Department of Mathematics, Washington University in St. Louis, St. Louis, MO 63130, USA e-mail: [email protected] E.T. Sawyer Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada e-mail: [email protected] S. Treil Department of Mathematics, Brown University, Providence, RI 02912, USA e-mail: [email protected] B.D. Wick School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA e-mail: [email protected] R.G. Douglas et al. (eds.), The Corona Problem, Fields Institute Communications 72, DOI 10.1007/978-1-4939-1255-1__1, © Springer Science+Business Media New York 2014

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1 Ancient History The idea of a Banach algebra was conceived by I. M. Gelfand in his thesis in 1936. A Banach algebra is a complex, normed algebra which is a complete Banach space in the norm metric. The theory of Banach algebras rapidly revealed itself to be a rich and powerful structure for attacking many different types of problems in analysis. It represented a beautiful marriage of analysis, functional analysis, algebra, and topology. It gave elegant, soft proofs of results in classical analysis (such as the Wiener Inversion Theorem, a particular case of Wiener’s Tauberian Theorem) that were quite difficult to prove by classical methods. In 1941, S. Kakutani posed the Corona Problem. The question concerned the maximal ideal space of the Banach algebra H 1 .D/, where D C is the unit disc. The only maximal ideals of this algebra that one can actually “write down” are the point evaluation functionals at z 2 D. The question was whether the point evaluation functionals are dense in the maximal ideal space (in the weak- topology). If the point evaluation functionals are not dense in the maximal ideal space of H 1 , then a big chunk of the maximal ideal space “sticks out” off the set of point evaluations (i.e., off the unit disc D), akin the sun’s Corona. People were fascinated by this question, but made little headway on it for twenty years or more. Some preliminary remarks on related ideas in function algebras appear in [32]. A lovely paper [57] was written in 1961 that laid the foundations for the study of the Corona Problem.1 Among the key results of [57] are the following: (a) The notion of fiber is introduced. The maximal ideals which are not point evaluations live in fibers over boundary points of the disc. (b) The paper gives a complete and explicit description of the S˘ılov boundary of H 1 .D/. (c) Even though any two fibers are homeomorphic, it is shown that the maximal ideal space less the disc (the point evaluations) is not the product of the circle with a fiber. (d) It is shown that each fiber contains a homeomorphic replica of the entire maximal ideal space. It is a good exercise in functional analysis to translate (recalling the definition of the topology in the maximal ideal space) the topological statement about density given above into the following algebraic Bezout formulation:

1 This paper is very entertaining because its author I. J. Schark is a fiction. “I. J. Schark” is actually an acronym for the authors Irving Kaplansky, John Wermer, Shizuo Kakutani, R. Creighton Buck, Halsey Royden, Andrew Gleason, Richard Arens and Kenneth Hoffman. The letters of “I. J. Schark” come from their first initials. The references to this paper, plus consultation with experts, show that virtually no work was done on the Corona Problem between 1941 and 1961.

A History of the Corona Problem

3

Suppose that f1 ; f2 ; : : : ; fk are bounded, analytic functions on the disc D that satisfy jf1 ./j C jf2 ./j C jfk ./j > ı > 0 8 2 D for some positive, real number ı. Then do there exist bounded, analytic functions g1 ; g2 ; : : : ; gk such that f1 ./g1 ./ C f2 ./g2 ./ C C fk ./gk ./ 1 ‹

It is a pleasure to thank Nikolai Nikolski for enlightening conversations about some of the topics of this paper. We also want to thank Ted Gamelin, Nessim Sibony, Richard Rochberg, and Tavan Trent for reading an early draft of this history and providing many interesting comments that ultimately improved the paper.

2 Modern History The Corona Problem was finally answered by Lennart Carleson in his seminal paper [19], which built on foundational work in [18]. Carleson’s paper was important not only for his main theorem, but for the techniques that he introduced to solve the problem. In particular, one of the main tools used in Carleson’s solution was the idea of Carleson measure, an idea that has become of pre-eminent importance in function theory and harmonic analysis. Carleson uses these measures to control the lengths of certain curves in the disc that wind around the zeros of a bounded analytic function. This construction was very clever and quite involved and has proved to be useful in other areas of mathematics. In particular, as pointed out by Peter Jones, [38]: The corona construction is widely regarded as one of the most difficult arguments in modern function theory. Those who take the time to learn it are rewarded with one of the most malleable tools available. Many of the deepest arguments concerning hyperbolic manifolds are easily accessible to those who understand well the corona construction. In the mid-1960s, Edgar Lee Stout [61] and Norman Alling [3] proved that the Corona Theorem remains true on a finitely-connected Riemann surface. By contrast, Brian Cole [29] gave an example of an infinitely connected Riemann surface on which the Corona Theorem fails. Cole’s counterexample was built by exploiting the connections between representing measures and uniform algebras. Around the same time, Kenneth Hoffman [35] showed that there is considerable analytic structure in the fibers of the maximal ideal space of H 1 . It may be mentioned that the paper [57] also constructs analytic discs in the fibers. In the remarkable paper [36], Lars Hörmander introduced a new method for studying the Corona Problem. His approach was first to construct a preliminary non-analytic solution of the Bezout equation, and then “correct” it to get an analytic one by solving an appropriate inhomogeneous @-equations. He used the Koszul complex to find these equations, although in the one complex variable case (he also

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considered some algebras of analytic functions in several variables) one can use elementary methods to deduce the equations, especially after the equations are already known. In the unit disc he used the result that the equation @w D ; has a bounded solution if is a Carleson measure (this fact can be easily proved by duality). To construct the preliminary solution, given a Carleson measure in the right hand side of the @-equation, Hörmander used the Carleson contours from the original Carleson construction, so the main technical difficulty remained. However, the main advantage of the new approach was that it allowed one to solve the Bezout equation with n functions (generators). The original Carleson construction gave a method of solving the equation with two generators. To move from two to n generators Carleson used a clever trick. This trick was based on the Riemann Mapping Theorem, so there was no hope to generalize it to higher dimensions. Hörmander’s paper raised hopes that the Corona Theorem can be generalized to higher dimensions, but the hopes for an easy generalization were squashed by N. Varopoulos [82], who had shown that in the unit ball in C2 , the Carleson measure condition on the right hand side does not imply existence of a bounded solution of the @-equation. In 1979 T. Wolff presented a new proof of the Corona Theorem, which followed Hörmander’s approach with one critical difference. Wolff used a different condition for the existence of a bounded solution of the @-equation @w D G, based on a “second order” Green’s formula. P That was crucial because, for P the trivial nonanalytic solution gk D f k = j jfj j2 of the Bezout equation k fk gk 1, the right sides of the corresponding @-equations satisfied this condition. In the remaining sections we discuss some of the major contributions of this history in more detail.

3 The Corona Theorem On the Disc Theorem 1 (Carleson, [19]). Suppose that f1 ; : : : ; fn 2 H 1 .D/ and there exists a ı > 0 such that ˇ ˇ 1 max fˇfj .z/ˇg ı > 0: 1j n

Then there exists g1 ; : : : ; gn 2 H 1 .D/ such that 1 D f1 .z/g1 .z/ C C fn .z/gn .z/ 8z 2 D

(1)


5

and gj 1 C.ı; n/ H .D/

8j D 1; : : : ; n:

An obvious remark isPthat the condition on the functions fj is clearly necessary. In [19] the assumption nkD1 jfk .z/j ı > 0 was used, but if one is not after sharp estimates, it does not matter what `p norm we use.

3.1 Carleson Embedding Theorem One of Carleson’s powerful and influential ideas in this development is the notion of Carleson measure, which was introduced earlier in [18] in connection with the interpolation problem. A nonnegative measure on the unit disc D that satisfies .S / C `

(2)

for any set S of the form S D fre i W r 1 `; 0 0 C `g is called a Carleson measure. Theorem 2 (Carleson Embedding Theorem). For any p > 0 the estimate (embedding) Z D

jf .z/jp d.z/ C1 kf kH p

8f 2 H p .D/

(3)

holds if and only if the measure is Carleson (i.e., satisfies (2)). Moreover, the best constant C1 is the same for all p > 0, and the best C1 is equivalent to the best C in (2) in the sense of two-sided estimates: A1 C C1 AC; where A is an absolute constant. The Carleson Embedding Theorem is now a staple of harmonic analysis, and many different proofs exist in the literature. The fact that the best C1 is the same for all p > 0 is an easy corollary of the Nevanlinna factorization of H p functions. Definition. The best constant C1 in (3) is called the Carleson norm of the measure . Note that sometimes the best constant C in (2) is used for the Carleson norm: since C1 and C are equivalent. If one does not look for exact constants, it does not matter what definition is used.

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3.2 Carleson’s Original Proof and Interpolation As was acknowledged in [19], Carleson’s original strategy for attacking the Corona Problem comes from D.J. Newman [47]. Newman had shown that the Corona Theorem follows from a certain interpolation result, which Carleson then proved. According to L. Ehrenpreis (see his recollection in the paper [86] dedicated to D.J. Newman’s memory) Newman’s contribution to the Corona Theorem was more significant than the credit he received. The original Carleson proof worked as follows. First, using a standard normal family argument one can assume without loss of generality that all the generators fk are holomorphic in a slightly bigger disc. Then a pretty straightforward (although not completely trivial) argument allows one to assume that one of the fk s, say fn , is a finite Blaschke product with simple zeroes. The next step reduced solution of the Bezout equation with two generators to a solution of an interpolation problem. Namely, if one wants to find g1 and g2 such that f1 g1 C f2 g2 1, and one of the generators, say f2 , is a Blaschke product with simple zeroes then finding a bounded solution of the interpolation problem g1 ./ D 1=f1 ./;

8 2 Z.f2 /

(4)

(Z.f / denotes the zero set of f ) solves the Bezout equation: one just needs to define g2 WD .1 f1 g1 /=f2 . Existence of a bounded solution of the above interpolation problem follows from the following theorem, which is the technical crux of the proof. Theorem 3. Let A be finite Blaschke product with simple zeroes. Assume that ı < 1=2 and that F is a holomorphic function in fz 2 D W jA.z/j < ıg bounded by 1. Then the interpolation problem g./ D F ./ ;

8 2 Z.A/ ;

(5)

has a solution g 2 H 1 with kf k1 C.ı/ < 1. To see how the solution of the interpolation problem follows from the above theorem, note that the condition (1) implies that jf2 .z/j 2ı whenever jf1 .z/j < 2ı . Therefore, applying Theorem 3 with 1=f2 on fz 2 D W jf1 .z/j < 2ı g being the (rescaled) function F , we get that (4) has a solution g1 2 H 1 , kg1 k1 1ı C.ı/ To prove the case of n generators, the following clever induction trick was used. Assuming that the theorem holds for n 1 generators, one can show that there exist bounded analytic functions p1 ; : : : ; pn1 defined on ˝ WD fz 2 D W jfn .z/j < 2ı g such that n1 X kD1

fk .z/pk .z/ 1;

8z 2 ˝:

(6)


7

Indeed, max

kD1;:::;n1

jfk .z/j

ı 2

8z 2 ˝:

By the maximum principle, connected components of ˝ are simply connected (and so conformally equivalent to D), therefore by the induction hypothesis the Bezout equation (6) has a bounded solution in each connected component of ˝. Finding bounded solutions (in all of D) of the interpolation problems gk ./ D pk ./;

8 2 Z.fn /

P and defining gn WD .1 n1 kD1 fk gk /=fn (recall that fn is a finite Blaschke product with simple zeroes), we get a bounded solution of the Bezout equation. The main technical part of the Carleson proof is the proof of Theorem 3. To prove this theorem, Carleson constructed what was later named the Carleson contour. Namely, he proved that, for ı < 1=2, there exists ".ı/, 0 < ".ı/ ı such that, for any A 2 H 1 and for any ı, 0 < ı < 1=2, there exists a domain ˝ D ˝A;ı such that 1. fz 2 D W jA.z/j < ".ı/g ˝ fz 2 D W jA.z/j < ıg; 2. the arclength on @˝ is a Carleson measure (with the norm depending only on ı). The boundary @˝ is now known as the Carleson contour. Remark. In Carleson’s paper ".ı/ D ı and the Carleson norm of the contour was estimated by C ı 1 . In [16] J. Bourgain constructed a Carleson contour (for an inner function) with the Carleson norm not depending on ı. The construction of the Carleson contour was rather technical and was based on the stopping moment technique—which was “stolen” from probability. The stopping moment technique is now a commonplace in harmonic analysis, and now people often refer to a decomposition obtained using stopping moment technique (similar to that used by Carleson, but often significantly more involved) as the “Corona decomposition.” After the Carleson contour is constructed the proof of Theorem 3 is fairly straightforward. Namely, any solution g of the interpolation problem (5) can be represented as g D g0 C Ah;

h 2 H 1;

where g0 is one of the solutions (the Lagrange interpolating polynomial, for example). By the Hahn–Banach Theorem, the smallest norm kgk1 D kg=Ak1 is the norm of the linear functional on H 1 Z Z Z Z fg d z fg0 d z fg0 d z fF dz D D D f 7! T B 2 T B 2 @˝ B 2 @˝ B 2

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(all the equalities above can be verified by the Residue Theorem). Since the arclength on @˝ is a Carleson measure, the last integral can be estimated using the Carleson Embedding Theorem for H 1 .

3.3 Hörmander’s Construction Suppose we constructed some functions 'k , that are bounded in the unit disc D and also solve the Bezout equation X

'k fk 1:

k

These functions are in general not analytic, and so to make them analytic we must “correct” them. To do this we define gj .z/ D 'j .z/ C

n X

aj;k .z/fk .z/

kD1

where the functions aj;k .z/ D ak;j .z/ are to be determined. The above alternating condition implies that n X j D1

fj .z/gj .z/ D

n X

fj .z/'j .z/ C

j D1

n X n X

aj;k .z/fj .z/fk .z/ D 1;

j D1 kD1

so the functions gk defined as above always solve the Bezout equation. To have the alternating condition aj;k D ak;j we set aj;k D bj;k .z/ bk;j .z/ for some yet to be determined functions. If we chose the functions bj;k , to be solutions to the following @ problem: @bj;k D 'j @'k WD Gj;k ; then @gj D @'j C

n X

fk @aj;k

kD1

D @'j C

n X

fk @bj;k @bk;j

kD1

D @'j C

n X kD1

fk 'j @'k 'k @'j

(7)


9 n X

D @'j C 'j @

! fk 'k @'j

kD1

n X

fk 'k

kD1

D @'j C 'j @1 @'j 1 D 0; so the functions gj are analytic. P Thus, finding an H 1 solution of the Bezout equation k fk gk 1 is reduced to finding a bounded solution of the @-equation (7). Note that, since at the end we are getting analytic functions gk , it is sufficient to prove that the solutions bj;k are bounded on the circle T.2 Hörmander used the fact that the equation @w D has a bounded on T solution whenever the variation jj is a Carleson measure; this fact can be easily obtained by a duality argument and using the Carleson Embedding Theorem. Note that, if the right side is a function G, then we just require that the measure GdA.z/ is a Carleson measure. For the bounded solutions 'k Hörmander chose 1 1˝k 'k WD fk

n X .1 1˝k /

!1 ;

kD1

where @˝k is the Carleson contour for fk with ı=n for ı. The derivative @'k in the definition of Gk can understood in the sense of distributions. If one wants to avoid the technical difficulties of working with distributions, one can consider smoothing out the characteristic functions 1˝k . Hörmander in [36] only sketched the proof, but did not give any details. A reader interested in all the details should look at J. Garnett’s monograph [30, Ch. VIII, Sect 5] where Hörmander’s construction was “smoothed out.” It should be mentioned that Hörmander’s approach to the corona problem, and perhaps Tom Wolff’s solution (discussed below) inspired Peter Jones [67] to come up with an interesting new way to attack the problem. Jones constructs, on the upper halfplane, a nonlinear solution to the @ problem and uses that together with the Koszul complex to effect a corona solution.

2 Of course there are some technical details about interpreting the boundary values of the function bj;k , but one can avoid such difficulties by first assuming that the corona data is analytic in a bigger disc and then using a standard normal family argument.

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3.4 Wolff’s Solution T. Wolff’s solution followed Hörmander’s construction with the (simplest possible) bounded non-analytic solutions 'k given by fk : 2 j D1 jfj j

'k WD Pn

(8)

To prove the existence of bounded solutions of the @-equations (7) on T, he introduced the following theorem. Theorem 4 (Wolff, [7, 30, 40, 42, 48]). Suppose that G.z/ is bounded and smooth on the closed disc D. Further assume that the measures jGj2 log

1 dA.z/ jzj

j@Gj log

and

1 dA.z/ jzj

are Carleson measures with Carleson norms B1 and B2 respectively. Then the equation @b D G has a solution b 2 C 1 .D/ \ C.D/ and, moreover, p kbkL1 .T/ C1 B1 C C2 B2 where C1 and C2 are absolute constants. The proof of this theorem is a clever application of Green’s Theorem together with the Carleson Embedding Theorem. If one does not care about sharp estimates, the Corona Theorem follows from Theorem 4 almost immediately. Namely, it is an easy exercise to show that, if f 2 1 H 1 , then the measure jf 0 .z/j2 ln jzj dA.z/ is Carleson with the Carleson norm at most C kf k1 . Then the Corona Theorem follows immediately if one notes that, for G D Gj;k D 'j @'k with 'k defined by (8), we have jGj2 ; j@Gj C

n X

jfl j2 :

lD1

To get sharper estimates, the following lemma attributed to A. Uchiyama can be used.


11

Lemma 1 (Uchiyama’s lemma). Let u, 0 u 1, be a subharmonic function. 1 Then the measure u.z/ ln jzj dxdy is a Carleson measure with Carleson norm at most 2e, meaning that Z jf .z/j2 u.z/ ln D

1 dxdy 2ekf k2H 2 jzj

8f 2 H 2 :

The proof of the lemma is, again, a clever application of Green’s formula. It can be found, for example, in the monograph [48, Lemma 6 in Appendix 3].

3.5 Estimates and Corona Theorems with infinitely many generators In 1980 M. Rosenblum [55] and V. Tolokonnikov [64], using a modification of T. Wolff’s proof, independently proved that with correct normalization the Corona Theorem holds for infinitely many generators fk . Theorem 5 (M. Rosenblum [55], V. Tolokonnikov [64]). Let fk 2 H 1 , k 2 N satisfy 1

1 X

!1=2 jfk .z/j

ı>0

2

8z 2 D:

(9)

kD1

Then there exist functions gk 2 H 1 , such that 1 X

P k

fk gk 1 and such that

!1=2 jgk .z/j

2

C.ı/

kD1

In both [55] and [64], C.ı/ D C ı 4 . Later A. Uchiyama, in an unpublished but extremely influential preprint [81], proved the above theorem with C.ı/ D C ı 2 ln ı 1 for small ı, which remains the best known to date (even in the case of 2 generators). For the proof of this result the reader could look at [48, Appendix 3]. The main idea is to use Wolff’s method, estimating aj;k D bj bk not separately, but estimating instead the Hilbert–Schmidt norm of the matrix .aj;k /1 j;kD1 . To do that one treats the system of @-equations as a vector-valued equation and uses Lemma 1 with appropriately chosen functions u. As in the proof of Theorem 4 the norm of the solution is estimated by duality.

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In [81] A. Uchiyama also proved that the Corona Theorem with the `1 normalization of the corona data as in (1) also holds with infinitely many generators. Namely, he proved that if the functions fk 2 H 1 satisfy the condition 1 max jfk .z/j ı > 0

8z 2 D;

kD1;:::;n

then there exist gk 2 H 1 such that 1 X

P1 kD1

fk gg 1 and

jgk .z/j C.ı/

8z 2 D:

kD1

To prove his result he used a modification of the Carleson–Hörmander construction with Carleson contours. Note that, using the above result and Theorem 5 it is easy to show, see [39], that for any p 2 and for any sequence of functions fk 2 H 1 satisfying 1

1 X

!1=p jfk .z/jp

ı>0

8z 2 D;

kD1

there exist gk 2 H 1 such that

P

1 X

k

fk gk 1 and !1=p0

jgk .z/j

p0

C.ı/;

kD1

where 1=p C 1=p 0 D 1. Uchiyama’s estimate C.ı/ D C ı 2 ln.1=ı/ in Theorem 5 is close to optimal, if not optimal. Namely, V. Tolokonnikov [65] has shown that the estimate cannot be better than C ı 2 , even in the case of two generators. This result was later improved by S. Treil [69], who had shown that the estimate cannot be better than C ı 2 ln ln.1=ı/. This looks like a silly “improvement,” but it allowed the author to solve T. Wolff’s problem [34, Problem 11.10] about ideals of H 1 ; for more details see Section 7 below.

3.6 Matrix and Operator Corona Theorems The Corona Problem admits the following interpretation/generalization. Let F be a bounded n m matrix, n > m with H 1 entries, which has a bounded left inverse. The question is whether F has a bounded and analytic left inverse? The left invertibility (in H 1 ) of bounded analytic matrix- or operator-valued functions,


13

play an important role in operator theory (such as the angles between invariant subspaces, unconditionally convergent spectral decompositions, computation of spectrum, etc.). The condition that F has a bounded left inverse means that F .z/F .z/ ı 2 I > 0;

8z 2 D:

(C)

In the case when F is a column (m D 1), this is exactly the assumption (9) of Theorem 5. Using a simple linear algebra argument. P. Fuhrmann [27] deduced the positive answer to the above question (for the case of finite matrices) from the Corona Theorem. The generalization of this problem is the so-called Operator Corona Problem, dealing with the space HE1!E .D/ of bounded analytic functions on D whose values are bounded operators acting between separable Hilbert spaces E and E. The question is that, given F 2 HE1!E .D/ satisfying (C), does there exist G 2 1 HE!E .D/ such that GF I ? This problem was posed by Sz.-Nagy in in 1978, [62], in connection with problems in operator theory (see also [33, Problem S4.11] for this problem and some comments). Fuhrmann’s result gives the positive answer in the case dim E ; dim E < 1. Theorem 5 says that it is also true if dim E D 1, dim E D 1. Using Theorem 5 and a modification of Fuhrmann’s proof, V. Vasyunin was able to extend this result to the case dim E < 1, dim E D 1; see also a paper [78] by T. Trent, where better estimates were obtained. In the general situation dim E D dim E D 1, the condition (C) does not imply the existence of a left inverse in H 1 : a corresponding counterexample, showing that the estimates on the norm of the left inverse G blow up as dim E ! 1, was constructed by S. Treil see [71] or [70]. Later in [68] he presented a different counterexample, giving better lower bounds for the norm of the solution. Note that the lower bounds in [68], obtained for the case dim E D n, dim E D n C 1 were very close to the upper bounds obtained by T. Trent in [78] for the general case dim E D n, dim E > n. While, as we discussed above, the Corona Theorem (i.e., the fact that the condition (C) implies the existence of a bounded analytic left inverse) fails in the general (infinite dimensional) case, it still holds in some particular cases. For example, the Operator Corona Theorem holds if the range F .D/ is relatively compact. Some particular results in this direction were obtained by P. Vitse [83,84]; in full generality it was proved recently by A. Brudnyi [17], using tools of complex geometry. Another partial result belongs to S. Treil [66], who proved in particular that the Operator Corona Theorem holds for functions F which are “small” perturbation of left invertible H 1 functions. For example, it holds if F D F0 C F1 , where F0 ; F1 2 HE1!E .D/, F0 is left invertible in H 1 , and the Hilbert–Schmidt norm of F .z/, z 2 D is uniformly bounded (by an arbitrary large constant).

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We can also mention a result of S. Treil and B. Wick [72], who introduced a curvature condition which together with (C), guarantees the existence of an H 1 left inverse. Namely, let ˘.z/ be the orthogonal projection onto Ran F .z/. It was proved in [72] that if k@˘.z/k C =.1 jzj/ and the measure k@˘.z/k2 log

1 dA.z/ jzj

is Carleson, then condition (C) implies the left invertibility in H 1 . In the case dim E < 1, the above curvature conditions easily follow from (C) (in fact they are equivalent to (C) under an extra assumption that the function F is co-outer). In the case codim Ran F .z/ < 1, the curvature conditions follow from the left invertibility (in H 1 ) of F , so [72] solves the Operator Corona Problem in the case of finite codimension. The proofs in [72] used the following surprising lemma discovered by N. Nikolski, which connects the solvability of the Corona Problem (in a general complex manifold ˝) with the geometry of the family of subspaces Ran F .z/, z 2 ˝. Lemma 2 (Nikolski’s Lemma). F Let F 2 HE1 !E .˝/ satisfy F .z/F .z/ ı 2 I;

8z 2 ˝:

1 .˝/ such that Then F is left invertible in HE1 !E .˝/ (i.e., there exists G 2 HE!E 1 GF I ) if and only if there exists a function P 2 HE!E .˝/ whose values are projections (not necessarily orthogonal) onto F .z/E for all z 2 ˝. Moreover, if such an analytic projection P exists, then one can find a left inverse 1 G 2 HE!E .˝/ satisfying kGk1 ı 1 kPk1 .

4 Other Domains The situation with the Corona Theorem on domains other than the unit disc can be summarized as follows: There is no domain in the complex plane for which the H 1 Corona Theorem is known to fail. There is no domain in Cn , n > 1, for which the H 1 Corona Theorem is known to hold.

4.1 Several Complex Variables In several variables it is trivial to construct a counterexample to the Corona Theorem, because of the phenomena of forced analytic continuation. Of course,


15

such a trivial “counterexample” would be cheating; the natural question to ask is whether the Corona Theorem holds for domains of holomorphy.3 In 1973, Nessim Sibony [59] produced a startling result in the context of several complex variables. He gave an elementary construction to produce a domain of holomorphy U in C2 with the property that any bounded holomorphic function f on U analytically continues to a strictly larger open domain UO . Here UO does not depend on f . The domain U does not have smooth boundary. Sibony’s ideas were later developed and generalized by Berg [15], Jarnicki and Pflug [37], and Krantz [41]. In the paper [60], Sibony modified his construction so as to produce a counterexample to the corona in C3 with smooth boundary. It is notable that this domain is strongly pseudoconvex except at one boundary point. In the paper [26], Fornæss and Sibony produce an example in C2 . However, if one requires ˝ to be a strictly pseudoconvex domain, no counterexamples are known; no positive results are known either, so the question of whether the H 1 Corona Theorem holds for such domains remains wide open. Nothing is known even in the simplest case when ˝ is the unit ball Bn in Cn . One might think that the product structure of the polydisc Dn could provide us with a better understanding of the Corona Problem there, but again, nothing is known in this case. As indicated before, N. Varopoulos [82] crushed the hopes of easily transferring the techniques for the disc to the case of several complex variables by showing that in the unit ball in C2 the condition that the measure is Carleson does not imply existence of a bounded solution of the @-equation @u D . In the positive direction, however, he was able to get BMO solutions of the Bezout equation f1 g1 Cf2 g2 1 with 2 generators in bounded pseudoconvex domains in Cn (assuming, of course, that the Corona data satisfy the Corona condition jf1 j C jf2 j ı > 0). Recently, Costea, Sawyer, and Wick [24] have extended this result to the case of k generators, or even infinitely many generators (but only in the ball in Cn ). Other positive results in several variables include the solution of the so-called H p Corona Problems, see Section 5 below.

4.2 Planar domains and Riemann surfaces For planar domains the situation is in a sense opposite to the one in the case of several variables: there are some positive results, but no counterexamples. Of course, the Riemann Mapping Theorem implies that the Corona Theorem holds for any simply connected domain. As we mentioned above, Edgar Lee Stout [61] and Norman Alling [3] proved in the mid-1960s that the Corona Theorem remains true on a finitely-connected Riemann surface (so in particular for finitely

3 That is, a domain supports a (unbounded) holomorphic function that cannot be analytically continued to any larger domain.

16

R.G. Douglas et al.

connected planar domains). Moreover, T. Gamelin proved in [28] that the solution of the Corona Bezout equation on finitely connected Riemann surfaces can be estimated by C.m; n; ı/, where m is the number of boundary components, n is the number of generators. In other words, the estimates in the Corona Theorem on Riemann surfaces can be made uniform on surfaces of fixed connectivity. In contrast, Brian Cole gave an example of an infinitely connected Riemann surface on which the Corona Theorem fails. Cole did not publish his counterexample, but it can be found in Gamelin’s book [29]. Cole’s counterexample was built by exploiting the connections between representing measures and uniform algebras. Cole’s construction does not work for planar domains, and the question of whether Corona Theorem is true for an arbitrary planar domain remains open. For planar domains the deepest result is probably the one by J. Garnett and P. Jones [31], who proved that the Corona Theorem holds for Denjoy domains. Recall that Denjoy domains are connected domains ˝ C such that C n ˝ R.

5 Toeplitz and H p Corona Theorems In the 70’s the importance of the Corona Theorem in operator theory was well understood, and people were trying to get an operator-theoretic proof of the theorem. This program was not completely realized, and only a weaker form, the so-called Toeplitz Corona Theorem, was proved in 1978 by Schubert [58]; see also a slightly earlier paper by B. Arveson [13]. Unpublished proofs of this fact by Ron Douglas and Bill Helton were also circulating about the same time. Recall that, for F 2 HE1 !E .D/, the analytic Toeplitz operator TF W HE2 ! HE2 and its adjoint TF are given by TF f D Ff;

TF f D PC .Ff /;

where PC is the orthogonal projection in L2 .T/ onto H 2 . 1 Theorem 6 (Toeplitz Corona Theorem). Suppose that F 2 HE!E .D/. Then F 1 1 is right invertible in H (i.e., there exists G 2 HE !E .D/ such that F G I ) if and only the Toeplitz operator TF is right invertible, or equivalently, the operator TF is left invertible. Moreover, if

kTF ukH 2 ı 2 kuk2H 2 2

E

8u 2 HE2 .D/;

(10)

then one can find a right inverse G 2 HE1 !E satisfying kGk1 1=ı. Remark. Theorem 6 is stated as a theorem about right invertibility of F in H 1 , while the Operator Corona Problem, see Section 3.6, deals with left invertibility of F . But, fixing orthonormal bases in E and E and replacing the matrix of F by its


17

transpose, we interchange the right and left invertibility, so the above theorem is a statement about the Operator Corona Problem. If the reader prefers to work in coordinate-free notation, replacing F by F ] , F ] .z/ WD .F .z// also switches right and left invertibility. This theorem was initially proved for the case dim E D 1, when F D .f1 ; f2 ; : : : ; fn/, but the proof is essentially the same for the general operator-valued case. The proof is not hard: it essentially an easy reduction to the Commutant Lifting Theorem, which is an extremely important (but not hard to prove) fact in operator theory. Theorem 6 looks like a huge success, but unfortunately it does not give an operator-theoretic proof of the Corona Theorem. There is no easy way to check that, in the case dim E D 1, dim E < 1 (so F D .f1 ; f2 ; : : : ; fn /), that the Corona Condition (9) implies the left invertibility of TF (equivalently, right invertibility of TF ). Note that, for F D .f1 ; f2 ; : : : ; fn /, the left invertibility of TF together with the estimate (10) can be rephrased as the following statement: For all h 2 H 2 .D/, there exists k1 ; : : : ; kn 2 H 2 .D/ such that n X j D1

fj kj h

and

n X 2 kj 2 ı 2 khk2 2 : H H

(11)

j D1

The fact that the Corona Condition (9) implies the above statement is known in the literature as the H 2 Corona Theorem. Unfortunately, for the disc its proof is not simpler than the proof of the Corona Theorem itself; one has to do essentially the same estimates as in the proof of the Corona Theorem. So, in the unit disc, the proof of the Corona Theorem can be split into two parts: an “easy” operator-theoretic one (the Toeplitz Corona Theorem) and a “hard” one (the H 2 Corona Theorem), with the proof about as difficult as the proof of the original Corona Theorem. The situation becomes quite different in several complex variables. Namely, the “hard” part (the H 2 , and even H p , 1 < p 1 Corona Theorems) can be generalized to several variables for appropriate domains, but no generalization of the “easy” Toeplitz Corona Theorem is known for several variables. Here by a generalization we mean not just some generalization, but one that together with the H p Corona Theorem gives a proof of the full Corona Theorem. As for the H p Corona Theorem, the following statement holds if ˝ is a ball (or a general strictly pseudoconvex domain in Cn ) or a polydisc. ˇ2 ˇ P Theorem 7. Suppose that f1 ; : : : ; fm 2 H 1 .˝/ with 1 nj D1 ˇfj .z/ˇ ı > 0 for all z 2 ˝ and let k 2 H p .˝/. Then there are h1 ; : : : ; hm 2 H p .˝/ such that Pm (i) fj hj D k; PjmD1 p (ii) j D1 hj H p .˝/ C.ı; m; p; ˝/.

18

R.G. Douglas et al.

As we mentioned above, this theorem holds for ˝ D D, and the proof is essentially the proof of the classical Corona Theorem. Extension of this theorem to several variables turned out to be possible, partially because the spaces Lp , 1 < p < 1 are “nice,” unlike the space L1 . Note, that the extension to several variables, while using some ideas from the one-dimensional case, is are far from trivial: new techniques and ideas were introduced to prove this result. In particular, using the Koszul complex one gets not only the first order @-equations, but the higher order equations as well. There has been substantial work on questions of this type; the interested reader should consult [5,8–12,23,43,46,49,50,73,75,78] for the case of H p type questions, as well as results in more general spaces of functions. In particular, one can consider the question when the data are given by multipliers on certain spaces of analytic functions. We now focus on a couple of the results above. In [5] Amar and Bruna proved the corresponding H p result on the unit ball. A similar result was obtained in the case of weakly pseudoconvex domains by Andersson in [8,9]. The case of the polydisc was first studied by Lin in [43]. The interested reader should also consult more recent treatments in [73, 75]. We exaggerate a bit in saying that no Toeplitz Corona Theorem is known for several variables. In fact, the following theorem shows that the solvability of the Bezout equation in H 1 is equivalent to left invertibility of a family of Toeplitz operators. Although the Agler–McCarthy theorem and its proof were initially restricted to n D 2, Amar managed to extend it to Dn and to Bn . See [4] for this extension. Theorem 8 (Agler–McCarthy [2], Amar [4]). Let ˝ denote either Dn or Bn and m 1 1 let ffj gm j D1 H .˝/. Then there exist fgj gj D1 H .˝/ with m X

fj gj 1 and sup

m X

jgj .z/j2

z2˝ j D1

j D1

1 ı2

if and only if TF .TF / ı 2 I

for all probability measures on @˝.

In the above theorem, TF is the Toeplitz operator with symbol F on the space H 2 .˝I /. Note that both the Agler-McCarthy and the Amar theorems are suggesting that one attempt to prove a family of Hilbert space Corona-type problems. In work of Trent and Wick, [80], it is shown how to reduce the family of Hilbert space problems that need to be considered. Additional interesting results have been obtained in [25, 52].


19

6 Corona Theorems in Multiplier Algebras The idea of splitting the Corona Problem into Toeplitz Corona and H 2 Corona Problems did not yield a solution of the Corona Problem for the classical domains (ball or polydisc or a strictly pseudoconvex domain). But the Corona Problem makes sense for different algebras of analytic function. Of course, for algebras where polynomials are dense (like the algebra of analytic functions continuous to the boundary) the Corona Theorem is trivially true, since it can be easily seen that the maximal ideal space in this case just the closure (in Cn ) of the domain. From the operator theory point of view the algebra H 1 is just the multiplier algebra of the Hilbert space H 2 , i.e., any bounded operator on H 2 commuting with the multiplication by independent variables is given by multiplication by a function ' 2 H 1 , and the norm of this operator coincides with k'k1 . From the point of view of operator theory it is interesting to consider the Corona Problem for multiplier algebras of different spaces of analytic functions. And it turns out that, for some multiplier algebras, the above program can be carried through: a corresponding Toeplitz Corona Theorem can be obtained using operator-theoretic method, and an appropriate analogue of the H p Corona Theorem could be proved by using “hard” harmonic analysis.

6.1 Toeplitz Corona Theorem and Complete Nevanlinna-Pick Kernels It turns out that the Toeplitz Corona Theorem holds for reproducing kernel Hilbert spaces with complete irreducible Nevanlinna–Pick kernels (see [1] for the definition). For a Hilbert space H of holomorphic functions in an open set ˝ in Cn , the set of multipliers MH can be defined as the collection of analytic functions f on ˝ such that the operator Mf , Mf h D f h is a bounded operator in H . The norm of f in MH is just the norm of the operator Mf ; endowed with this norm, MH is a Banach algebra. We also need to consider a direct sum ˚N 1 MH of the multiplier algebras: for N f D .f˛ /N 2 ˚ H we again define its norm kf kMH D kf kMult.H ;˚N H / as ˛D1 1 the norm of the multiplication operator Mf W H ! ˚N 1 H ;

Mf h D .f˛ h/N ˛D1 ;

h2H:

Note that v uN uX Mf 2 ; max Mf˛ MH kf kMult.H ;˚N H / t ˛ MH

1˛N

˛D1

20

R.G. Douglas et al.

so kf kMult.H ;˚N H / is equivalent to one of the standard norms on the direct sum of Banach spaces. The following Toeplitz Corona Theorem is due to Ball, Trent and Vinnikov [14] (see also Ambrozie and Timotin [6] and Theorem 8.57 in [1]). Theorem 9 (Toeplitz Corona Theorem). Let H be a Hilbert function space in an open set ˝ in Cn with an irreducible, complete Nevanlinna-Pick kernel. Let N ı > 0 and N 2 N, and let f D .f˛ /N ˛D1 2 ˚1 MH . The following statements are equivalent: N (i) f D .f˛ /N ˛D1 2 ˚1 MH satisfies the following “Baby Corona Property,” satisfies namely for every h 2 H , there are g1 ; : : : ; gN 2 H such that

g1 .z/ f1 .z/ C C gN .z/ fN .z/ D h .z/ ;

z 2 ˝;

and kf1 k2H C C kfN k2H

1 khk2H ; ı

(12)

N (ii) f D .f˛ /N ˛D1 2 ˚1 MH satisfies the following “Multiplier Corona Property,” N i.e., there exists ' D .'˛ /N ˛D1 2 ˚1 MH such that

g1 .z/ '1 .z/ C C gN .z/ 'N .z/ D 1;

z 2 ˝;

and k'kMult.H ;˚N H /

1 : ı

(13)

This theorem is an abstraction of Theorem 6; the Hardy space H 2 .D/ is a reproducing kernel Hilbert space with the complete irreducible Nevanlinna-Pick kernel. One can say that such spaces in many respects behave like the Hardy space H 2 .D/. The proof of this theorem uses purely operator-theoretic methods, as one can guess from the equality of the constants. Unfortunately, as the reader could guess, the Hardy space H 2 .Bn / on the ball Bn in Cn , n 2 does not possess a complete Nevanlinna–Pick kernel, so the above theorem does not apply to H 1 .Bn /. A similar problem persists in the polydisc Dn .

6.2 Baby Corona Theorem for Besov–Sobolev spaces and the Corona Theorem for their multipliers A well-known class of functions with complete irreducible Nevanlinna–Pick kernels is the Besov-Sobolev space of analytic functions, B2 .Bn / with 0 1=2.


21

Recall that the space B2 .Bn / is the collection of holomorphic functions f on the unit ball Bn such that kf kB2 .Bn /

( m1 ˇ2 Z X ˇˇ ˇ D ˇf .k/ .0/ˇ C kD0

Bn

) 12 ˇ ˇ2 mC ˇ ˇ ˇ 1 jzj2 f .m/ .z/ˇˇ d n .z/ < 1; ˇ

n1 where d n .z/ D 1 jzj2 d V .z/ is the invariant measure on Bn and m is any n integer satisfying m C > 2 (the space does not depend on the choice of m; by choosing a different m one gets an equivalent norm). This is a reproducing kernel Hilbert space with kernel given by k .z/ D

1 .1 z/2

:

In particular note that B2 recovers the well known spaces of analytic functions such as the Hardy space, the Bergman space, and the Dirichlet space. For example choosing D 0 corresponds to the Dirichlet space, D n2 is the Hardy space, corresponds to the Bergman space, while D 12 is the Drury-Arveson D nC1 2 Hardy space. The notion of the mth derivative taken in the definition is also purposely vague. Any reasonable choice of mth derivative will provide the same function space. The “Baby Corona Property” (statement (i) of the above Toeplitz Corona Theorem) looks very much like the conclusion of the H p Corona Theorem (with H instead of H 2 and MH instead of H 1 ). As it was recently demonstrated by Costea, Sawyer and Wick, [24] the Corona Condition (14) below implies the “Baby Corona property” for the whole scale of Besov–Sobolev spaces. Theorem 10 (Costea, Sawyer, Wick, [24]). Let 0 and 1 < p < 1. Given f1 ; : : : ; fN 2 MBp .Bn / satisfying 0 0:

kD1

It is easy to see that both (i) and (ii) each individually imply (iii), the important part of the argument is to show that (iii) implies (i) and (ii). The proof of Proposition 1 follows the lines of Wolff’s proof of the Corona Theorem for H 1 .


23

Note that an alternative proof can be given by using the Toeplitz Corona Theorem 9, and the proof that (iii) implies (ii). This was the strategy employed by Trent in [77] to handle the case of an infinite number of generating functions. We also mention that results of this type were studied much earlier by Tolokonnikov, [65] and was essentially obtained via connections with Carleson’s Corona Theorem. While these results present a class of function spaces for which the Corona Theorem is true, we also want to point out the following very interesting result of Trent, [74], where he demonstrates that there are reproducing kernel Hilbert spaces for which the corresponding Corona question fails for the multiplier algebra. Theorem 12 (Trent, [74]). There exists a domain ˝, a reproducing kernel space H˝ over the domain ˝ and a function g 2 MH˝ so that g1 2 H˝ and jg.x/j

for all x 2 ˝, but g1 … MH˝ .

7 The Ideal Problem An alternative, algebraic, viewpoint to the Carleson’s Corona Theorem, Theorem 1 is to ask for necessary and sufficient conditions so that the function 1 is in the ideal generated by the functions ffj gN j D1 . This ideal can be written as I .f1 ; : : : ; fN / D

8 N <X :

fj gj W gj 2 H 1 .D/

j D1

9 = ;

and, once 1 2 I.f1 ; : : : ; fN /, then we immediately have H 1 .D/ D I .f1 ; : : : ; fN /. A natural extension of Carleson’s Corona Theorem is the following question: Question 1. Suppose that '; f1 ; : : : ; fN 2 H 1 .D/. Give necessary and sufficient conditions so that the function ' 2 I .f1 ; : : : ; fN /. This question was raised by Rubel and Shields in [56]. It is obvious that a necessary condition to answer this question is that N X ˇ ˇ ˇfj .z/ˇ C j' .z/j ;

z 2 D:

(15)

j D1

As a counterexample of Rajeswara Rao [53] shows, the condition (15) is, in general, not sufficient for ' 2 I .f1 ; : : : ; fN /. The above condition (15) is sufficient for ' to be in the ideal in some particular cases, for example when ' is an interpolating Blaschke product: see the work of K.C. Lin, [44, 45], and V. Tolokonnikov [63]. But since in general the condition is not sufficient, the following question is a very natural one to ask.

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R.G. Douglas et al.

Question 2. Let h be a continuous increasing function on Œ0; 1/. Let N 2 and suppose that f1 ; : : : ; fN ; ' 2 H 1 .D/ and satisfy 1 N X ˇ ˇ ˇfj .z/ˇA j'.z/j ; h@ 0

z 2 D:

(16)

j D1

For which functions h does this condition always imply that ' belongs to the ideal generated by the f1 ; : : : ; fN , i.e., ' 2 I .f1 ; : : : ; fN /? The function h.x/ D x p is probably the most natural choice for the gauge function h, so the Question 2 for these functions is of particular interest. One can ask a similar question for the closed ideals (in the norm of H 1 ). Question 3. Let h be a continuous increasing function on Œ0; 1/. Let N 2 and suppose that f1 ; : : : ; fN ; ' 2 H 1 .D/ and satisfy (16). For which functions h does this condition always imply that ' belongs to the closed (in the norm of H 1 ) ideal generated by the f1 ; : : : ; fN ? Question 3 was settled by J. Bourgain [16]: he showed that if limt!0C h.t/ D 0, t then (16) implies that ' is in the closed ideal generated by f1 ; : : : ; fn , and that for h.t / t the implication fails. In the same paper [16] J. Bourgain had shown using a modification of Rao’s counterexample that, for h.t / t p , p < 2 the condition (16) is not sufficient for ' to be in the ideal generated by f1 ; : : : ; fN . On the other hand, modifying T. Wolff’s proof of the Corona Theorem, it is possible to show that condition (16) with h.t / D t p , p > 2 implies that ' 2 I .f1 ; : : : ; fn /. The question of what happens in the case p D 2, i.e., whether the condition (16) for h.t / t 2 implies the ' 2 I .f1 ; : : : ; fn / is essentially the famous question of T. Wolff, see [34, Problem 11.10]. We say here “essentially,” because his question was about a related, but formally weaker, statement. Namely, he asked whether the condition (15) implies that ' 2 2 I .f1 ; : : : ; fn /. Both questions remained open for almost 2 decades, until they were answered negatively by S. Treil in [69]. After the above mentioned result by Treil, Problem 2 is well understood for the scale of gauge functions h.t / t 2 : the answer is positive for p > 2 and negative for p 2. So the natural question is to consider a finer scale of gauge functions near the critical exponent p D 2, for example the functions h.t / t 2 =.ln t 2 /˛ , or even include higher order logarithms. In fact, investigation in this direction started before the question about what happens at the critical exponent p D 2 was settled in [69]. For example, it was shown by K. C. Lin [44] that the answer is positive for h.t / t 2 =.ln t 2 /˛ , ˛ > 3=2. This result was improved by U. Cegrell [20, 21] who had shown that the answer is affirmative for the function h.t / D

t2 .ln t 2 /3=2 .ln ln t 2 /3=2

ln ln ln t 2

;


25

and later by J. Pau [51] who had shown that the function h.t / D

t2 .ln t 2 /3=2

ln ln t 2

works as well (see also the paper [76] by T. Trent, where a similar, but a bit weaker, result was proved). If the exponent 3=2 on the ln t 2 looks suspicious to the reader, it indeed is just an artifact of the methods used to prove these results. As was shown by S. Treil [67], the answer is positive for any function h of form h.t / D Rt 2 .ln t 2 /, where 1 W RC ! RC is a bounded non-increasing function satisfying 0 .x/ dx < 1. In particular, any of the functions behaving near 0 as 2 2 1C" : : : ln : : : ln h.t / D t 2 =..ln t 2 /.ln ln t 2 / : : : .ln … t /.ln … t / /; „ lnƒ‚ „ lnƒ‚ m times

">0

mC1 times

works. Thus, ˛ D 3=2 is definitely not the critical exponent, the critical exponent ˛ is at least 1. It was conjectured in [67] that the above result is sharp, meaning that for any bigger function h the answer is negative. The nice closed form of the result supports this conjecture, but no supporting results were obtained to date. The above results dealt with description of the “smallness” condition guaranteeing that all the functions ' satisfying this conditions belong to I .f1 ; : : : ; fN /. It would also be very interesting to obtain necessary and sufficient conditions for an individual ' to be in I .f1 ; : : : ; fN /; the condition could, for example, be in terms of curvature like the conditions appearing in [72]. The ideal problem in several variables has been largely unstudied, mostly due to lack of progress on the Corona Problem in several complex variables. However, some preliminary results in several variables were obtained in articles [22, 54]. We also point out that the question of ideals has essentially not been explored in the case of multiplier algebras either, though there is recent work by Trent and Banjade [79] examining this matter.

8 Concluding Remarks This history article has surveyed the points of the theory of the Corona Problem with which the authors are most familiar. And, while we have endeavored to give proper credit and to reference all the relevant literature, there likely are some unintentional omissions. We apologize in advance for any such lapses. This survey unfortunately must omit the numerous connections between the Corona Problem and H 1 control theory; this is an essential and interesting story in which applied mathematics interacts powerfully with complex function theory. We have also omitted the story of more refined versions of the Corona Problem that can be formulated and the connections with the stable rank of the

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algebra H 1 . The interested reader can find these connections and information through a MathSciNet search. Additionally, the references listed in this article point to numerous other interesting paper related to the Corona Problem, some of which we unfortunately weren’t able to discuss at all. Acknowledgements Research of Sergei Treil supported in part by National Science Foundation DMS grant # 0800876. Research of Brett Wick supported in part by National Science Foundation DMS grant # 955432.

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Corona Problem for H 1 on Riemann Surfaces Alexander Brudnyi

Abstract In this paper we survey some results and methods related to the famous corona problem for algebras H 1 of bounded holomorphic functions on Caratheodory hyperbolic Riemann surfaces. Keywords Corona theorem • Maximal ideal space • Bounded holomorphic function • Covering • Riemann surface of finite type Subject Classification: Primary 30D55; Secondary 30H05

1 Corona Problem Let X be a complex manifold and H 1 .X / be the Banach algebra of bounded holomorphic functions on X equipped with the supremum norm. We assume that X is Caratheodory hyperbolic, that is, the functions in H 1 .X / separate the points of X . The maximal ideal space M D M .H 1 .X // is the set of all nonzero linear multiplicative functionals on H 1 .X /. Since the norm of each 2 M is one, M is a subset of the closed unit ball of the dual space .H 1 .X // . It is a compact Hausdorff space in the Gelfand topology, the weak topology induced by .H 1 .X // . There is a continuous embedding i W X ,! M taking x 2 X to the evaluation homomorphism f 7! f .x/, f 2 H 1 .X /. The complement to the closure of i.X / in M is called the corona. The corona problem is: given X to determine whether the corona is empty. It was first posed in the case of the unit disc D in C by S. Kakutani in 1941. The complement of the closure of i.D/ in M was called the corona by D. Newman [N] (as in this case there would have been a set of maximal ideals

A. Brudnyi () Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N 1N4 e-mail: [email protected] R.G. Douglas et al. (eds.), The Corona Problem, Fields Institute Communications 72, DOI 10.1007/978-1-4939-1255-1__2, © Springer Science+Business Media New York 2014

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suggestive of the sun’s corona if the complement failed to be empty). Newman showed that the corona problem in the disk is equivalent to a certain interpolation problem [C1, p. 548]. The latter was solved by L. Carleson [C1] in 1962. The proof of the Carleson corona theorem was subsequently simplified by L. Hörmander [Ho] N who used the Koszul complex technique to reduce it to a @-problem on D. Along these lines the simplest proof of the corona theorem was obtained by T. Wolff, see, e.g., [Ga]. Following the appearance of Carleson’s proof, a number of authors have proved the corona theorem for finite bordered Riemann surfaces, e.g., [A1, A2, EM1, F, St1, St2, St3, HN, O2]. The corona problem was first solved for a class of infinitely connected domains (the, so-called, “roadrunner” domains) by M. Behrens [Be1, Be2] (his results were further extended in [D, DW, Nar]), and for a class of finitely sheeted covering Riemann surfaces by M. Nakai [Nak2]. Also, it was shown that there are non-planar Riemann surfaces for which the corona is non-trivial (see, e.g., [JM], [G1], [BD, L] and references therein). This is due to a structure that in a sense makes the surface seem higher dimensional. So there is a hope that the restriction to the Riemann sphere might prevent this obstacle. However, the general problem for planar domains is still open as is the problem in several variables for the ball and polydisk. (In fact, there are no known examples of domains in Cn , n 2, without corona.) In this direction, Gamelin [G2] has shown that the corona problem for planar domains is local in the sense that it depends only on the behavior of the domain locally about each boundary point. At present, one of the strongest corona theorems for planar domains is due to Moore [M]. It states that the corona is empty for any domain with boundary contained in the graph of a C 1C function. This result is the extension of the earlier result of Jones and Garnett [GJ] for a Denjoy domain (i.e., a domain with boundary contained in R). Among other results, it is worth mentioning recent results of Handy [Han] establishing the corona theorem for complements of certain square Cantor sets and of NewDelman [ND] who proves the corona theorem for the complement of a subset of a Lipschitz graph of homogeneous type (in the lines of Carleson’s result [C2] on Denjoy domains). The corona problem can be equivalently reformulated as follows, see, e.g., [Ga]: A collection f1 ; : : : ; fn of functions from H 1 .X / satisfies the corona condition if 1 max jfj .x/j ı > 0 for all x 2 X: 1j n

(1)

The corona problem being solvable (i.e., the corona is empty) means that for all n 2 N and f1 ; : : : ; fn satisfying the corona condition, the Bezout equation f1 g1 C C fn gn 1

(2)

has a solution g1 ; : : : ; gn 2 H 1 .X /. We refer to max1j n jjgj jj1 as a “bound on the corona solutions“. (Here jj jj1 is the norm on H 1 .X /.) In the present paper we survey some results and methods related to the corona problem for H 1 on Riemann surfaces.

Corona Problem for H 1 on Riemann Surfaces

33

2 Forelli Projections In this part we describe one of the methods allowing to prove corona theorems for a wide class of Riemann surfaces. Let r W XQ ! X be an unbranched covering of a connected complex manifold X . Q we denote the pullback by r of the algebra H 1 .X /. By r .H 1 .X // H 1 .X/ Q ! r .H 1 .X // satisfying A bounded linear projection P W H 1 .X/ P .fg/ D P .f /g

for any

Q f 2 H 1 .X/

and

g 2 r .H 1 .X //

(3)

is called a Forelli projection. For X a connected Caratheodory hyperbolic Riemann surface and r W D ! X the universal covering such a projection P , if exists, solves the corona problem for H 1 .X /. Indeed, if f1 ; : : : ; fn 2 H 1 .X / satisfy (1), then their pullbacks r f1 ; : : : ; r fn 2 H 1 .D/ satisfy (1) on D. According to the Carleson corona theorem there exist corona solutions g1 ; : : : ; gn 2 H 1 .D/ of the equation r f1 g1 C C r fn gn 1

(4)

with a bound C.n3=2 ı 2 C n2 ı 3 / for an absolute constant C , see, e.g., [Ga, Ch. VIII]. Applying to (4) the projection P we solve, due to (3), the Bezout equation (2) on X with the bound on the corresponding solutions C kP k.n3=2 ı 2 Cn2 ı 3 /. If X is an annulus, then the fibre of the covering r W D ! X is naturally identified with the group of integers Z. Since this group is amenable and abelian, there exists a nonnegative bounded linear functional of norm one on `1 .Z/ invariant under translation by group elements. Then the Forelli projection P is given by the formula P .f /.z/ WD .fz /;

z 2 D;

f 2 H 1 .D/;

where fz .g/ WD f .g z/, g 2 Z and Z D ! D, g z 7! g z, is the holomorphic action on D of the deck transformation group Z of the covering r W D ! X . By means of this projection the corona theorem for an annulus was established independently by S. Scheinberg [Sc] and E. L. Stout [St2]. If X is a generic connected noncompact Riemann surface, then the fundamental group 1 .X / of X is free with the number of generators 2. In particular, it is not amenable and the previous averaging method is not applicable to the universal covering r W D ! X in this case. Nevertheless, for certain Riemann surfaces X projections P W H 1 .D/ ! r .H 1 .X // satisfying (3) still exist. Forelli [F] was the first to discover that such projections P exist for X being a finite bordered Riemann surface. Later, his construction made explicit by Earle

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and Marden [EM1, EM2]. Subsequently, existence of Forelli projections for certain infinitely connected Riemann surfaces were established by Carleson [C2] for complements in C of compact homogeneous subsets E of R (i.e., for which there exist " > 0 such that mes .x r; x C r/ \ E "r for all r > 0 and all x 2 E) and by Jones and Marshall [JM] for Riemann surfaces X such that critical points of Green’s functions on X or their preimages in D form an interpolating sequence for H 1 .X / or H 1 .D/, respectively. (In particular, the latter class contains surfaces considered previously in [F] and [C2].) In these approaches the projection can be constructed explicitly by means of a function h 2 H 1 .D/ with the properties: P O (a) h.z/ WD w2r 1 .z/ jh.w/j, z 2 X , is a continuous function on X satisfying O C < 1; sup h PX (b) w2r 1 .z/ h.w/ D 1 for all z 2 X . Having such a function one defines a Forelli projection P W H 1 .D/ ! r .H 1 .X // of norm kP k C by the formula X h.w/f .w/; y 2 r 1 .z/; f 2 H 1 .D/: (5) P .f /.y/ WD

w2r 1 .z/

Note that P induces weak continuous linear functionals on `1 .r 1 .z//, z 2 X , while in the approach with the invariant mean on the deck transformation group such induced functionals are bounded but not weak continuous. In [Br4] existence of weak continuous Forelli projections of the form (5) was established by the author for a wide class of (not necessarily one-dimensional) complex manifolds. The construction is more abstract than the previous ones and uses some techniques of the theory of coherent Banach sheaves over Stein manifolds developed by L. Bungart [B]. Specifically, it was shown in [Br4, Th. 1.1] that such a projection P W H 1 .XQ / ! r .H 1 .X // exists for XQ and X being unbranched coverings of a relatively compact domain N in a connected Stein manifold M such that inclusion N ,! M induces an isomorphism of fundamental groups 1 .N / Š 1 .M /. Moreover, kP k is bounded by a constant depending on N only. In particular, this condition is valid for N being a bordered Riemann surface. Thus, due to the covering homotopy theorem, the weak continuous Forelli projection P W H 1 .D/ ! r .H 1 .X // exists for X being a domain in an unbranched covering R of a bordered Riemann surface N such that inclusion X ,! R induces a monomorphism of the fundamental groups, and kP k C.N /, [Br4, Cor. 1.3]. Here is a simple example of such an X : Example. Consider the standard action of the group Z C i Z on C by translations. The fundamental domain of this action is the square Q WD fz D x C iy 2 C W maxfjxj; jyjg 1g:


35

By Qt we denote the square homothetic to Q with sidelength t . Let O be the orbit of 0 2 C with respect to the action of Z C i Z. For any x 2 O we will choose some t .x/ 2 Œ 12 ; 34 and consider the square Q.x/ WD x C Qt.x/ centered at x. Let V C be a simply connected domain satisfying the property: there exists a subset fxi gi2I O such that V \

[ x2O

! Q.x/ D

[

Q.xi /:

i2I

We set X WD V n.[i2I Q.xi //. Then X satisfies the required conditions. In fact, the quotient space C=.ZCi Z/ is a torus CT. Let S be the image of Q1=3 in CT. Then X is a domain in the regular covering R of N WD CT n S with the deck transformation group Z C i Z. The condition that the embedding X ,! R induces a monomorphism of fundamental groups follows from the construction of X . Let us recall that a connected non-parabolic Riemann surface X with a Green function Go is of Widom type if Z 1 b.t / dt < 1; 0

where b.t / is the first Betti number of the set fx 2 X W Go .x/ > t g. This means that the topology of X grows slowly as measured by the Green function. Widom type surfaces are the only infinitely connected ones for which the Hardy theory has been developed to any extent. They have many bounded holomorphic functions. In particular, such functions separate points and directions. We refer to [Ha] for an exposition. It was noted by Jones and Marshall [JM, p. 295] that using results in [W] and [P] it is possible to show that if the Forelli projection P W H 1 .D/ ! r .H 1 .X // exists for a connected Caratheodory hyperbolic Riemann surface X , then X is of Widom type.

3 Riemann Surfaces with Corona The first example of a (non-planar) Riemann surface for which the corona theorem fails was found by B. Cole, see, e.g., [G1]. He constructed a sequence of finite .k/ .k/ 2 H 1 .R.k/ / with bordered Riemann surfaces R.k/ and functions f1 ; f2 .k/ .k/ .k/ .k/ maxj jfj .z/j ı > 0 for all z 2 R , but where any solution of f1 g1 C .k/ .k/

.k/

.k/

f2 g2 1 must satisfy supk .kg1 k1 C kg2 k1 / D 1. (Recently, modifying Cole’s example, B. Oh [O1] constructed explicitly Riemann surfaces with large bounds on corona solutions in an elementary way.) Constructing such a sequence for planar domains is equivalent to the failure of the corona theorem for a planar domain, as shown in [G2]. Later, it was shown in [Nak1] that Cole’s example can be modified to be of Widom type. Also, it was observed in [JM] that modifying the construction in Gamelin [G2] it is possible to show that if the corona theorem fails for a planar domain, it must fail for a planar domain of Widom type.

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Other examples being found later by Barrett and Diller [BD] and Larusson [L]. The construction of the latter paper is based on an interpolation result for holomorphic functions of exponential growth on unbranched coverings of complex projective manifolds which is proved by means of certain vanishing theorems for L2 cohomology groups on the coverings. In the case of H 1 functions this result was strengthen in [Br3] to produce the following: Let Y be an unbranched covering of a complex connected projective manifold M CPN of dimension n 2. Let C be intersection with M of at most n 1 generic hypersurfaces of degree d in CPN . By the Lefschetz theorem, the preimage X of C in Y is a connected submanifold. Then the restriction H 1 .Y / ! H 1 .X / is an isometry for d large enough. For instance, this can be applied to Y being either an open Euclidean ball or polydisk in Cn , n 2, to find a compact Riemann surface Sn and its regular covering r W SQn ! Sn such that SQn ,! Y and this embedding induces an isometry of the H 1 spaces. Thus maximal ideal spaces M .H 1 .SQn // and M .H 1 .Y // are homeomorphic. In particular, the covering dimension of M .H 1 .SQn // is at least n and the corona theorem fails for SQn . A similar L2 cohomology technique was used in [Br6] by the author to prove the solvability of the corona problem for H 1 on unbranched coverings of connected Caratheodory hyperbolic Riemann surfaces of finite type.

4 Projective Freeness and Hermiteness for H 1 on Riemann Surfaces In this part we describe a matrix version of the corona problem. A commutative ring with identity R is said to be projective free if every finitely generated projective R-module is free. Recall that if M is an R-module, then 1. M is called free if M Š Rd for some integer d 0; 2. M is called projective if there exists an R-module N and an integer d 0 such that M ˚ N Š Rd . In terms of matrices, the ring R is projective free iff every square idempotent matrix F is conjugate (by an invertible matrix) to a matrix of the form

Ik 0 ; 0 0

see [Co1, Prop. 2.6]. From the matricial definition, it follows that any field k is projective free, since matrices F satisfying F 2 D F are diagonalizable over k. Quillen and Suslin (see [La]) proved, independently, that the polynomial ring over a projective free ring is again projective free. Thus, the polynomial ring kŒx1 ; : : : ; xn is projective


37

free. Also, if R is any projective free ring, then the formal power series ring RŒŒx

in a central indeterminate x is again projective free [Co2, Th. 7]. Hence, the ring of formal power series kŒŒx1 ; : : : ; xn

is also projective free. Less is known about projective freeness of topological rings arising in analysis. For instance, from Grauert’s theorem [Gra] one obtains that the ring H.X / of holomorphic functions on a connected reduced Stein space X satisfying the property that any complex vector bundle of finite rank on X is topologically trivial, is projective free (cf. the proof of Theorem 1.2 in [BS]). This is the case if, e.g., the space X is contractible or if it is biholomorphic to a connected noncompact (possibly singular) Riemann surface. For R being a complex commutative unital Banach algebra, certain topological conditions on its maximal ideal space M .R/ (e.g., its contractibility) implying projective freeness of R are presented in [T2] and [BS]. Also, it was established in [BS, Th. 1.5], by means of some results of [Br5], that H 1 .X / is projective free for X being a domain in an unbranched covering S of a bordered Riemann surface N such that inclusion X ,! S induces a monomorphism of the fundamental groups (cf. Example in Sect. 2). As a particular case of this result one obtains that H 1 .X / is projective free for every connected Caratheodory hyperbolic Riemann surface X of finite type. Earlier projective freeness of H 1 .D/ was proved in [Q] by a different method. The concept of a Hermite ring is weaker than that of a projective free ring. A commutative ring R with identity is said to be Hermite if every finitely generated stably free R-module is free. Recall that a R-module M is called stably free if there exist nonnegative integers n; d such that M ˚ Rn D Rd . Clearly every stably free module is projective, and so every projective free ring is Hermite. A complex commutative unital Banach algebra R is Hermite if and only if for all k; n 2 N, k < n, each k n matrix with entries in R such that all its minors of order k do not belong together to a maximal ideal of R can be completed to an invertible n n matrix with entries in R. Moreover, R is Hermite if and only if the algebra C.M .R// of complex continuous functions on the maximal ideal space of R is Hermite (see, e.g., [Li, Th. 3], [T2, No, Th. 4], [Ta, pp. 179, 196]). If X is a Riemann surface for which the corona theorem is valid, Hermiteness of H 1 .X / is equivalent to the following statement (which is stronger than just the solvability of the Bezout equation (2) in the corona theorem for H 1 .X /): For all k; n 2 N, k < n, each k n matrix Ak;n with entries in H 1 .X / such that all its minors of order k satisfy the corona condition (1) can be completed to an n n matrix AQn;n of determinant one with entries in H 1 .X /. Hermiteness of H 1 .D/, with estimates of norm supz2D kAQn;n .z/k`n2 !`n2 of the extension AQn;n of Ak;n depending on ı (in the corona condition for minors of order k of Ak;n ) and k; n only, was first proved by V. Tolokonnikov [T1, Th. 4]. Later, in [T2] he proved Hermiteness of H 1 .X /, where X is a bordered Riemann surface, with similar estimates of the norm of the extension AQn;n of Ak;n . The proof is based

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on the Beurling-Lax-Halmos [Beu,Lax,NF], Forelli [F] and Grauert [Gra] theorems. Recently, in [Br5] Hermiteness of H 1 .X / was proved by the author for X being a domain in an unbranched covering S of a bordered Riemann surface N such that inclusion X ,! S induces a monomorphism of the fundamental groups. The proof uses analogs of the Beurling-Lax-Halmos theorem [Br5, Th. 1.7], Grauert theorem [Br5, Th. 1.5] and Forelli theorem [Br4] and allows to estimate the norm of the extension AQn;n of Ak;n by a constant depending on the corresponding data ı, k; n for Ak;n and on N only (in particular, this extends the result of [T2]). Finally, in [Br7] using a topological method, Hermiteness of H 1 .X / was proved by the author for X being an unbranched covering of a connected Caratheodory hyperbolic Riemann surface of finite type N with the norm of the extension AQn;n of Ak;n bounded by a constant depending on the corresponding data ı, k; n for Ak;n and on N only.

5 Remarks about Bounds on the Corona Solutions All known examples of Caratheodory hyperbolic Riemann surfaces X for which the corona theorem is valid allow bounds on the corona solutions of Bezout equations (2) depending on n and ı (in the corona condition (1) on X ), and some inessential parameters only. For such X this, in particular, implies much more stronger corona theorems. Indeed, let X WD fXs gs2S be the family of Caratheodory hyperbolic Riemann surfaces. We define XS WD ts2S Xs . Then XS is a one-dimensional complex manifold (not necessarily second-countable). The Banach algebra H 1 .XS / of bounded holomorphic functions on XS is well defined and separates the points of XS . It is easy to see that If for each n 2 N and all n-tuples of functions in H 1 .Xs /, s 2 S , satisfying corona conditions (1), Bezout equations (2) with these functions admit solutions bounded by a nonnegative function in variables n and ı, then the corona theorem is valid for XS , that is, i.XS / is dense in the maximal ideal space M .H 1 .XS // (here i.XS / consists of the evaluation homomorphisms at points of XS ). This is valid, e.g., for a family X WD fXs gs2S , where each Xs is either D or an open annulus. In turn, if each Xs is a domain in an unbranched covering R of a (fixed) bordered Riemann surface N such that inclusion X ,! R induces a monomorphism of the fundamental groups, then H 1 .XS / is Hermite, see estimates in [Br5]. The same is true if each Xs is an unbranched covering of a fixed Caratheodory hyperbolic Riemann surface of finite type, see [Br7]. Further, if all Xs D D in the definition of X , then estimates obtained in [Tr4] show that H 1 .XS / has stable rank one, etc. Note that for XS being the disjoint union of all bounded finitely connected domains in C, the corona problem is still open and is equivalent to the general corona problem for planar domains, see, e.g., [G2, Han, JM, Nak1, ND] and references therein. In connection with that problem, it might be of interest to study analogs of Gleason parts of points in M .H 1 .XS // (defined similarly to those in M .H 1 .D//, see Sect. 7 below).


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6 Operator Corona Problem Let H 1 .L.X; Y // be the Banach space of holomorphic functions F on D with values in the space of bounded linear operators X ! Y between complex Banach spaces X; Y with norm kF k WD supz2D kF .z/kL.X;Y / . As usual, L.X / WD L.X; X /; by IX we denote the identity operator X ! X . The operator corona problem is a tentative to generalize the Carleson corona theorem to operator valued functions. It was posed by Sz.-Nagy in 1978 in the following form [Na]: Suppose that F 2 H 1 .L.H1 ; H2 //, H1 ; H2 are separable Hilbert spaces, satisfies kF .z/xk ıkxk for all x 2 H1 , z 2 D, where ı > 0 is a constant. Does there exist G 2 H 1 .L.H2 ; H1 // such that G.z/F .z/ D IH1 for all z 2 D? This problem is of importance in operator theory (angles between invariant subspaces, unconditionally convergent spectral decompositions) and in control theory (the stabilization problem). It is also related to the study of submodules of H 1 and to many other subjects of analysis, see [Ni1,Ni2,Tr2,Tr5,Vi] and references therein. Obviously, the condition imposed on F is necessary since it implies existence of a uniformly bounded family of left inverses of F .z/, z 2 D. The question is whether it is sufficient for the existence of a bounded holomorphic left inverse of F . In general, as it was shown by S. Treil, the answer is negative (see [Tr1, Tr3, Tr6, TW] and references therein). But in some specific cases it is positive. In particular, the Carleson ffi gniD1 H 1 .D/ Pncorona theorem [C1] stating that for n the Bezout equation iD1 gi fi 1 is solvable with fgi giD1 H 1 as soon as max1in jfi .z/j > ı > 0 for every z 2 D means that the answer is positive when dim H1 D 1, dim H2 D n < 1. Later, using some ideas from the T. Wolff’s proof of the Carleson corona theorem, M. Rosenblum [R], V. Tolokonnikov [T1] and Uchiyama [U] independently proved that the operator corona problem is solvable if dim H1 D 1, dim H2 D 1. Using a simple linear algebra argument, P. Fuhrmann [Fu] proved that the operator corona problem is solvable if dim H1 ; dim H2 < 1, and further V. Vasyunin (see [T1] for the proof) extended this result to the case dim H2 D 1 (but still dim H1 < 1). Recently, it was established by S. Treil in [Tr7] that the operator corona problem is solvable as soon as F is a “small” perturbation of a left invertible function F0 2 H 1 .L.H1 ; H2 // (for example, if F F0 belongs to H 1 .L.H1 ; H2 // with values in the class of Hilbert Schmidt operators). For a long time there were no positive results in the case dim H1 D 1. The first positive results were obtained by P. Vitse in [Vi] where the following more general problem was studied. Let X1 ; X2 be complex Banach spaces and F 2 H 1 .L.X1 ; X2 // be such that for each z 2 D there exists a left inverse Gz of F .z/ satisfying supz2D kGz k < 1. Does there exist G 2 H 1 .L.X2 ; X1 // such that G.z/F .z/ D IX1 for all z 2 D? Since in this general setting the answer is negative, it was suggested in [Vi] 1 to investigate the problem for the case of F 2 Hcomp .L.X1 ; X2 //, the space of

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holomorphic functions on D with relatively compact images in L.X1 ; X2 /. (The 1 notation Hcomp has been introduced in this paper as well.) In particular, it was shown, see [Vi, Th. 2.1], that Pthe answer is positive for F that can be uniformly approximated by finite sums fk .z/Lk , where fk 2 H 1 .D/ and Lk 2 L.X1 ; X2 /. 1 The question of whether each F 2 Hcomp .L.X1 ; X2 // can be obtained in that form is closely related to the still open problem about the Grothendieck approximation property for H 1 .D/. (The strongest result in this direction [BR, Th. 9] states that H 1 .D/ has the approximation property “up to logarithm”.) Another, not involving the approximation property for H 1 .D/, approach was 1 proposed by the author and led to the solution of the above problem for Hcomp 1 spaces, i.e., for F 2 Hcomp .L.X1 ; X2 // satisfying the hypothesis of the problem 1 the required left inverse G of F exists and belongs to Hcomp .L.X2 ; X1 // (see [Br8, Th. 1.5]). In fact, in [Br8] a particular case of the following operator completion problem was investigated (compare with Hermiteness of H 1 .D/): Let X1 ; X2 be complex Banach spaces and F 2 H 1 .L.X1 ; X2 // be such that for each z 2 D there exists a left inverse Gz of F .z/ satisfying supz2D kGz k < 1. Do there exist functions H 2 H 1 .L.X1 ˚ Y; X2 // and G 2 H 1 .L.X2 ; X1 ˚ Y //, Y WD Ker G0 , such that H.z/G.z/ D IX2 , G.z/H.z/ D IX1 ˚Y and H.z/jX1 D F .z/ for all z 2 D. Seemingly much stronger, this problem is equivalent to the operator corona problem for X1 ; X2 being separable Hilbert cases. This result, known as the Tolokonnikov lemma, is proved in full generality by S. Treil in [Tr7]. (Thus in this case the operator completion problem has a positive solution as soon as the operator corona problem has it.) In general, the operator completion problem is much more involved. Some 1 sufficient conditions for its solvability for Hcomp spaces were given by the author in [Br8, Th. 1.3]. Specifically, it was shown that 1 If F 2 Hcomp .L.X1 ; X2 // satisfies the conditions of the problem and the group GL.Y / of invertible elements of L.Y / is connected, then the required H and G exist 1 1 and belong to Hcomp .L.X1 ˚ Y; X2 // and Hcomp .L.X2 ; X1 ˚ Y //, respectively. For instance, this is valid in one of the following cases (see [Br10, Mi]): (a) dimC Y < 1; (b) X2 is isomorphic to a Hilbert space or c0 or one of the spaces `p , 1 p 1; (c) X2 is isomorphic to one of the spaces Lp Œ0; 1 , 1 < p < 1, or C Œ0; 1 and X1 is not isomorphic to X2 . Moreover, it is proved by the author in [Br10, Th. 2.5] that if GL.Y / is not connected, then the operator completion problem is not solvable in the class of 1 Hcomp spaces. Proofs of these results are based on an analog of the Cartan-Oka theory for Banach-valued holomorphic functions on the maximal ideal space M .H 1 .D// developed by the author in [Br9]. In this approach one uses some topological properties of this space described in the next section.


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7 Topology of the Maximal Ideal Space of H 1 Recall that the pseudohyperbolic metric on D is defined by ˇ ˇ ˇ zw ˇ ˇ; ˇ .z; w/ WD ˇ 1 wz N ˇ

z; w 2 D:

For x; y 2 M .H 1 .D// the formula .x; y/ WD supfjfO.y/j I f 2 H 1 .D/; fO.x/ D 0; kf k 1g gives an extension of to M .H 1 .D// (here O W H 1 .D/ ! C.M .H 1 .D/// stands for the Gelfand transform). The Gleason part of x 2 M .H 1 .D// is then defined by .x/ WD fy 2 M .H 1 .D// I .x; y/ < 1g. For x; y 2 M .H 1 .D// we have .x/ D .y/ or .x/ \ .y/ D ;. Hoffman’s classification of Gleason parts [H] shows that there are only two cases: either .x/ D fxg or .x/ is an analytic disk. The latter case means that there is a continuous one-to-one and onto map Lx W D ! .x/ such that fO ı Lx 2 H 1 .D/ for every f 2 H 1 .D/. Moreover, any analytic disk is contained in a Gleason part and any maximal (i.e., not contained in any other) analytic disk is a Gleason part. By Ma and Ms we denote the sets of all non-trivial (analytic disks) and trivial (one-pointed) Gleason parts, respectively. It is known that Ma M .H 1 .D// is open. Hoffman proved that .x/ Ma if and only if x belongs to the closure of some interpolating sequence in D. In [S1] D. Suárez established that the covering dimension of M .H 1 .D// is 2 ˇ and the Cech cohomology group H 2 .M .H 1 .D/// D 0 (the latter also follows directly from projective freeness of H 1 .D/). One of the important ingredients of his proof is a theorem of S. Treil [Tr4] stating that the stable rank of H 1 .D/ is one. Later, in [S2] D. Suárez proved, in addition, that the set Ms of trivial Gleason parts of M .H 1 .D// is totally disconnected (see, e.g., [Nag] for basic topological definitions). The proof uses a modified version of the construction of Garnett and Nicolau [GN] who proved that the interpolating Blaschke products generate H 1 .D/ as a uniform algebra. Further, in [Br2] the author described the set Ma as a fibre bundle over a fixed compact Riemann surface S of genus 2 with fibre an open dense subset of the ˇ Stone Cech compactification of the fundamental group of S (in particular, this shows that each bounded uniformly continuous with respect to the metric on D function is extended to a continuous function on Ma ). This and the result of [S2] have been used in [Br8, Br9] in the proofs of the basic facts of the theory of Banach-valued holomorphic functions on M .H 1 .D//. It is worth noting that one can use that the covering dimension of M .H 1 .D// is 2 and H 2 .M .H 1 .D/// D 0 to give another proofs of projective freeness and Hermiteness of H 1 .X / for X being a Riemann surface of finite type (see, e.g., [Br1, BS, T2]).

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Acknowledgements Research supported in part by NSERC. I thank the anonymous referee for useful remarks improving the presentation of the paper.

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[Tr3] S. Treil, Geometric methods in spectral theory of vector-valued functions: Some recent results. In Toeplitz Operators and Spectral Function Theory, Oper. Theory Adv. Appl., vol. 42, Birkhäuser Verlag, Basel 1989, pp. 209–280. [Tr4] S. Treil, The stable rank of H 1 equals 1. J. Funct. Anal. 109 (1992), 130–154. [Tr5] S. R. Treil, Unconditional bases of invariant subspaces of a contraction with finite defects. Indiana Univ. Math. J. 46 (1997), no. 4, 1021–1054. [Tr6] S. Treil, Lower bounds in the matrix corona theorem and the codimension one conjecture. GAFA 14 5 (2004), 1118–1133. [Tr7] S. Treil, An operator Corona theorem. Indiana Univ. Math. J. 53 (2004), no. 6, 1763–1780. [TW] S. Treil and B. Wick, Analytic projections, Corona Problem and geometry of holomorphic vector bundles. J. Amer. Math. Soc. 22 (2009), no. 1, 55–76. [U] A. Uchiyama, Corona theorems for countably many functions and estimates for their solutions, (1980) (preprint). [Vi] M. Vidyasagar, Control system synthesis: a factorization approach. MIT Press Series in Signal Processing, Optimization, and Control 7, MIT Press, Cambridge, MA, 1985. [W] H. Widom, H p sections of vector bundles over Riemann surfaces. Ann. of Math. 94 (1971), 304–324.

Connections of the Corona Problem with Operator Theory and Complex Geometry Ronald G. Douglas

Abstract The corona problem was motivated by the question of the density of the open unit disk D in the maximal ideal space of the algebra, H 1 .D/, of bounded holomorphic functions on D. In this note we study relationships of the problem with questions in operator theory and complex geometry. We use the framework of Hilbert modules focusing on reproducing kernel Hilbert spaces of holomorphic functions on a domain, ˝, in Cm . We interpret several of the approaches to the corona problem from this point of view. A few new observations are made along the way. Keywords Corona problem • Hilbert modules • Reproducing kernel Hilbert space • Commutant lifting theorem Subject Classification: 46A15, 32A36, 32A70, 30H80, 30H10, 32A65, 32A35, 32A38

1 Introduction Frequently, questions in abstract functional analysis lead to very concrete problems in “hard analysis” with results on the latter having stronger implications in the larger abstract context. That is the case with the corona problem. Describing some of these larger relationships is the main goal of our note. In this problem in analysis, however, the mix of related topics is quite broad and includes aspects of complex geometry and function theory as well as operator theory.

R.G. Douglas () Texas A&M University, College-Station, TX 77843-3368, USA e-mail: [email protected] R.G. Douglas et al. (eds.), The Corona Problem, Fields Institute Communications 72, DOI 10.1007/978-1-4939-1255-1__3, © Springer Science+Business Media New York 2014

47

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Almost all the results mentioned are due to others, as we will try to make clear, and the novelty, if any, is in the organization and point of view. There are a few observations in Sect. 6, 7, 8, and 9 which may be new. In particular, we show if the multiplier algebra for a reproducing kernel Hilbert module is logmodular, then the Toeplitz corona property implies the usual corona theorem. Further, for many reproducing kernel Hilbert modules, knowing that the Toeplitz corona theorem holds for all cyclic submodules, with a uniform lower bound, implies the corona theorem. The corona problem springs from the study of uniform algebras and, in particular, the description of their maximal ideal spaces. Following Gelfand’s fundamental results on commutative Banach algebras, calculating the space of maximal ideals for particular commutative Banach algebras became a central issue in functional analysis. In 1941, Kakutani raised the question (cf. [15]) for the algebra, H 1 .D/, of bounded holomorphic functions on the open unit disk D and asked whether or not the open disk D is dense in its maximal ideal space. One quickly reduces the latter problem to showing for f'i gniD1 H 1 .D/, satisfying n X

j'i .z/j2 2 > 0

iD1

for z 2 D, that there exists an n-tuple f n X

n i giD1

'i .z/

H 1 .D/ such that

i .z/

D1

iD1

for z 2 D. N of D in C or are in If f'i gniD1 consists of continuous functions on the closure D n N N satisfying C.D/, then it is not hard to show that an n-tuple f i giD1 exists in C.D/ n X

'i

i

N D 1 in C.D/:

iD1

If one could control the supremum norms fk i k1 gniD1 in terms of the norms fk 'i k1 gniD1 and , then a standard normal families argument from function theory N doesn’t yield would establish the result for H 1 .D/. But the easy proof for C.D/ bounds. (A slightly more general result for holomorphic functions continues on the closure of a bounded domain, with proof, appears in Sect. 7.) However, even if one N to H 1 .D/ in some other way, one quickly were able to extend this result for C.D/ sees that in most applications of the solution to the corona problem, one needs to discuss such bounds anyway. All known solutions of the corona problem involve techniques from harmonic analysis and function theory. Here we focus on connections the problem has with operator theory and complex geometry. For a recent overview of the analysis picture, cf. [9]. An earlier note relating the corona problem to operator theory was given by Trent [22].

Connections of the Corona Problem with Operator Theory and Complex Geometry

49

There is a closely related problem in which matrix-valued holomorphic functions are considered to which many of the ideas in this note apply. Trent in [21] showed how to relate this family of problems to the ones considered in this note but we will not consider that here. In this note we will cite the relevant sources for the main results discussed but refer the reader to [9] for more detailed references and historical remarks.

2 The Corona Problem Although the corona problem could be formulated more generally (cf. Sect. 8), here we restrict our attention to algebras of bounded holomorphic functions. Let ˝ be a bounded connected domain in Cm for some positive integer m, and let H 1 .˝/ denote the algebra of bounded holomorphic functions on ˝. Define the supremum norm by k ' k1 D sup j'.!/j for ' 2 H 1 .˝/: !2˝

Then H 1 .˝/ is a commutative Banach algebra for the pointwise algebraic operations and hence there is a compact Hausdorff space, MH 1 .˝/ , of maximal ideals. Recall, following Gelfand, that a maximal ideal I in the commutative Banach algebra H 1 .˝/ can be identified with a multiplicative linear functional on H 1 .˝/. This follows since H 1 .˝/=I Š C as algebras by Gelfand’s theorem. The family of all such maps lies in the unit ball of the Banach space dual of H 1 .˝/ and, in the weak*-topology, is a compact Hausdorff space denoted by MH 1 .˝/ , called the maximal ideal space. For ! 2 ˝, the set I! D f' 2 H 1 .˝/ W '.!/ D 0g is a maximal ideal in H 1 .˝/ with the corresponding multiplicative linear functional defined by evaluation at !. Thus one has an imbedding, ˝ MH 1 .˝/ , of ˝ in MH 1 .˝/ . The corona problem asks: Is ˝ dense in MH 1 .˝/ ? Let clos ˝ denote the closure of ˝ in MH 1 .˝/ . The complement, MH 1 .˝/ n clos ˝, if non empty, is said to be the corona for H 1 .˝/. Hence, the corona problem asks if a corona exists. However, there is a small problem. There exist connected bounded domains ˝1 ˝2 Cm so that H 1 .˝1 / D H 1 .˝2 / in the sense that every ' 2 H 1 .˝1 / extends uniquely to a bounded holomorphic function on ˝2 . In such cases, not all points in MH 1 .˝1 / n clos ˝1 should be considered to be in the corona; in particular, the points in ˝2 n ˝N 1 should not be viewed as being in the corona. Hence, we should, and do, restrict attention to bounded domains ˝ for which no such larger domain exists. Hence, we make the following assumption from now on: ˝ is a bounded connected domain in Cm for which no super domain exists supporting the same algebra of bounded holomorphic functions. Domains such as the unit ball, Bm , or the polydisk, Dm , have this property but, for example, the domain between two concentric spheres in Cm does not. For a bounded domain ˝ Cm , consider the map ˝ W MH 1 .˝/ ! Cm such that

50

R.G. Douglas

˝ .x/ D .Oz1 .x/; : : : ; zOm .x//, for x 2 MH 1 .˝/ , where fOzi gm iD1 are the Gelfand 1 transforms of the functions fzi gm H .˝/. Then the open subset of MH 1 .˝/ iD1 on which ˝ is locally one-to-one defines the largest domain in Cm containing ˝ supporting the same algebra of bounded holomorphic functions. Since in our note this topic is only peripheral, we won’t consider the problem further of characterizing the property. We now return to the corona problem. (0) If f'i gniD1 H 1 .˝/ satisfies n X

j'i .!/j2 2 > 0

iD1

for ! 2 ˝ and ˝ is dense in MH 1 .˝/ , then inequality (0) extends to all of MH 1 .˝/ . Therefore, the ideal ( J D

n X

) 'i

i

Wf

n i giD1

1

H .˝/ ;

iD1

generated by the set f'i gniD1 , is not contained in any proper maximal ideal. Hence the function 1 2 J , or equivalently, (1) There exists f

n i giD1

H 1 .˝/ such that n X

'i .!/

i .!/

D1

iD1

for ! 2 ˝ or n X

'i

i

D 1 in H 1 .˝/:

iD1

Thus, we can restate the corona problem: For f'i gniD1 H 1 .˝/, does (0) imply (1)? It is clear that (1) implies (0). In fact, a simple application of the Cauchy-Schwarz inequality yields 1Dj

n X

'i .!/

i .!/j

n X

iD1

! 12 j'i .!/j

2

iD1

(

n X iD1

n X

! 12 j

i .!/j

2

iD1

) 12 ( j'i .!/j2

n X iD1

) 12 k

i

k21


51

or n X iD1

( j'i .!/j 1= 2

n X

) k

2 i k1

:

iD1

In 1962 Carleson settled the corona problem for H 1 .D/ in the affirmative using techniques from harmonic analysis and function theory [2]. Since then, many results have been obtained, in both the affirmative and negative, for the corona problem for a variety of domains in Cm (cf. [9]). We will not pursue these results or their proofs in this note. Rather we want to explore various connections and relationships of the corona problem with operator theory and complex geometry.

3 Hilbert Modules and the Corona Problem Although there is no Hilbert space mentioned in the statement of the corona problem, there is a natural way to relate the corona problem for function algebras to operator theory. Recall that a Hilbert space R of holomorphic functions on a bounded, connected domain ˝ of Cm is said to be a reproducing kernel Hilbert space (RKHS) if R O.˝/, and the evaluation map ev! .f / D f .!/ for f 2 R is bounded and onto for ! 2 ˝, where O.˝/ denotes the space of holomorphic functions on ˝. One obtains the usual “two-variable” kernel function, K.z; !/, on ˝ ˝ by setting K.z; !/ D evz ev! 2 L.C; C/ Š C for z; ! 2 ˝. The unique function k! satisfying ev! f D< f; k! >R is obtained by setting k! .z/ D K.z; !/. (The Riesz representation theorem is used implicitly here in defining the adjoint ev! .) If zi R R for i D 1; 2; : : : ; m, then R is a Hilbert module over the algebra of polynomials in m variables, CŒz1 ; : : : ; zm . In this case we say that R is a reproducing kernel Hilbert module (RKHM). Although it is not necessary, to simplify matters we assume that 1 2 R which implies CŒz1 ; : : : ; zm R. In general, CŒz1 ; : : : ; zm is not dense in R but it is in many natural examples, such as for R the Hardy space, H 2 .@Bm /, on the unit ball, Bm , in Cm or the Bergman space L2a .˝/, where the former space can be defined as the closure of CŒz1 ; : : : ; zm in L2 .@Bm / for Lebesgue measure on the unit sphere, @Bm , and the latter space is the closure of CŒz1 ; : : : ; zm in L2 .˝/ for Lebesgue measure on ˝. (Actually, one might want to put some restrictions on ˝ and, perhaps, let L2a .˝/ be the closure of H 1 .˝/, but we do not go into any detail here.) For R a RKHM, a function 2 O.˝/ is said to be a multiplier for R if R R. The set of multipliers, M.R/, forms a commutative Banach algebra which one can show is contained in H 1 .˝/.

52

R.G. Douglas

We can formulate a corona problem in the context of a RKHM R as follows: • The n-tuple f'i gniD1 M.R/ satisfies (0) or n X

j'i .!/j2 2 > 0, for ! 2 ˝:

iD1

• Statement (1) holds or there exists f n X

n i giD1

'i .!/

M.R/ such that

i .!/

D1

iD1

for ! 2 ˝. The corona problem for M.R/ asks if (0) implies (1). As before, (1) implies (0). An immediate connection with operator theory concerns the quotient Hilbert module sequence (2)

M˚

˚

0 ! R ! R ˝ Cn ! R˚ ! 0;

which is exact if range M˚ is closed, where R˚ is the quotient Hilbert module R= range M˚ and ˚ is the quotient map. If (1) holds, then range M˚ is closed. Here, M˚ W R ! R ˝ Cn is defined M˚ f D

n X

'i f ˝ ei ;

iD1

where fei gniD1 is the standard orthonormal basis for Cn . N We will call a RKHM R subnormal if there exists a probability measure on ˝, the closure of ˝ in Cm , such that R L2 ./ isometrically. Although many RKHM are subnormal, not all are. Perhaps the simplest example of this phenomenon is the Dirichlet space on D. Another family of recent interest is the Drury-Arveson space Hm2 on Bm with the kernel function K.z; !/ D .1 hz; !iCm /1 . An alternate description of Hm2 is that it is the symmetric Fock space. The first question concerning the exactness of (2) is the exactness at R, the left-most module. If ˝ is connected and ˚ ¤ 0, then M˚ is one-to-one. Hence, exactness comes down to the closeness of the range of M˚ . This issue is transparent in the subnormal case. Proposition 1. For f'i gniD1 M.R/ with R a subnormal RKHM on ˝ for the N statement, (0) implies that range M˚ is probability measure supported on ˝, closed or that the sequence .2/ is exact at R and hence (2) is a short exact sequence.


53

Proof. For f 2 R we have k M˚ f k2R˝Cn D

n X

k 'i f k2R D

iD1

Z D

n X ˝N

Z X n ˝N iD1

! j'i j

j'i f j2 d

jf j2 d 2 k f k2R ;

2

iD1

N Thus M˚ is bounded below where R L2 ./ for the probability measure on ˝. and hence has closed range. Remark 1. Note that since M˚ is one-to-one for ˚ ¤ 0, the assumption that range M˚ is closed implies that M˚ is onto and vice-versa. The relationship between the module sequence .2/ and the corona problem is strong. Remark 2. Suppose f

n i giD1

M.R/ satisfies n X

'i

i

D 1:

iD1

Then, as mentioned above, this implies (0). If one defines the operator N W R ˝ Cn ! R such that N .

n X

fi ˝ ei / D

iD1

n X

i fi for

iD1

n X

fi ˝ ei 2 R ˝ Cn ;

iD1

then N M˚ D IR and hence range M˚ is closed. Thus the failure of range M˚ to be closed is an obstruction to an affirmative answer to the corona problem for M.R/. We can ask further whether the closeness of range M˚ implies (1). The question is especially relevant for the non-subnormal case. (We will have more to say about this matter in Sect. 8.) To investigate questions such as this one, it is useful to know what module maps look like between modules of the form R ˝ Ck and R ˝ Cl for positive integers k and l. Lemma 1. Let R be a RKHM over CŒz1 ; : : : ; zm and k; l be positive integers. An operator X W R ˝ Ck ! R ˝ Cl is a module map over CŒz1 ; : : : ; zm (that is, one that satisfies .M ˝ ICl /X D X.M ˝ ICk / for 2 CŒz1 ; : : : ; zm ) iff there exists the matrix f'ij glj D1 kiD1 M.R/ such that X

k X iD1

! fi ˝ ei

D

l X k X j D1 iD1

M'ij fi ˝ ej

54

R.G. Douglas

for k X

fi ˝ ei 2 R ˝ Ck :

iD1

Proof. Standard linear algebra calculations yield the existence of f'ij gkiD1 lj D1 M.R/: Although this result was doubtless known to many, one can find it in [14]. Proposition 2. Let f'i gniD1 M.R/ for the RKHM R over the bounded domain ˝ Cm such that M˚ has closed range and, hence, (2) is exact. Then the following statements are equivalent to (1): (3) M˚ has a left module inverse N W R ˝ Cn ! R. (4) There exists a right module inverse ˚ for ˚ . (5) There exists a module idempotent E on R ˝ Cn with range E D range M˚ . Proof. For a short exact sequence of modules in the algebra category, (3), (4) and (5) are always equivalent and this fact carries over to our context. But let us provide a complete proof since the techniques are relevant to later issues. If X is a left module inverse for M˚ , then by Lemma 1 there exists f i gniD1 M.R/ such that X D N . Moreover, XM˚ D IR implies that n X

'i

i

D 1:

iD1

Hence, N M˚ D IR and we see that (1) and (3) are equivalent. If X is a left module inverse for M˚ , then E D M˚ X is a module idempotent on R ˝ Cn with range E D range M˚ . Therefore, (3) implies (5) and one can define the right module inverse Y D E˚1 W R˚ ! R ˝ Cn . Conversely, if Y is a right module inverse for ˚ , then E 0 D Y ˚ is a module idempotent on R ˝ Cn and we can define the left module inverse X D M˚1 .IR˝Cn E 0 / W R ˝ Cn ! R for M˚ since range .IR˝Cn E 0 / D range M˚ . But the existence of E 0 is enough to define X or (5) implies (3) and this completes the proof. Remark 3. Note that in these constructions, E C E 0 D IR˝Cn . Remark 4. This proposition is closely related to results by Foias, Sarkar and the author in [7]. Remark 5. Note that since is onto, it always has a right inverse. However, a right inverse is, in general, not a module map. Remark 6. Rather than Hilbert modules, one could consider the sequence of Banach M˚

spaces 0 ! H 1 .˝/ ! H 1 .˝/ ˝ Cn ! .H 1 .˝/ ˝ Cn /= range M˚ ! 0 and an analogue of Proposition 2 holds. However, there doesn’t seem to be any way to use this structure in the non Hilbert space setting.


55

4 Module Idempotents and Sub-Bundles The previous results make clear the importance of constructing or identifying module idempotents on R ˝ Cn for R a RKHM over a bounded domain ˝ Cm with range equal to range M˚ for ˚ given by f'i gniD1 M.R/ assuming range M˚ is closed. If E is such a module idempotent on R ˝ Cn , then by Lemma 1 there exists f ij gniD1 nj D1 M.R/ such that E

n X iD1

! fi ˝ ei

D

n X n X iD1 j D1

M ij fi ˝ ej for

n X

fi ˝ ei 2 R ˝ Cn :

iD1

If for ! 2 ˝ we set E.!/ D ij .!/niD1 nj D1 2 L.Cn /, then the fact that E 2 D E implies E.!/2 D E.!/ and vice versa. Since the ij are holomorphic functions and multipliers for R, it follows that E.!/ is a bounded holomorphic matrix-valued function (and more) on ˝. The range of E.!/ defines a holomorphic sub-bundle EO of ˝ Cn with fiber equal to range 0 E.!/ at ! 2 ˝. Moreover, E D I E is the complementary idempotent and it 0 0 O defines a complementary sub-bundle EO of ˝ Cn . That is, ˝ Cn D EO u E, where “u” denotes a (skew) linear direct sum of vector sub-bundles. In particular, 0 EO and EO are not necessarily orthogonal in ˝ Cn . The converse is also true but we must be careful to state it correctly, which involves a mixture of operator theory and complex geometry. Proposition 3. For R a RKHM over a bounded domain ˝ Cm and a submodule M of R ˝ Cn , the following are equivalent: (i) There exists a submodule N of R ˝ Cn such that R ˝ Cn D M u N . (ii) There exists a module idempotent E on R ˝ Cn such that M D range E. Proof. The proof follows basically from algebra and by invoking the closed graph theorem to conclude the boundedness of E. In particular, if (ii) holds, then setting N D .IR˝Cn E/.R ˝ Cn / yields the complementary submodule N . Conversely, if (i) holds, then one can define a bounded module idempotent E with range M by setting Ex D y1 , where x D y1 C y2 is the unique decomposition of x with y1 2 M and y2 2 N . Remark 7. If R ˝ Cn D M u N , then one can define sub-bundles E and F of ˝ Cn such that ˝ Cn D E u F and, most important, M D ff 2 R ˝ Cn W f .!/ 2 E.!/; ! 2 ˝g; where E.!/ is the fiber of E at ! 2 ˝. The same is true for F and N . However, there is no simple converse to this relationship. In particular, if one expresses ˝

Cn D E u F , where E and F are holomorphic sub-bundles, it need not be the case that

56

R.G. Douglas

(iii) R ˝ Cn D ff 2 R ˝ Cn W f .!/ 2 E.!/; ! 2 ˝g u ff 2 R ˝ Cn W f .!/ 2 F .!/; ! 2 ˝g: If we add this assumption for sub-bundles E and F of ˝ Cn , then (iii) is equivalent to (i) and (ii) of Proposition 3. These relationships are at the heart of Nikolski’s lemma as presented by Treil and Wick [20] in their approach to the corona theorem as follows. We begin by placing their framework in the context of the short exact sequence .2/. Assume that f'i gniD1 M.R/ for some RKHM R over the bounded domain ˝ Cm , satisfies (0). For ! 2 ˝ let P .!/ denote the orthogonal projection of Cn onto range ˚.!/ Cn . Since (0) holds, rank ˚.!/ D 1 for ! 2 ˝ which implies that q!2˝ range P .!/ defines a Hermitian holomorphic line bundle over ˝. However, the function P .!/ is holomorphic only when it is constant. But there are other idempotent-valued functions E.!/ on ˝ with range E.!/ D range P .!/ D range ˚.!/ for ! 2 ˝. Moreover, one of them might be both bounded and holomorphic on ˝ (and in M.R/ which is what would be required if M.R/ ¤ H 1 .˝/) to define the module idempotent, denoted by E, needed in the proof of Proposition 2. We capture this fact in the following statement which is a form of Nikolski’s lemma. Proposition 4. For f'i gniD1 M.R/ for the RKHM R over the bounded domain ˝ Cn with M.R/ D H 1 .˝/ satisfying (0), statement (1) is equivalent to (6) There exists a bounded, real-analytic function V .!/ W ˝ ! L.Cn / such that (a)

E.!/ D P .!/ C V .!/ is bounded and holomorphic on ˝ and

(b)

V .!/ D P .!/V .!/.ICn P .!// for ! 2 ˝:

Proof. If (1) holds, then by Proposition 2 there exists a module idempotent E on R ˝ Cn such that range P .!/ D range ˚.!/ D range E.!/. But the latter implies that range P .!/ D rangeE.!/ or that (a) holds. Since E.!/ is an idempotent with range equal to range P .!/, it follows that V .!/ D E.!/ P .!/ satisfies (b) which concludes the proof that (1) implies (6). Conversely, if such a function V .!/ exists, then by setting E.!/ D P .!/CV .!/ we obtain a bounded holomorphic idempotent map on ˝ such that range E.!/ D range ˚.!/ for ! 2 ˝. Therefore, since M.R/ D H 1 .˝/, we can define using E a module idempotent, which we also denote by E, such that range E D range M˚ . Returning to Proposition 2, we see that (4) holds which implies (1). Remark 8. Note that (1) implies (6) does not require that M.R/ D H 1 .˝/ but for the implication (6) implies (1), one needs somehow to conclude that the function E.!/ defines a multiplier on R ˝ Cn .


57

One can obtain another geometrical interpretation of this result by restating Proposition 3 on the existence of the idempotent E in terms of a bounded, holomorphic module idempotent with range M˚ . Proposition 5. Let f'i gniD1 M.R/ for the RKHM R on the bounded domain ˝ Cm satisfying (0) and let E˚ be the holomorphic sub-bundle of ˝ Cn with fiber E˚ .!/ D range ˚.!/ Cn . Then (1) is equivalent to (7) There exists a complementary holomorphic sub-bundle F of ˝ Cn such that R ˝ Cn D ff 2 R ˝ Cn W f .!/ 2 E.!/; ! 2 ˝g u ff 2 R ˝ Cn W f .!/ 2 F .!/; ! 2 ˝g or (8) There exists a complementary submodule S of R ˝ Cn such that R ˝ Cn D range M˚ C S. Remark 9. There is another formulation of the required condition on the complementary holomorphic vector sub-bundle, at least when MR D H 1 .˝/. In particular, recall for subspaces X and Y of Cn , the notion of the angle, ang.X ; Y/, between X and Y, which is defined: inffarc cos.hx; yi =jjxjj jjyjj/ W 0 ¤ x 2 X ; 0 ¤ y 2 Yg: Then one has (7) is equivalent to (7’) There exists a complementary holomorphic sub-bundle F of ˝ Cn such that there exists > 0 such that ang.E.!/; F .!// for ! 2 ˝. The proof is an adaptation of arguments in [8] along the following lines. (1) One notes for ! 2 ˝ that the decomposition Cn D E.!/ u F .!/ determines a unique idempotent on Cn with a bound determined by ang.E.!/F .!//. (2) Using local holomorphic frames for E and F and Cramer’s Rule, one shows that there is a holomorphic idempotent-valued function defined locally on ˝. (3) By the uniqueness of the idempotent at each !, this function at ! 2 ˝ must agree with the one determined in (1) by the decomposition Cn D E.!/uF .!/. Since ang.E.!/F .!// is uniformly bounded below, one obtains a bounded, holomorphic idempotent-valued function on ˝ with range function equal to E which completes the proof. Remark 10. Note that if E is a Hermitian holomorphic sub-bundle of ˝ Cn for a bounded domain ˝ Cm and !0 2 ˝, then there exists a neighborhood V of !0 and a sub-bundle F of V Cn so that EjV u F D V Cn . The issue in .70 / is not a local one but the question of whether one can find F defined for V D ˝ such that ang.E.!/; F .!// is bounded below.

5 Comparing the Hörmander and Treil-Wick Approaches The approach of Treil-Wick to the corona problem is related to the earlier one due to Hörmander [13]. In this case we work on R D L2a .m/, the Bergman space for Lebesque measure m on the unit ball Bm . If f'i gniD1 H 1 .Bm / satisfies (0), then one can write down an L1 .m/-solution to the corona problem as follows:

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One defines j .z/ D 'j .z/=

n X

j'i .z/j2 for z 2 Bm ; and j D 1; 2; : : : ; n

iD1

so that n X

'i .z/i .z/ D 1 for z 2 Bn :

iD1

But, as in the first step leading to Nikolski’s lemma, this solution is unlikely to be an n-tuple of holomorphic functions. To remedy this deficiency, Hörmander seeks C 2 -functions f˛j gnj D1 on Bm so that (a) n X

'i .z/˛i .z/ D 0 for z 2 Bn

iD1

and (b) f i gniD1 H 1 .Bn /, where

i

D ˛i C i for i D 1; : : : ; n.

N This amounts to solving what is referred to as a “@-problem with L1 -bounds”. Let us explore a little more the relationship between the Hörmander and TreilWick approaches. Consider the operator NA .z/ W Cn ! C defined for z 2 Bn such that NA .z/c D

n X

˛i .z/ci

for c D .c1 ; : : : ; cn / 2 Cn :

iD1

Next, we define NA W L2 .m/ ˝ Cn ! L2 .m/ such that NA

n X iD1

! fi ˝ ei

D

n X iD1

˛i fi

for

n X

fi ˝ ei 2 L2 .m/ ˝ Cn :

iD1

Now setting EQ D M˚ .NA CN /, we obtain a module idempotent EQ on L2 .m/˝Cn Q such that E.z/ is an idempotent with range ˚.z/ D range P .z/ for z 2 Bm . Hence, Q range E.z/ D range P .z/ for z 2 Bm . Thus one is seeking a modification, A.z/, of .z/ so that NA C N is bounded, A.z/ C .z/ is holomorphic, and range .NA C N / D range N , which are essentially the same conditions required for P C V in


59

the Treil-Wick approach. By (b) it follows that .NA C N /.R ˝ Cn / R ˝ Cn , Q R˝Cn defines a module idempotent on R ˝ Cn . thus; EQQ D Ej In particular, by (b) we have M˚ NA D 0 and hence EQQ D M˚ .N C NA / is a module idempotent on R ˝ Cn . Moreover, clearly range EQQ is contained in range QQ ˝ Cn /. Since EQQ D Ej Q R˝Cn , we see that range EQQ D range EQ R˝Cn D M˚ D EQ E.R range M˚ which shows that (a) and (b) yield the module idempotent EQQ on R ˝ Cn which establishes (1). In other words, in both approaches, one has a module idempotent on L2 .m/ ˝ n C with the appropriate range function and one seeks to modify it to another idempotent-valued map keeping the range function the same but making the perturbed function bounded and holomorphic. Both Hörmander and Treil-Wick complete the proof for the case Bm D D or m D 1, using nontrivial methods from harmonic analysis involving the @N problem and the Koszul complex for Hörmander and Hankel forms for Treil-Wick. We will not consider them. (A considerably simpler version of the Hörmander approach has been given by Wolff (cf. [9]).)

6 Toeplitz Corona Problem and the CLT Property Another way to relate the corona problem to operator theory is by weakening the conditions on the solution functions by requiring that they lie in R as opposed to lying in M.R/, which is contained in R since 1 2 R. A multiplier 2 M.R/ for a RKHM R over a bounded domain ˝ Cm defines the Toeplitz operator T R 2 L.R/ such that T R f D f for f 2 R. Let f'i gniD1 M.R/ for some RKHM R over a bounded domain ˝ Cm . One says that there is a weak solution to the corona problem if (9) There exists an n-tuple ffi gniD1 R such that n X

'i .!/fi .!/ D 1 for ! 2 ˝

iD1

and a solution to the Toeplitz corona problem if (10) For every f 2 R there exists ffi gniD1 R such that n X

'i .!/fi .!/ D f .!/ for ! 2 ˝:

iD1

Since this statement involves R, one sometimes speaks of the R-Toeplitz corona problem.

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There are various relationships between statement (0)–(10). We explore some of them that emphasize a somewhat weaker version of the corona problem introduced by several authors including Arveson and Schubert, (cf. [9]) in the seventies which is related to the classical Toeplitz operators. Proposition 6. For f'i gniD1 M.R/ for a RKHM R over a bounded domain ˝ Cm , (1) implies (10) which in turn implies (9). Moreover, (10) is equivalent to either (11) N˚ is onto or (12) n X

k T'Ri f k2 2 k f k2 for f 2 R for some > 0:

iD1

Proof. The equivalence of (10) and N˚ being onto follows from the definition of N˚ and basic operator theory. Similarly, since N˚ f D

n X

T'Ri f ˝ ei for f 2 R;

iD1

N˚ being onto is equivalent to N˚ being bounded below which is statement (12). Note that the constants in both (0) and (12) are the same. The question of whether (11) or (12) implies (1) is what is usually referred to as the Toeplitz corona problem for the algebra M.R/ for some RKHM R for a bounded domain ˝ Cm . Schubert in [16] showed that (12) implies (1) for R D H 2 .D/ and hence (9)–(12) are all equivalent in this case. Thus for a RKHM R one can “divide” the corona problem for M.R/ into two parts: For f'i gniD1 satisfying .0/, first show that (11) holds for some RKHM R with f'i gniD1 M.R/ and second, show that (11) for R implies (1). Note that although the strategy doesn’t involve R at the conclusion - only M.R/ 0 0 if (1) holds, then (11) must hold for any RKHM R with M.R / D M.R/ with the same . For the special class of RKHM, which satisfy the commutant lifting theorem or have the (CLT) property, one has (11) implies (1) (cf. [1]). Note that since H 2 .D/ satisfies the CLT, this shows that (11) implies (1) in this case. Definition 1. A RKHM R is said to have the CLT property if for submodules S1 and S2 of R˝Cm and R˝Cn , respectively, and a module map X W R˝Cm =S1 ! R˝ Cn =S2 , there exists a module map XN W R ˝ Cm ! R ˝ Cn such that XS1 D S2 XN and k XN kDk X k, where Si is the quotient map from R ˝ Cn onto R ˝ Cn =Si for i D 1; 2, respectively. Theorem 1. Let R be an RKHM over a bounded domain ˝ Cm having the CLT property and let f'i gniD1 M.R/ satisfy (11). Then (1) holds.


61

Proof. Set S D ker N˚ R ˝ Cn and then consider Y W .R ˝ Cn =S / ! R such that N˚ D Y S . Then (11) implies that X D Y 1 is a bounded module map and N Since S D XN˚ , the CLT property yields XN W R ! R ˝ Cn so that X D S X. N N N one has X D S X D XN˚ X , which implies N˚ X D IR since X is invertible. By Lemma 1, XN D M for some f i gniD1 M.R/ or n X

'i .!/

i .!/

D 1 for ! 2 ˝;

iD1

which completes the proof. Remark 11. There is another operator theory question related to the corona problem - the similarity of certain quotient Hilbert modules of a RKHM R over a bounded domain ˝ Cm to R ˝ Ck for the appropriate multiplicity k. For ˝ D D, this question goes back to the canonical models for contraction operators of Sz-Nagy and Foias [17]. More recently, this question was considered for R the Drury-Arveson space over the unit ball Bm [3]. In both cases, one is considering a RKHM with the CLT property.

7 Toeplitz Corona Problem and Logmodular Algebras There is another method of showing that (11) implies (1) for some RKHM R which was developed by Agler-McCarthy, Amar and culminating in [22] by Trent-Wick (cf. [9]), and then extended modestly in [11]. The proof given in the latter paper can be used, without change, to prove the following result. Theorem 2. Let R be a subnormal RKHM over a bounded domain ˝ Cm for the probability measure on ˝N such that (i) fk! g!2˝ M.R/; (ii) range TkR! is closed for ! 2 ˝; and N N (iii) for f!j gN j D1 ˝ and fj gj D1 C ; there exists g 2 R such that jg.!/j2 D

N X

jj j2 jk!j .!/j2 ;

a.e.

j D1

Then for f'i gniD1 M.R/ satisfying (0), (13) an affirmative answer to the Toeplitz corona problem (10) for all submodules of R of the form range T R for 2 M.R/ with the same , implies an affirmative answer to the corona problem (1). The Hardy module H 2 .D/ on the unit disk satisfies (i), (ii) and (iii) since the kernel functions are continuous on @D and all nonnegative bounded measurable functions on @D are the boundary values of a function in H 1 .D/. In [23], Trent and

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Wick show that the Hardy modules H 2 .Bm / and H 2 .Dm / also satisfy (i), (ii) and (iii) although the proof requires somewhat more effort. The idea here is to consider all the solutions to (10) in R and show, using the von Neumann min-max theorem, that one can bound the values at all finite sets of points in ˝. A normal family argument completes the proof. However, the Toeplitz corona solution for the family of submodules must have the same as is expressed in (13). In the proof one considers submodules S of R that are the range of Toeplitz operators defined by multipliers with closed range which is what conditions (i) and (ii) provide. Condition (iii) allows one to replace a kind of “convex combination” of such submodules by one submodule. As indicated above, the proof of the theorem and the earlier versions of it along these lines rely on an affirmative answer to the Toeplitz corona problem not just for a subnormal RKHM R over a bounded domain ˝ Cm for a probability measure N but for a whole family of RKHM. The family can be restricted supported on ˝, to cyclic submodules where the generator can be taken to be invertible in L1 ./. In some cases it suffices to assume (11) for just R which is what happens in the classical case of R D H 2 .D/, since by Beurling’s theorem all (cyclic) submodules of H 2 .D/ are isometrically isomorphic to H 2 .D/ itself. That doesn’t happen very often (cf. [10]). Although it does if one has a subalgebra of M.R/ which is a Dirichlet algebra. But one can get by with less. Definition 2. A subnormal RKHM R over a bounded domain ˝ Cm for the probability measure supported on ˝N is said to be invertibly approximating in modulus if for every nonnegative function 2 L1 ./ such that the operator M W L2a ./ ! L2 ./ has closed range and ı > 0, there exists 2 M.R/1 such that j.z/ j .z/jj < ı a.e. A logmodular algebra is invertibly approximating in modulus but the converse is unclear. In any case, the former notion has the following implication for the cyclic submodules of the RKHM R defined by a Toeplitz operator with closed range. The closeness of the range of M on L2a ./ will depend on being bounded below near the boundary of ˝, in some sense. Definition 3. Two Hilbert modules M1 and M2 over a bounded domain ˝ Cm are said to be almost isometric if for every ı > 0 there exists a module isomorphism X W M1 ! M2 such that max .k X k; k X 1 k/ < 1 C ı. Our interest in these two notions lies in the following results. 0

Proposition 7. If R and R are almost isometric RKHM over the bounded domain 0 ˝ Cm , then R and R are isometrically isomorphic Hilbert modules. Proof. We adapt the isomorphism result in [8] that establishes curvature as a complete unitary invariant for operators in the class B1 .˝/ (cf. [5]). Since there is an anti-holomorphic cross-section for the bundle ER for the RKHM R over the bounded domain ˝ Cm , the result will follow once we show that the curvatures


63

for ER and ER0 are equal. The latter will follow once we show that the restrictions 0 of the adjoints of the module actions of CŒz1 ; : : : ; zm on R and R to the invariant subspaces R.!/ D

\ '.!/D0

0

ker.T'R2 / and R .!/ D

\

0

ker.T'R 2 /;

'.!/D0 0

respectively, are unitarily equivalent for ! 2 ˝. But R and R almost isometric implies that they are isomorphic with a similarity Xı satisfying maxfk Xı k; k 0 Xı1 kg 1 C ı for ı > 0. Since the spaces R.!/ and R .!/ have dimension 0 m C 1, it follows that R.!/ and R .!/ are unitarily equivalent for ! 2 ˝ which completes the proof. Remark 12. It seems unlikely that all pairs of almost isometric Hilbert modules are in fact unitarily equivalent. However, that is the case for a pair of Hilbert modules 0 M and M over CŒz1 ; : : : ; zm for which the common eigenspaces for each module 0 are finite dimensional and they span M and M , respectively. Proposition 8. Let R be the subnormal RKHM L2a ./ over a bounded domain ˝ Cm and the probability measure supported on ˝N such that M.R/ D H 1 .˝/. If H 1 .˝/ is invertibly approximating in modulus, then for ' 2 H 1 .˝/ such that 0 0 T'R has closed range R , R and R are isometrically isomorphic. Proof. Since range T'R is closed, it follows for ı > 0 there exists 2 M.R/1 such that k j'j j j k1 < ı. Assuming ı 0 is arbitrary, we 0 0 have that R and R are almost isometric. Thus by the previous result, R and R are isometrically isomorphic. Theorem 3. Let R be a subnormal RKHM over a bounded domain ˝ Cm for the probability measure supported on ˝N such that M.R/ D H 1 .˝/ and such that H 1 .˝/ is invertibly approximating in modulus. Then (6) for R implies it for all cyclic submodules that are the range of T R for some 2 H 1 .˝/ such that R T has closed range. Proof. By hypothesis, the multiplier algebra is invertibly approximating in modulus and the rest follows by combining the previous two propositions. From this result it follows for a subnormal RKHM R over a bounded domain ˝ Cm such that (i) if H 1 .˝/ has the invertibly approximating in modulus property, (ii) satisfies the hypotheses in Theorem 1, and (iii) has the Toeplitz corona property (12), then the corona property (1) is valid.

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Remark 13. Although H 1 .Bm / satisfies the hypotheses of Theorem 1, unfortunately, it is NOT invertibly approximating in modulus. For example, while there is a function 2 H 1 .Bm / such that j ./j2 D 1 C j1 j2 on @Bm a.e. (cf. [23]), for m > 1, cannot be chosen to be invertible. In particular, the function z1 ! .z1 ; : : : ; zm / would be outer for each .z2 ; : : : ; zm / 2 Bm1 and since it has constant modulus it must be constant. Further consideration shows that j .z/j2 D 1 C jzj2 for z 2 Bm which is not possible. It seems possible that results in [7] can be used to show that the assumption that H 1 .˝/ is invertibly approximating in modulus implies that m D 1 and that ˝ is conformally equivalent to D. Theorem 4. Suppose R is a subnormal RKHM over a bounded domain ˝ Cm for the probability measure supported on ˝N such that H 1 .˝/ D M.R/ is an invertibly approximating in modulus algebra. Then (12) for R implies it for all cyclic submodules of R for the same . In particular, if the Toeplitz corona property (6) holds for R, then it holds for all submodules of R that are the range of a T R for some 2 M.R/. Corollary 1. If, in addition to the hypotheses of Theorem 4, the hypotheses of Theorem 1 hold, then (12) implies (1). Remark 14. In [12] Hamilton-Raghupathi show that one can use another technique in operator theory, factorization in dual algebras, to prove some general Toeplitz corona theorems. Combining these results with some recent results of Prunaru yields Toeplitz corona theorems for Bergman spaces over certain domains in Cm . This topic merits further exploration.

8 Taylor Spectrum and the Corona Problem In [18] Taylor introduced a notion of joint spectrum for n-tuples of commuting elements in a Banach algebra which also applies to an n-tuple of commuting operators .T1 ; : : : ; Tn / on a Hilbert space H. The Taylor spectrum, TAY .T1 ; : : : ; Tn /, is a nonempty compact subset of Cn for which there is a good holomorphic functional calculus. Moreover, Taylor shows that the existence of an n-tuple .S1 ; : : : ; Sn / of and the fTi gniD1 satisfying Pnoperators on H which commute with each other TAY .T1 ; : : : ; Tn /. In particular, iD1 Ti Si D IH implies that 0 D .0; : : : ; 0/ … for f'i gniD1 M.R/ for a RKHM R over a bounded domain ˝ Cn , a necessary condition for (1) to hold or for the corona problem to have an affirmative solution is for (13) 0 … TAY .T'R1 ; : : : ; T'Rn /. The origin 0 is not in the Taylor spectrum precisely when the Koszul complex, built from the n-tuple .T'R1 ; : : : ; T'Rn /, is exact. We won’t recall the definition of the full Koszul complex but only for the sequence at the first and the last nonzero


65

N˚

M˚

modules which are 0 ! R ! R ˝ Cn and R ˝ Cn ! R ! 0. Hence, (13) or exactness of the Koszul complex implies that the range of M˚ is closed and that N˚ is onto. In particular, the assumption of exactness implies (2) and (10) or that the Toeplitz corona problem has an affirmative solution. These considerations suggest the following: Question 1. Does the exactness of the Koszul complex for .T'R1 ; : : : ; T'Rn / for some RKHM R over a bounded domain ˝ Cn imply (1) or that the corona problem has an affirmative solution? 0

If that were true, one would know that for any two RKHM R; R over a bounded 0 domain ˝ Cm with M.R/ D M.R /, one would have TAY .T'R1 ; : : : ; T'Rn / D 0

0

TAY .T'R1 ; : : : ; T'Rn /. Appealing to Theorem 2, we have a positive result if R has the CLT property. Corollary 2. For a RKHM R over a bounded domain ˝ Cm with the CLT TAY property and f'i gnzD1 M.R/; 0 … TM.R/ .T'i ; : : : ; T'n / iff (1) holds or there n exists f i giD1 M.R/ such that n X

i 'i

D 1:

iD1

In [19] Taylor considers in more detail the possible implication of when 0 … TAY .T'R1 ; : : : ; T'Rn /, implies (1). We prove the following related result after introducing some notions and propositions. Theorem 5. Let f'i gniD1 M.R/ for the RKHM R over a bounded domain ˝ Cm and set ˚.!/ D Œ'1 .!/; : : : ; 'n .!/ 2 Cn for ! 2 ˝: If 0 … Pol.˚.˝// D closed polynomial convex hull of f˚.!/ W ! 2 ˝g Cn , then (1) holds or the corona problem has an affirmative solution. For a bounded subset X Cn , its polynomial convex hull, Pol.X /, is defined Pol.X / D fz0 2 Cn W jp.z0 /j sup jp.z/j for p 2 CŒz1 ; : : : ; zn g: z2X

Let P .X / denote the function algebra obtained by completing the restriction of CŒz1 ; : : : ; zn to X in the supremum norm. (Note that if XN is the closure of X in Cm , N D P .X /.) One way in which the polynomial then Pol.XN / D Pol.X / and P .X/ convex hull arises is in the following well-known result: Proposition 9. For a bounded subset X of Cn ; MP .X/ D Pol.X /: Proof. First, note that for z0 2 Pol.X /, it follows that evaluation at z0 defines a bounded multiplicative linear functional on P .X / which yields an embedding of Pol.X / into MP .X/ . Second, if L is a bounded multiplicative linear functional

66

R.G. Douglas

on P .X /, then .L.z1 /; : : : ; L.zn // 2 Cn determines a point in Pol.X / at which evaluation is identical to L. This identification completes the proof. Proof (Proof of Theorem 6.). If ˚ D .'1 ; : : : ; 'n / W ˝ ! Cn , then the assumption that 0 … Pol.˚.˝// implies the existence of functions f i gniD1 P .˚.˝// satisfying n X

zi i .z1 ; : : : ; zn / D 1 for .z1 ; : : : ; zn / 2 Pol.˚.˝//:

iD1

This follows because the ideal generated by z1 ; : : : ; zn in Pol.X / can’t be proper or evaluation at 0 would define a multiplicative linear functional on P .X /. Hence, the ideal contains the constant function 1. Now setting i .!1 ; : : : ; !m /; D

i .'1 .!1 ; : : : ; !m /; : : : ; 'n .!1 ; : : : ; !m //

for .!1 ; : : : ; !m / 2 ˝ yields functions f n X

'i .!/

i .!/

n i giD1

H 1 .˝/ satisfying

D 1 for ! 2 ˝;

iD1

which completes the proof. Remark 15. If the basic algebra is not H 1 .˝/ but is some other Banach algebra of holomorphic functions A, then it seems likely that one could adapt the argument in Taylor [19] to reach the same conclusion with the functions constructed being in A. Corollary 3. For f'i gniD1 M.R/ D H 1 .˝/ for the RKHM R over a bounded domain ˝ Cm , if there is a p.z/ 2 CŒz1 ; : : : ; zn such that jp.0/j > sup jp.'1 .!/; : : : ; 'n .!//j; !2˝

then (1) holds or the corona problem has an affirmative solution. As indicated in the previous remark, it is likely that one doesn’t need to assume M.R/ D H 1 .˝/.

9 Taylor Spectrum For the Drury-Arveson Algebra Since the multiplier algebra M.Hm2 / for the Drury-Arveson space Hm2 is contained in H 1 .Bm /, there exists a continuous map W MH 1 .Bm / ! MM.Hm2 / . Using the result of Costea, Sawyer and Wick [4 ] establishing the corona theorem for M.Hm2 /, one sees that is onto. As a consequence, one has the following result.


Theorem 6. For f'i gniD1

H 2 .@Bm /

M.Hm2 /, HTAY 1 .Bm / .T'1

67 H 2 .@Bm

; : : : ; T'n

/

D

Hm2 Hm2 TAY M.H 2 / .T'1 ; : : : ; T'n /: m

It would seem that the fibers fx 2 MH 1 .Bm / W .x/ D !g for ! 2 MM.Hm2 / are, in general, not trivial. The corona problem for H 1 .Bm / reduces to subtle questions of the structure of these fibers, but this point of view doesn’t seem to simplify the problem. Remark 16. A related issue concerns the Taylor spectrum TAY , or the essential Taylor spectrum, eTAY , respectively for an n-tuple of Toeplitz operators fT'Ri gniD1 for f'i gniD1 in M.R/ for R a RKHM over a bounded domain ˝ Cm . (Here, the essential Taylor spectrum refers to the Taylor spectrum for the n-tuple modulo the ideal of compact operators or, in the Calkin algebra, Q.R/.) For R D H 2 .Bm /, the Hardy space, or on Hm2 , the Drury-Arveson space, respectively, both over the unit ball, Bm in Cm , Proposition 5 (iii) in [6] shows that eTAY .T'R1 ; : : : ; T'Rn / D \f˚.U \ Bm / W U open; closU @Bm g: Here one needs to know that a function holomorphic on a neighborhood of closBm is in MH 2 .Bn / and MHm2 which follow using the holomorphic functional calculus. The related result for the Taylor spectrum with R D Hm2 has been obtained in [3] using the authors’ corona theorem for MHm2 . Remark 17. One can consider another pair of algebras and the corresponding mapping W MH 1 .D/ ! MM.D/ , where D is the Dirichlet space. Here we don’t know that is onto which is equivalent to the corona theorem for D.

References 1. Joseph A. Ball, Tavan T. Trent, and Victor Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert space. In: Operator Theory and Analysis, Amsterdam, 1987, Oper. Thy. Adv. Appl. pp. 122, Birkhauser, Basel (2001), 89–138. 2. Lennart Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. Math. (2) 76 (1962), 547–559. 3. Serban Costea, Eric Sawyer, and Brett Wick, The Corona Theorem for the Drury-ArvesonHardy space and the holomorphic Besov-Sobolev spaces on the unit ball in Cn , Anal. & PDE 4 (2011), 499–550. 4. Serban Costea, Eric Sawyer, and Brett Wick, The Taylor spectrum of analytic Toeplitz Tuples on Besov-Sobolev spaces. Preprint. 5. M. J. Cowen and Ronald G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), 187–261. 6. Ronald G. Douglas and Jörg Eschmeier, Spectral inclusion theorems. Oper. Thy. Adv. Appl. 222 (2012), 113–128. 7. Ronald G. Douglas, Ciprian Foias, and Jaydeb Sarkar, Resolutions of Hilbert modules and similarity, J. Geom. Anal. 22 (2012) no. 2, 471–490. 8. Ronald G. Douglas, Yun-Su Kim, Hyun-Kyoung Kwon and Jaydeb Sarkar, Curvature invariants and generalized canonical operator models - II, J. Funct. Anal. 266 (2014), 2486–2502.

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9. Ronald G. Douglas, Steve Krantz, Eric Sawyer, Sergei Treil, and Brett Wick, A History of the Corona Problem, pre-print. 10. Ronald G. Douglas and Jaydeb Sarkar, On unitary equivalent submodules, Indiana U. Math. J. 57 (2008), no. 6, 2729–2743. 11. Ronald G. Douglas and Jaydeb Sarkar, Some remarks on the Toeplitz corona problem, Hilbert spaces of analytic functions, CRM Proc. Lecture Notes 51, Amer. Math. Soc. Providence, RI (2010), 81–89. 12. Ryan Hamilton and Mrinal Raghupathi, The Toeplitz corona problem for multipliers on Nevanlinna-Pick space, Indiana U. Math. J. 61 (2012) 1393–1405. 13. Lars Hörmander, An introduction to complex analysis in several variables, van Nostrand, Princeton, 1966. 14. C. Jiang and Z. Wang, Structure of Hilbert Space Operators, World Scientific, Singapore (2006). 15. I. J. Schark, Maximal ideals in an algebra of bounded analytic functions, J. Math. Mech. 10 (1961), 735–746. 16. C. F. Schubert, The corona theorem as an operator theorem, Proc. Amer. Math. Soc. 69 (1978), no. 1, 73–76. 17. Bela Sz-Nagy and Ciprian Foias, Harmonic Analysis of Operators on Hilbert Space, North Holland, Amsterdam (1970). 18. Joseph L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970), 172–191. 19. Joseph L. Taylor, The Analytic Functional Calculus for Several Commuting Operators, Acta Math. 125 (1970), 1–38. 20. Sergei Treil and Brett D. Wick, Analytic projections, corona problem and geometry of holomorphic vector bundles, J. Amer. Math. Soc. 22 (2009), no. 1, 55–76. 21. Tavan T. Trent, The matrix-valued H p -corona problem in the disc and polydisk, J. Funct. Anal. 226 (2005), no. 1, 138–172. 22. Tavan T. Trent, Operator Theory and the Corona Problem on the Bidisk, Current Trends in Operator Theory and its Applications, Oper. Thy.: Adv. Appl., Volume 149 (2004), pp. 503–588. 23. Tavan T. Trent and Brett D. Wick, Toeplitz corona theorems for the polydisk and the unit ball, Complex Anal. Oper. Theory 3 (2009), no. 3, 729–738.

On the Maximal Ideal Space of a Sarason-Type Algebra on the Unit Ball Jörg Eschmeier

Abstract In this note we study the maximal ideal space and invertibility problems in a Sarason-type algebra A on the unit ball. By definition A consists of all bounded measurable functions on the unit sphere for which the associated Hankel operator on the Hardy space is compact. We determine a natural closed subset of the maximal ideal space of A and apply our results to show that the essential Taylor spectrum of all Toeplitz tuples with symbols in A is connected. Keywords Sarason algebra • Maximal ideal space • Toeplitz tuples • Essential spectrum Subject Classifications: Primary 47A13; Secondary 47B35, 46J15

1 Introduction It was shown by Sarason that the space H 1 C C L1 .T/ is a closed subalgebra on the unit circle T and that its maximal ideal space is obtained from the maximal ideal space of H 1 by deleting the unit disc (see e.g. [15]). Here C is the space of all continuous functions on T and H 1 consists of the boundary values of all bounded holomorphic functions on the unit disc. In [13] Rudin proved that H 1 .S / C C.S / L1 .S / is a closed subalgebra on the unit sphere S D @Bn Cn and McDonald [11] observed that Sarason’s result on the maximal ideal space remains true in this setting. A classical result of Hartman says that on the unit circle the equality J. Eschmeier () Fachrichtung Mathematik, Universität des Saarlandes, Postfach 15 11 50, 66041 Saarbrücken, Germany e-mail: [email protected]

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H 1 C C D ff 2 L1 .T/I Hf is compactg holds, where Hf is the Hankel operator with symbol f . It was observed by Davie and Jewell [2] that the result of Hartman is no longer true on the unit sphere. While it is elementary to see that A D ff 2 L1 .S /I Hf is compactg L1 .S / is still a closed subalgebra it turns out that the inclusion H 1 .S / C C.S / A is strict in every dimension n > 1. A number of recent results indicates that for many questions the Banach algebra A is the appropriate substitute of H 1 C C in the multivariable case. For instance, generalizing a one-variable result of Davidson [1], Ding and Sun [5] (see also [3]) showed that on the unit sphere the essential commutant of all analytic Toeplitz operators consists precisely of the compact perturbations of all Toeplitz operators with symbol in A . This result immediately implies that the quotient T .A /=K .H 2 .S // of the Banach algebra T .A / generated by all Toeplitz operators with symbol in A modulo the compact operators is a maximal abelian subalgebra of the Calkin algebra on the Hardy space H 2 .S /. In the present note, we show that via the Poisson transform the point evaluations at points in the corona ˇ.B/nB of the Stone-C˘ech compactification ˇ.B/ of the open unit ball B Cn define a closed subset of the maximal ideal space A . This can be seen as an extension of the well known classical result saying that the Poisson transform is asymptotically multiplicative on H 1 C C . As a consequence we show that the cluster set of the Poisson transform PŒf of every function f 2 A is contained in the spectrum of f in the Banach algebra A . Since A .f / coincides with the essential spectrum e .Tf / of the Toeplitz operator Tf 2 L.H 2 .S //, this gives at least a lower bound for the essential spectrum of Toeplitz operators with symbol in A . Using the observation that the Banach algebra A contains no non-trivial idempotents, we show that the essential Taylor spectrum of every Toeplitz tuple Tf D .Tf1 ; : : : ; Tfm / 2 L.H 2 .S //m with symbols fi 2 A is connected. Thus we partially extend one-variable results of Douglas [6] on the invertibility of functions in H 1 CC and the essential spectrum of Toeplitz operators to the symbol algebra A on the unit sphere. For n D 1 the Corona theorem implies that there is a natural surjection from ˇ.B/ n B onto the character space of A representing each character as a point evaluation at a suitable point in the corona of B in ˇ.B/. The question whether this result holds in general, and the closely related question whether the spectrum A .f / of every function f 2 A coincides with the cluster set of its Poisson transform, are left open in this note.

2 Main Results Let be the surface measure on the topological boundary S D @B of the unit ball B D fz 2 Cn I jzj < 1g Cn . Define K W B B ! C; K.; z/ D

1 .1 h; zi/n

On the Maximal Ideal Space of a Sarason-Type Algebra on the Unit Ball

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1

and kz W S ! C; kz ./ D K.z; z/ 2 K.; z/ for z 2 B. Then KjB B is the reproducing kernel of the Hardy space Z H .B/ D ff 2 O.B/I kf k2 D sup .

jf .r/j2 d ./ /1=2 < 1g:

2

0 0 for every open neighbourhood W of wg is contained in the cluster set C.F / of F . Indeed since F D f almost everywhere, for a given element w 2 R.f /, we can choose a sequence .zk /k1 in @B with jf .zk / wj < k1 and limr"1 F .rzk / D f .zk / for every k. Hence there is a sequence k

k

.k /k1 D .rk zk /k1 in B with .jk j/ ! 1 and .F .k // ! w. Let us denote by QC D ff 2 A I f 2 A g the largest C -subalgebra of A . Corollary 1. For f 2 QC and F D PŒf , we have R.f / D L1 . / .f / D A .f / D QC .f / D C.F /: Proof. Fix a function f 2 L1 . /. It is well known and elementary to check that R.f / D L1 . / .f /. Since QC is a C -algebra and QC A L1 . /, we find that R.f / D L1 . / .f / A .f / QC .f / D L1 . / .f /: By Theorem 4 and the remarks preceding the corollary we know that R.f / C.F / A .f /: Hence all these sets coincide. It was observed by J. Xia in [17] that the C -algebra QC consists precisely of those functions f 2 L1 . / which have vanishing mean oscillation (Sect. 6 in [17]). The equality R.f / D C.F / for functions with vanishing mean oscillation on the unit sphere was proved before using a different method by J. H. Shapiro in [16]. As a direct consequence of Corollary 1 we obtain that the Banach algebra A contains no non-trivial idempotents (cf. Proposition 5.3 in [17]). Indeed, if p 2 A is an idempotent, then R.p/ A .p/ f0; 1g. Hence p 2 QC and R.p/ D C.PŒp / f0; 1g: Since intersections of downwards directed families of compact connected sets are connected, the cluster set C.PŒp / is connected. Therefore p D 0 or p D 1. As another application we can show that the essential Taylor spectrum of Toeplitz tuples with symbols in A is connected. For the definition and properties of the essential Taylor spectrum of almost commuting operator tuples, we refer the reader to [9].

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Corollary 2. For f 2 A m , the essential Taylor spectrum e .Tf / of the Toeplitz tuple Tf D .Tf1 ; : : : ; Tfm / 2 L.H 2 . //m is connected. Proof. For simplicity we write H D H 2 . //. By definition (Sect. 10.3 in [9]) the essential Taylor spectrum e .Tf / of Tf coincides with the Taylor spectrum .LTf ; C .H // of the commuting tuple LTf D .LTf1 ; : : : ; LTfm / 2 L.C .H //m of left multiplication operators LTfi W C .H / ! C .H /; ŒX 7! ŒTfi X : Assume that e .Tf / D 1 [ 2 is a disjoint union of two non-empty compact subsets i . Let p 2 O. 1 [ 2 / be an analytic function with p D 0 near 1 and p D 1 near 2 . Applying Taylor’s analytic functional calculus we obtain an operator p.LTf / 2 L.C .H // in the bicommutant of LTf (Theorem 2.5.7 in [9]). Since all right multiplication operators on C .H / belong to the commutant of LTf , there is an operator C 2 L.H / such that p.LTf /ŒX D ŒCX

.X 2 L.H //:

Since A Š T .A /=K .H / is a commutative algebra, we conclude that ŒC Tg D p.LTf /LTg Œ1 D LTg p.LTf /Œ1 D ŒTg C holds for all g 2 A . Hence ŒC 2 .T .A/=K .H //0 D T .A/=K .H /. Choose a function g 2 A with ŒTg D ŒC . Then p.LTf / D LTg leaves T .A/=K .H / invariant and A .g/ D T .A /=K .H / .ŒTg / D .p.LTf // D f0; 1g: Hence A would contain a non-trivial idempotent. This contradiction shows that e .Tf / is connected.

References 1. K. Davidson, On operators commuting with Toeplitz operators modulo the compact operators, J. Funct. Anal. 24 (1977), 291–302. 2. A.M. Davie, N.P. Jewell, Toeplitz operators in several variables, J. Funct. Anal. 26 (1977), 356–368. 3. M. Didas, J. Eschmeier, K. Everard, On the essential commutant of analytic Toeplitz operators associated with spherical isometries, J. Funct. Anal. 261 (2011), 1361–1383. 4. M. Didas, J. Eschmeier, Derivations on Toeplitz algebras, Canad. Math. Bull., to appear. 5. X. Ding, S. Sun, Essential commutant of analytic Toeplitz operators, Chinese Science Bulletin 42 (1997), 548–525. 6. R.G. Douglas, Toeplitz and Wiener-Hopf operators in H 1 C C , Bull. Amer. Math. Soc. 74 (1968), 895–899.

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7. R.G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Math., Vol. 49, New York, 1972. 8. R.G. Douglas, J.Eschmeier, Spectral inclusion theorems, Oper. Theory Adv. and Appl., Vol. 222, 113–128, Springer, Basel, 2012. 9. J. Eschmeier, M. Putinar, Spectral decompositions and analytic sheaves, LMS Monograph Series, Didas and Eschmeier meanwhile appeared in Canad. Math. Bull. 57 (2014), 270–276. 10. G. McDonald, Fredholm properties of a class of Toeplitz operators, Indiana Univ. Math. J. 26 (1977), 567–576. 11. G. McDonald, The maximal ideal space of H 1 C C on the unit ball, Can. J. Math. 31 (1979), 79–86. 12. J.R. Munkres, Topology. A first course, Prentice Hall, Inc., Englewood Cliffs, NJ, 1975. 13. W. Rudin, Spaces of type H 1 C C , Ann. Institut Fourier 25 (1975), 99–125. 14. W. Rudin, Function theory in the unit ball of Cn , Springer, Berlin, 2003. 15. D. Sarason, Algebras of functions on the unit circle, Bull. Amer. Math. Soc. 79 (1973), 286–299. 16. J. L. Shapiro, Cluster set, essential range, and distance estimates in BMO, Michigan Math. J. 34 (1987), 323–336. 17. J. Xia, Bounded functions of vanishing mean oscillation on compact metric spaces, J. Funct. Anal. 209 (2004), 444–467. 18. D. Zheng, Toeplitz operators and Hankel operators on the Hardy space of the unit sphere, J. Funct. Anal. 149 (1997), 1–24

A Subalgebra of the Hardy Algebra Relevant in Control Theory and Its Algebraic-Analytic Properties Marie Frentz and Amol Sasane

Abstract We denote by A0 C APC the Banach algebra of all complex-valued functions f defined in the closed right halfplane, such that f is the sum of a holomorphic function vanishing at infinity and a “causal” almost periodic function. We give a complete description of the maximum ideal space M.A0 C APC / of A0 C APC . Using this description, we also establish the following results: 1. The corona theorem for A0 C APC . 2. M.A0 C APC / is contractible (which implies that A0 C APC is a projective free ring). 3. A0 C APC is not a GCD domain. 4. A0 C APC is not a pre-Bezout domain. 5. A0 C APC is not a coherent ring. The study of the above algebraic-anlaytic properties is motivated by applications in the frequency domain approach to linear control theory, where they play an important role in the stabilization problem. Keywords Maximal ideal space • Corona theorem • Contractability • GCD • Pre-Bezout • Coherence Subject Classifications: Primary 30H80; Secondary 46J20, 93D15, 30H05

M. Frentz • A. Sasane () Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK e-mail: [email protected]; [email protected] R.G. Douglas et al. (eds.), The Corona Problem, Fields Institute Communications 72, DOI 10.1007/978-1-4939-1255-1__5, © Springer Science+Business Media New York 2014

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1 Introduction The aim in this paper is to study the algebraic-analytic properties of a new Banach algebra which is relevant in control theory as a class of stable transfer functions. This Banach algebra, denoted by A0 C APC , which is defined below, is a bigger Banach algebra than the well known algebra AC used widely in control theory since the 1970s (see [4, 5]), for systems described by PDEs and delay-differential equations. Recall that AC consists of all Laplace transforms of complex Borel measures on R with support contained in the half-line Œ0; C1/, and such that does not have a singular non-atomic part. Let C0 WD fs 2 C W Re.s/ 0g. Then by the Lebesgue decomposition, it follows that X 8 fk e stk .s 2 C0 /; ˆ .s/ O D fba .s/ C ˆ < k0 AC D O W C0 ! C W fa 2 L1 Œ0; C1/; .fk /k0 2 `1 ; ˆ ˆ : t0 D 0 < t1 ; t2 ; t3 ; :

9 > > = > > ;

:

AC is a Banach algebra with pointwise operations, and the norm taken as the total variation of the measure, namely kk O AC WD jj.Œ0; C1// D kfa kL1 Œ0;C1/ C k.fk /k0 k`1 ;

O 2 AC :

The algebra AC is relevant in control theory as a class of stable transfer functions because it maps Lp inputs to Lp outputs for all 1 p C1: indeed, if O 2 AC , 1 p C1, u 2 Lp Œ0; C1/, then y WD u 2 Lp Œ0; C1/ and moreover, sup 0¤u2Lp

kykp kk O AC : kukp

In fact, if p D 1 or p D C1, then we have equality above. On the other hand, another widely used Banach algebra in control theory serving as a class of stable transfer functions is the Hardy algebra H 1 of the half-plane, consisting of all bounded and holomorphic functions f defined in the open right half-plane C>0 WD fs 2 C W Re.s/ > 0g; again with pointwise operations, but now with the supremum norm: kf k1 D sup jf .s/j; s2C>0

f 2 H 1:

Algebraic-Analytic Properties of A0 C APC

87

It is well-known that if f 2 H 1 and u 2 L2 Œ0; C1/, then y defined by y.s/ O WD f .s/Ou.s/ (s 2 C>0 ) is such that y 2 L2 Œ0; C1/. Moreover, we have sup 0¤u2L2

kyk2 D kf k1 : kuk2

Clearly AC H 1 , but from the point of view of control theory, H 1 is an unnecessarily large object to serve as a class of stable transfer functions, since it includes functions that can hardly be considered to represent any physical system. In this article we consider the closure of AC in H 1 in the k k1 , and this will be our Banach algebra A0 C APC . We give the precise definition below. Definition 1. Let 9 8 ˆ f is holomorphic in C>0 > = < A0 WD f W C0 ! C W f is continuous on C0 ; > ˆ ; : lim f .s/ D 0 s!1

and let the set of causal almost periodic functions APC be the closure of spanfe t ; t 0g in the L1 -norm. In this paper we consider the following class of functions A0 C APC D ff W C0 ! C W f D fA0 C fAPC ; fA0 2 A0 ; fAPC 2 APC g; with pointwise operations and with the norm kfA0 C fAPC k1 : More precisely, we consider classes of unstable control systems with transfer functions belonging to the field of fractions F.A0 C APC /. Our first main result is the following corona theorem for A0 C APC : Theorem 1. Let n; d 2 A0 C APC . The following are equivalent; 1. There exist x; y 2 A0 C APC such that nx C dy D 1. 2. There exist ı > 0 such that jn.s/j C jd.s/j ı > 0 for all s 2 C0 . A similar theorem can also be shown for matricial data, but for the sake of simplicity, we just prove the result for a pair of functions. To prove this theorem, we characterize the maximal ideal space M.A0 C APC / of the Banach algebra A0 C APC ; see Theorem 4. The next main result is the following: Theorem 2. M.A0 C APC / is contractible. With the corona theorem for A0 CAPC , and the contractability of M.A0 CAPC /, we prove a number of results, all of them concerning algebraic properties of A0 C APC : 1. APC C A0 is a Hermite ring. 2. APC C A0 is projective free.

88


3. APC C A0 is not a GCD domain. 4. APC C A0 is not a pre-Bezout domain. 5. APC C A0 is not a coherent ring. For the definitions of the above algebraic properties, we refer the reader to Sect. 2. From the point of view of applications, we now briefly mention the relevance of these results in the solution of the stabilization problem in control theory: 1. Hermiteness: since the ring A0 C APC is Hermite, if a transfer function G has a left (or right) coprime factorization, then G has a doubly coprime factorization, and the standard Youla parametrization yields all stabilizing controllers for G; see [15, Theorem 66, p. 347]. 2. Projective freeness: since the ring A0 C APC is projective free, a plant is stabilizable if and only if it admits a right (or left) coprime factorization, see [11]. 3. Pre-Bezout property: Every transfer function p 2 F.R/ admits a coprime factorization if and only if R is a pre-Bezout domain, see [12, Corollary 4]. 4. GCD domain: Every transfer function p 2 F.R/ admits a weak coprime factorization if and only if R is a GCD domain, see [12, Corollary 3]. 5. Coherence: For implications of (the lack of) the coherence property we refer to [11]. The paper is organized as follows: 1. In Sect. 2, we introduce the relevant notation used throughout in the article. 2. In Sect. 3, we characterize the maximal ideal space of A0 C APC . 3. In Sect. 4, we use the characterization of M.A0 C APC / to prove a corona theorem for A0 C APC . 4. In Sect. 5, we prove that A0 C APC is contractible and as corollaries we get that A0 C APC is Hermite and projective free. 5. In Sect. 6, we prove that A0 C APC is not a GCD domain. 6. In Sect. 7, that A0 C APC is not a pre-Bezout domain. 7. Finally, in Sect. 8, we prove that A0 C APC is not coherent.

2 Preliminaries and Notation Throughout the article, we consider the Banach algebra A0 C APC , defined in Definition 1, with pointwise operations and supremum norm. We point out that elements fAPC in APC can be written X fAPC .s/ D fk e stk ; k0

where t0 D 0 < t1 ; t2 ; . To prove the corona theorem for A0 C APC , we shall use Kronecker’s approximation theorem, see for instance [6, Chap. 23].


89

Theorem 3. Let f1 ; ; n g be a set of rationally independent real numbers. Then, given " > 0 and .ı1 ; ; ın / 2 Rn ; there exist .m1 ; ; mn / 2 Zn and 2 R such that jıi i mi j < " for all i 2 f1; ; ng: We continue with a few definitions which explain the results presented in the introduction. Definition 2. A topological space X is said to be contractible if there exist a continuous map H W X Œ0; 1 ! X and an x0 2 X such that for all x 2 X; H.x; 0/ D x and H.x; 1/ D x0 : Definition 3. Let R be a ring with an identity element. A matrix f 2 Rnk is called left invertible if there exists a g 2 Rkn such that gf D Ik : The ring R is called a Hermite ring if for all k; n 2 N, k < n; and for all left invertible matrices f 2 Rnk , there exist F; G 2 Rnn such that GF D In and Fij D fij for all i 2 f1; ; ng; j 2 f1; ; kg. Definition 4. Let R be a commutative ring with identity. Then R is projective free if every finitely generated projective R-module is free. Recall that an R-module M is called 1. free if M Š Rd for some integer d 0; 2. projective if there exists an R-module N and an integer d 0 such that M ˚ N Š Rd . It can be shown that every projective-free ring is Hermite, see for example [3]. Definition 5. Let R be an integral domain, that is, a commutative unital ring having no divisors of zero. Then 1. An element d 2 R is called a greatest common divisor (gcd) of a; b 2 R if it is a divisor of a and b, and, moreover, if k is another divisor, then k divides d . 2. The integral domain R is said to be a GCD domain if for all a; b 2 R; there exists a greatest common divisor d of a; b. 3. The integral domain R is said to be a pre-Bezout domain if for every a; b 2 R for which there exists a gcd d , there exist x; y 2 R such that d D xa C yb: 4. The ring R is called coherent if for any pair .I; J / of finitely generated ideals in R, their intersection I \ J is finitely generated again. We also recall the definition of a bounded approximate identity, which will play an important role when we prove that A0 C APC is not a GCD domain, not a preBezout domain and not coherent. Definition 6. Let R be a commutative Banach algebra without identity element. Then R has a bounded left approximate identity if there exists a bounded sequence .en /n1 of elements en 2 R such that for all f 2 R, limn!1 ken f f k1 D 0.

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3 The Maximal Ideal Space of A0 C APC In this section we give a characterization of the maximal ideal space of A0 C APC . Before we state this theorem, we need to introduce some notation. For s 2 C0 let s denote point evaluation at s, that is, s.f / D f .s/, f 2 A0 C APC . Definition 7. W R ! C is called a character if j.t /j D 1 and .t C / D .t /. / for all t; 2 R. Theorem 4. If ' is in the maximal ideal space of A0 C APC , then one of the following three statements holds: (a) There exists an s 2 C0 such that '.f / D f .s/ for all f 2 A0 C APC . (b) We have ' D C1, that is, '.f / D C1.f / WD lim f .s/ D f0 ; s!1

for all f D fA0 C

X

fk e tk 2 A0 C APC , where fA0 2 A0 .

k0

(c) There exist a 0 and a character such that '.f / D

X

fk e tk .tk /;

k0

for all f D

X

fk e tk C fA0 2 A0 C APC ; with fA0 2 A0 .

k0

Proof. We note that via the conformal map s!

s1 ; sC1

we can map D onto C0 : In particular, this means that f 2 A.D/ ” f

s1 sC1

2 A0 C C:

Since the maximal ideal space of A.D/ is D (point evaluations on the closed unit disc; see for instance [13, p. 283]), it follows that the maximal ideal space of A0 C C is C0 [ fC1g: Suppose that ' 2 M.A0 C APC / n .C0 [ fC1g/: First, we shall show that if fA0 2 A0 , then '.fA0 / D 0. Assume on the contrary that '.fA0 / ¤ 0. Then 'jA0 CC 2 M.A0 C C / D C0 [ fC1g. Note that if 'jA0 CC D C1, then '.fA0 / D lim fA0 .s/ D 0; s!C1


91

which is a contradiction. So 'jA0 CC ¤ C1, and there exists s0 2 C0 such that 'jA0 CC D s0 . Moreover, we note that A0 is an ideal in A0 C APC . Thus for F 2 A0 C APC , we have '.F / D

.F f0 /.s0 / '.F f0 / D D F .s0 / D s0 .F /: '.f0 / f0 .s0 /

But this is a contradiction since ' 2 M.A0 C APC / n .C0 [ fC1g/. Hence 'jAPC is a nontrivial complex homomorphism. From the known characterization of the maximal ideal space of APC (see for example [1, Theorem 4.1]), we obtain that there exist a 0; and a character such that 'jAPC D ' ; , that is, 0 ' ; @

X k0

1 fk e tk A D

X

fk e tk .tk /:

k0

This completes the proof.

4 Corona Theorem for A0 C APC We will now give a proof of the corona theorem, Theorem 1. The proof relies on the characterization of the maximal ideal space of A0 C APC provided in Theorem 4. Proof (Proof of Theorem 1). Suppose that the corona condition holds, that is, that there exist ı > 0 such that jn.s/j C jd.s/j ı > 0 for all s 2 C0 : Assume that (1) in Theorem 1 does not hold. Then the ideal hn; d i generated by n; d is not the whole ring, and so there is a maximal ideal which contains it. Thus there exists ' 2 M.A0 C APC / such that hn; d i ker.'/ A0 C APC . In particular, '.n/ D '.d / D 0. By Theorem 4 we know that the maximal ideal space of A0 C APC consists of three different types of homomorphisms. However, ' cannot be of type (a), since then there exists s 2 C0 such that '.n/ D n.s/ D 0 and '.d / D d.s/ D 0; which contradicts the corona condition. Moreover, ' cannot be of type (b), since then '.n/ D lim n.s/ D 0 and '.d / D lim d.s/ D 0; s!C1

s!C1

92


which again violates the corona condition. Therefore, ' must be of type (c). Let n D nA0 C nAPC D nA0 .s/ C

X

nk e stk and

k0

d D dA0 C dAPC D dA0 .s/ C

X

dk e stk :

k0

There exist 0 and a character such that, 0 D '.n/ D

X

nk e tk .tk /; and

k0

0 D '.d / D

X

dk e k .k /:

k0

Since n, d 2 A0 C APC , for every choice of ı; M > 0; there exists N 2 N such that N X ı stk ; nk e nAPC < M kD0

1

N X sk dk e dAPC kD0

1

0 there exist a ˇ 2 R and real numbers m1 ; ; mN such that jˇi ıi mi j < ; for i D 1; ; N: Hence ˇ ! !ˇ K K ˇ ˇ X X ˇ ˇ ˛j k ık exp 2 iˇ ˛j k k ˇ j.tk / exp.2 iˇtk /j D ˇexp 2 i ˇ ˇ kD1

2

K X

kD1

j˛j k j:

kD1

Similarly, j.k / exp.2 iˇk /j 2

K X

j˛Q j k j. Let

kD1

C D

sup j 2f1;:::;N g

2

K X

maxfj˛j k j; j˛Q j k jg:

kD1

Then for small enough, ˇN ˇ N ˇX X ˇ ı ˇ 2 iˇtk ˇ : ak .tk / ak e jak j < ˇ ˇ C ˇ ˇ 2M kD0

kD0

Using this we obtain ˇ ˇ ˇN ˇ N ˇ ˇX ˇ X ˇ ˇ ˇ ˇ . Ci! /tk ˇ . Ci! /tk ˇ ˇnAP . C i ! /ˇ ˇˇnAP . C i ! / C n e n e ˇ ˇ ˇ k k C ˇ C ˇ ˇ ˇ kD0

ı ı 3ı C D : M 2M 2M

kD0

94


ˇ ˇ 3ı and also, ˇdAPC . C i ! /ˇ 2M . But the choice of M was arbitrary, and so, the above shows that for every ı > 0; there exists ! 2 R such that ˇ ˇ1 ˇ ı ˇX ˇ tk i! tk ˇ nk e e ˇ< ˇ ˇ 8 ˇ

ˇ ˇ1 ˇ ı ˇX ˇ k i! k ˇ and ˇ dk e e ˇ< : ˇ 8 ˇ

kD0

(1)

kD0

Hence jnAPC . Ci ! /jCjdAPC . Ci ! /j < ı=4: Now, we note that both nAPC . C i / and dAPC . C i / are almost periodic since ! 7! nAPC . C i !/ D

1 X

nQ k e i!tk

kD0

and ! 7! dAPC . C i !/ D

1 X

dQk e i!tk ;

kD0

for nQ k D nk e tk ; dQk D dk e k . But we recall the classical result that for any almost periodic function F , for every > 0 there exists L > 0 such that every interval of length L has a D .L/ such that for all ! 2 R jF .!/ F .! C /j < : We can use this fact to construct a sequence .!n /1 nD1 such that !n ! C1 as n ! 1 and ˇ ˇ ˇnAP . C i ! / nAP . C i !n /ˇ < ı ; C C 8 ˇ ˇ ı ˇdAP . C i ! / dAP . C i !n /ˇ < : C C 8

(2)

Moreover, by definition of A0 ; lim nA0 . C i !n / D lim dA0 . C i !n / D 0:

n!C1

n!C1

Thus, taking s D C i !n in the corona condition we get jn.s/j C jd.s/j jnA0 .s/j C jdA0 .s/j C jnAPC .s/j C jdAPC .s/j: „ ƒ‚ … „ ƒ‚ … I

II

(3)


95

But using (3) we can make I as small as we please, and using (1)–(2), II is smaller than ı=2: That is, jn.s/j C jd.s/j ı; which is a contradiction. Thus, (1) in Theorem 1 must hold whenever (2) in Theorem 1 (the corona condition) holds. That (1) implies (2) in Theorem 1 is obvious and this concludes the proof.

5 Contractability of M.A0 C APC / We will show that the maximal ideal space M.A0 C APC / of A0 C APC is contractible (Theorem 2). Then, as corollaries, we obtain that A0 C APC is Hermite and projective free. To prove that M.A0 C APC / is contractible, we will proceed in several steps, and we start with a few lemmas which mainly concern the topology of M.A0 C APC /. Lemma 1. The set M.A0 C APC /nC0 is closed in M.A0 C APC /. Proof. It is enough to show that C0 is open. Let O denote the Gelfand transform. As a consequence of Theorem 4, ' 2 C0 if and only if there exists fA0 2 A0 such that ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ O ˇfA0 .'/ˇ D ˇ' fOA0 ˇ > 0: Therefore, C0 D

[ n

ˇ ˇ o ˇ ˇ ' 2 M.A0 C APC / W ˇfOA0 .'/ˇ > 0 ;

fA0 2A0

and since the union of open sets is open, C0 is open, which completes the proof. Lemma 2. C0 is homeomorphic to C0 : Proof. The map W C0 ! C0 given by s 7! s is onto. Further it is injective since, if s1 D s2 , then s1

1 1Cs

1 1 D D D s2 1 C s1 1 C s2

1 1Cs

:

That is, s1 D s2 : Therefore, is invertible. Let .s˛ / be a net such that s˛ ! s0 : Since elements in A0 C APC are continuous in C0 by definition, it follows that, for f 2 A0 C APC ; f .s˛ / ! f .s0 /: That is s˛ .f / ! s0 .f /:

96


Since this holds for arbitrary f it follows that s˛ ! s0 in C0 . What remains to prove is that the inverse is continuous, but if s˛ ! s0 , then s˛

1 1Cs

1 1 D ! D s0 1 C s˛ 1 C s0

1 1Cs

:

Thus, s˛ ! s0 in C0 so the inverse is continuous and C0 is indeed homeomorphic to C0 : Lemma 3. If .s˛ / is a net in C0 which converges in M.A0 C APC / to a ' 2 M.A0 C APC /nC0 then s˛ ! 1: Proof. Note that, in particular, s˛ since

1 1Cs

1 1Cs

D

1 !' 1 C s˛

1 1Cs

D 0;

is not invertible. Therefore s˛ ! 1.

Lemma 4. Let ' 2 M.A0 C APC /nC0 and let .'˛ / be a net such that; 1. for all f 2 A0 ; '˛.f / ! 0: 2. for all T > 0; '˛ e sT ! ' e sT : Then '˛ ! ' in M.A0 C APC /nC0 : Proof. By hypothesis, for every fA0 2 A0 and for every exponential polynomial P .s/ D

N X

fk e stk ; 0 D t0 < t1 ; t2 ; : : : ; tN ;

kD0

we have that '˛ .fA0 C P / ! '.fA0 C P / since '˛ and ' are homomorphisms. Let f D fA0 C

X

fk e tk 2 A0 C APC :

k0

Then, for every > 0, we can chose an exponential polynomial P such that kf fA0 P k1

X tk D fk e P k0

1

: 4

(4)

Moreover, since '˛ .fA0 CP / ! '.fA0 CP / there exists ˛ such that for all ˛ > ˛ there holds ˇ ˇ ˇ'˛ .fA C P / '.fA C P /ˇ < : 0 0 2

(5)


97

Combining (4) and (5) ˇ ˇ j'˛ .f / '.f /j D ˇ'˛ .fA0 C P C f fA0 P / '.fA0 C P C f fA0 P /ˇ ˇ ˇ ˇ ˇ ˇ'˛ .fA0 C P / '.fA0 C P /ˇ C ˇ.'˛ '/.f fA0 P /ˇ " C k'˛ 'k kf fA0 P k1 2 " " C 2 D ": 2 4

That is '˛ .f / ! '.f / for all f 2 A0 C APC and consequently .'˛ / converges in the weak -topology on M.A0 C APC /; which completes the proof. With these tools available, we shall now prove that A0 C APC is contractible. Proof (Proof of Theorem 1.3). Recall that M.A0 C APC / consist of three kinds of homomorphisms, as stated in Theorem 4. We will define a map H W M.A0 C APC / Œ0; 1 ! M.A0 C APC / and show that this map satisfies the necessary properties in Definition 2. In particular, we define H as follows; 1. For s 2 C0 , define H.s; t / D s log.1 t / for t 2 Œ0; 1/, and H.s; 1/ D C1. 2. For C1 we define H.C1; t / D C1 for all t 2 Œ0; 1 : 3. For ' 2 M.A0 C APC /n.C0 [ C1/ there exists > 0 and a character such that ' D ' ; , where ' ; is defined as in (c) in Theorem 4. We define H.' ; ; t / D ' log.1t/; for t 2 Œ0; 1/ and H.s; 1/ D C1. From the definition of H it is obvious that the choice of x0 in Definition 2 should be C1. We shall now prove that H is continuous. Every net .'˛ ; t˛ /, with elements in M.A0 C APC / Œ0; 1 can be partitioned onto three subnets: 1ı one where .'˛ ; t˛ / D .s˛ ; t˛ / 2 C0 Œ0; 1 for each ˛; 2ı one where .'˛ ; t˛ / D .C1; t˛ / 2 fC1g

Œ0; 1 for each ˛; and

3ı one where .'˛ ; t˛ / D .' ˛ ;˛ ; t˛ / 2 M.A0 C APC /n.C0 [ C1/ Œ0; 1 for each ˛:

It is therefore enough to prove that, for these three kinds of nets, if .'˛ ; t˛ / converges to .'; t / in M.A0 CAPC / Œ0; 1 ; then .H.'a ; t˛ // converges to H.'; t / in M.A0 C APC /. We shall treat these cases separately. Case when .'˛ ; t˛ / D .s˛ ; t˛ / 2 C0 Œ0; 1. Firstly, if t˛ 1; then t D 1 by necessity, and H.s˛ ; t˛ / D H.s˛ ; 1/ D C1 D H.'; 1/ D H.'; t /; so we are done. Therefore, assume that each t˛ 2 Œ0; 1/; and thus H.s˛ ; t˛ / D s˛ log.1 t˛ /:

98


Now consider the following three cases. (1a) ' D s for some s 2 C0 : If t 2 Œ0; 1/; then H.s; t / D s log.1 t /: But, since s˛ ! s we know by Lemma 2 that s˛ ! s in C0 ; and since log.1 t / is continuous for t 2 Œ0; 1/; log.1 t˛ / ! log.1 t / and s˛ log.1 t˛ / ! s log.1 t / in C0 : Hence, using Lemma 2 once more we see that H.s˛ ; t˛ / ! H.s; t /: On the other hand, if t D 1, then H.s; t / D H.s; 1/ D C1; and since t˛ ! 1 we have that Re.s˛ log.1t˛ // ! 1: The elements f 2 A0 CAPC are given by X fk e tk ; (6) f D fA0 C k0

so f .s˛ log.1 ta // ! f0 when Re.s˛ log.1 t˛ // ! 1; which corresponds to evaluation at infinity. Since the choice of f 2 A0 C APC was arbitrary, also in this case H.s˛ ; t˛ / ! H.s; t /: (1b) ' D C1: In this case, H.'; t / D C1: Note that, by Lemma 3, if s˛ ! C1; then s˛ ! 1 and s˛ log.1 t˛ / ! 1: In particular, for all fA0 2 A0 fA0 .s˛ log.1 t˛ // ! 0 D C1.fA0 /; since elements in A0 have limit zero at infinity. Moreover, for all T > 0; s˛ .e sT / D e s˛ T !0 D C1.e sT / and s˛ log.1 t˛ /.e sT / ! 0 D C1.e sT /: Hence, using Lemma 4 we see that H.s˛ ; t˛ / ! H.C1; t /: (1c) ' D ' ; : Since s˛ ! ' ; 2 M.A0 C APC /nC0 we have that s˛ ! 1 by Lemma 3 and therefore s˛ log.1 t˛ / ! 1: Let f 2 A0 C APC with f as in (6) be given (and arbitrary). Then, arguing as above and for t D 1, we have H.'; t / D H.' ; ; 1/ D C1 and H.s˛ ; t˛ / ! f0 D C1.f /: Since this holds for all f 2 A0 C APC this implies that H.s˛ ; t˛ / ! H.' ; ; t /:


99

If t < 1; then for T > 0; H.' ; ; t /.e sT / D e . log.1t//T .T /: Since s˛ ! ' ; ; we have that s˛ .e sT / D e s˛ T ! e T .T / D ' ; .e sT /: In particular, this means that e .s˛ log.1t˛ //T ! e . log.1t˛ //T .T /; which by the definition of H implies that H.s˛ ; t˛ / ! H.' ; ; t /: Now, the cases (1a)–(1c) prove that H is continuous when .'˛ ; t˛ / D .s˛ ; t˛ /: Case when .'˛ ; t˛ / D .C1; t˛ / 2 fC1g Œ0; 1. This case is trivially satisfied, since ' D C1 in this case, and H.'˛ ; t˛ / D H.C1; t˛ / D C1 D H.'; t /. Case when .'˛ ; t˛ / D .'˛ ;˛ ; t˛ /. By Lemma 1 M.A0 C APC /nC0 is closed, which means that ' 2 M.A0 C APC /nC0 . Therefore H.'˛ ; t˛ /.fA0 / D 0 D H.'; t /.fA0 / for all fA0 2 A0 :

(7)

From here on, we therefore only consider the case when f 2 APC : Firstly, if t˛ 1; then t D 1 and H.'˛ ; t˛ / D H.'˛ ; 1/ D C1 D H.'; 1/ D H.'; t /: If each t˛ 2 Œ0; 1/, then '˛ .f / D

X

fk e ˛ tk ˛ .tk /;

k0

for f 2 APC : Moreover, for T > 0; H.'˛ ; t˛ /.e sT / D e . ˛ log.1t˛ //T ˛ .T /: We will now consider two separate cases. (2a) ' ¤ C1: Then '˛ ! ' ¤ C1 so for T > 0; '˛ .e sT / D e ˛ T ˛ .T / ! e T .T / D '.e sT /: If t < 1; then this implies that H.'˛ ; t˛ /.e sT / D e . ˛ log.1t˛ //T ˛ .T / ! e . log.1t//T .T / D H.'; t /;

100


and we are done. If t D 1 on the other hand, then e . ˛ log.1t˛ //T ˛ .T / ! 0 D C1.e sT /; and so H.'˛ ; t˛ /.e sT / ! H.'; t /.e sT /: (2b) ' D C1: Since '˛ ! C1; we have that '˛ .e sT / ! C1.e sT / D 0 for T > 0: That is, e ˛ T ˛ .T / ! 0: Therefore, e . ˛ log.1t˛ //T ˛ .T / ! 0; and H.'˛ ; t˛ /.e sT / ! H.'; t /.e sT /: Due to (7) this completes the proof in the case when .'˛ ; t˛ / D .' ˛ ;˛ ; t˛ /. We have now shown that H converges for all three types of subnets .'˛ ; t˛ / which shows that M.A0 C APC / is contractible. Corollary 1. A0 C APC is Hermite As mentioned before, this corollary is a direct consequence of [7, Theorem 3, p. 127]. Yet another result, which follows immediately from [3, Corollary 1.4], concerns projective freeness. Corollary 2. A0 C APC is a projective free ring.

6 A0 C APC is Not a GCD Domain The relevant definitions were presented in Sect. 2, and now we will prove that A0 C APC is not a GCD domain, by giving an example of two elements in A0 C APC which do not have a gcd. The method of proof is the same as in [10]. The two elements we will consider are F1 D

1 1 and F2 D e 1=s : .1 C s/ .1 C s/

(8)

Note that both F1 and F2 belong to A0 , and thereby to A0 CAPC . We shall now state some preliminary results used in the proof. We begin with Cohen’s factorization theorem; see for example [2, Theorem 1.6.5]. Proposition 1. Let R be a Banach algebra with a bounded left approximate identity. Then for every sequence .an /n1 in R converging to zero, there exists a sequence .bn /n1 in R converging to zero, as well as an element c 2 R such that an D cbn for all n 1: Lemma 5. The maximal ideal M0 WD ff 2 A0 C APC j f .0/ D 0g has a bounded left approximate identity.


101

Proof. Let en D

s : s C n1

We shall now prove that this is a bounded approximate identity for A0 C APC : First of all, we note that ! 1 1 en D 1 DW 1 C eAn0 2 A0 C APC : n s C n1 „ ƒ‚ … n DWeA 2A0 0

Furthermore, en .0/ D 0; and so in fact, en 2 M0 . Moreover, ken k1

1 D 1 n

1 sC

! 1 n

D 1;

1

so en is bounded. We need to prove that lim ken f f k1 D 0:

n!1

(9)

Now, fix n and let f 2 M0 , then en f f D .1 C eAn0 /f f D eAn0 f 2 A0 : Therefore, we only need to prove that lim eAn0 f 1 D 0:

n!1

Let " > 0 be given. We know that lim f .s/ D 0

jsj!0

since f 2 M0 ; so there exists ı > 0 such that jf .s/j < " for all s with jsj < ı. For jsj < ı; ˇ ˇ1 ˇ jeA0 f j D ˇ ˇn n

1 sC

! 1 n

ˇ ˇ ˇ f .s/ˇ < "; ˇ

and this is independent of our choice of n: When jsj ı, we have

102


ˇ ˇ1 ˇ jeA0 f j D ˇ ˇn n

!

1 sC

1 n

ˇ ˇ ˇ ˇ ˇ 1 ˇ ˇ f .s/ˇˇ : f .s/ˇ ˇˇ ˇ ın C 1

However, since f 2 A0 C APC , jf j is bounded. Hence, for n>

kf k1 "

1

ı

;

jeAn0 .s/f .s/j < " for all jsj ı; and hence, for all s: Therefore (9) holds, and .en / is a bounded approximate identity for M0 . Theorem 5. A0 C APC is not a GCD domain. Proof. We claim that F1 ; F2 in (8) have no gcd. Suppose, on the contrary that D is a gcd for F1 and F2 , so that F1 D DQ1 and F2 D DQ2 ; for some Q1 ; Q2 2 A0 C APC : Suppose that Q1 .0/ ¤ 0: Then, at least in a neighbourhood of zero, e 1=s D

Q2 .s/ F2 .s/ D : F1 .s/ Q1 .s/

However, this is impossible because lim e 1=.i!/ does not exist,

RÖ !!0

while lim

RÖ !!0

Q2 .i !/ Q2 .0/ D ; Q1 .i !/ Q1 .0/

since Q1 .0/ ¤ 0 and both Q1 and Q2 are continuous. Hence, Q1 .0/ D 0: Similarly, we suppose that Q2 .0/ ¤ 0: Then, in a neighbourhood of zero, e 1=s D

F1 .s/ Q1 .s/ D : F2 .s/ Q2 .s/

However, this is impossible because lim e 1=.i!/ does not exist,

RÖ !!0


103

while lim

RÖ !!0

Q2 .0/ Q2 .i !/ D Q1 .i !/ Q1 .0/

does. We have thus concluded that Q1 .0/ D Q2 .0/ D 0; and thus both Q1 and Q2 belong to the maximal ideal M0 WD ff 2 A0 C APC j f .0/ D 0g: By Lemma 5, M0 has a bounded approximate identity, and then, by Proposition 1 there exists a G 2 M0 such that G is a common factor for Q1 and Q2 . Hence, K WD DG is also a divisor of both F1 and F2 . Since D is a gcd for F1 and F2 , K must divide D; and so DG H D D; „ƒ‚…

(10)

DK

for some H 2 A0 C APC : By definition F1 ¤ 0 for s ¤ 1, so the same must hold for D. Therefore, by (10), GH D 1 for s 2 C0 nfC1g. But, G 2 M0 and H 2 A0 C APC is bounded, so lim G.s/H.s/ D 0;

s!0

which is a contradiction. Thus, F1 and F2 have no gcd in A0 C APC , and A0 C APC is not a GCD domain.

7 A0 C APC is Not a Pre-Bezout Domain The proof that A0 C APC is not a pre-Bezout domain relies on the notion of an approximate identity, Proposition 1 and the corona theorem. The method of proof is the same as in [10]. Theorem 6. A0 C APC is not a pre-Bezout domain. Proof. Consider the following two elements in A0 C APC : U1 D

1 and U2 D e s : sC1

As U1 is outer and U2 is inner in the Hardy algebra of the right half-plane, it can be seen from the inner-outer factorization of H 1 functions that the pair .U1 ; U2 / has 1 as a greatest common divisor in A0 C APC . Suppose that A0 C APC is a pre-Bezout domain. Then there exist X , Y in A0 C APC such that 1 D U1 X C U2 Y . Passing the limit as s ! C1, we arrive at the contradiction that 1 D 0. Hence A0 C APC is not a pre-Bezout domain.

104


8 A0 C APC is Not Coherent The proof we give in this section is based on the same method used to show the noncoherence of the causal Wiener algebra W C in [9]. We will begin with two lemmas, the first lemma is a special case of Nakayama’s lemma, see for instance [8, Theorem 2.2], while the second one can be proved using the same arguments as [14, Lemma 2.8]. Lemma 6. Let L ¤ 0 be an ideal in A0 C APC ; contained in the maximal ideal M0 . If L D LM0 ; then L cannot be finitely generated. Lemma 7. Let f1 ; f2 2 M0 ; and let ı > 0: Let Inv.A0 C APC / denote the set of all invertible elements in A0 CAPC : Then there exists a sequence .gn /n0 in A0 CAPC such that 1. gn 2 Inv.A0 C APC / for all n 0; 2. gn converges to a limit g 2 M0 in A0 C APC ; 1 3. jjgn1 fi gnC1 fi jjA ı=2n , for all n 0, i D 1; 2: Theorem 7. A0 C APC is not coherent. Proof. We will proceed much in line with [14, Theorem 1.3]. Let 1Ce s

p D .1 e s /3 ; S D e 1es : We note that 1Cz

.1 z/3 e 1z D

1 X

ak zk ; for z 2 D;

kD0

1 X

jak j < 1;

kD0

by [9, Remark, p. 224]. Since e s 2 D for s 2 C0 ; pS 2 A0 C APC ; and by definition p 2 M0 . Let I D .p/ and J D .pS / be the ideals, (finitely) generated by p respectively pS . To prove that A0 C APC is not coherent, we shall show that I \ J is not finitely generated. The first step is to characterize I \ J: Let, K D fpSf W f; Sf 2 A0 C APC g: We claim that K D I \ J: By definition, K .I \ J /; so we only need to show the reverse inclusion. Let g 2 I \ J , then there exist two functions f; h 2 A0 C APC such that g D ph D pSf: Since p ¤ 0 and since A0 C APC is an integral domain, h D Sf 2 A0 C APC and so g 2 K and K D I \ J .


105

Now we shall prove that K D I \ J is not finitely generated. To do this, we define L D ff 2 A0 C APC W Sf 2 A0 C APC g: Then K D pSL; and since S has a singularity at s D 0 we have that L M0 : We shall now prove that L D LM0 . Then by Lemma 6 L cannot be finitely generated, and neither can K, so A0 C APC is not coherent. To prove that L D LM0 ; let f 2 L be arbitrary. We want to find factors h 2 L; g 2 M0 such that f D hg. Let f1 D f 2 M0 , f2 D Sf 2 M0 : Then, by Lemma 7, for every ı > 0 there exists a sequence .gn /n1 in A0 C APC such that properties (1)–(3) of Lemma 7 hold. Let hn D gn1 f; Hn D gn1 Sf: Then hn ; Hn 2 M0 and (3) in Lemma 7 imply that .hn /n1 ; .Hn /n1 are Cauchy sequences in A0 C APC : Since M0 is closed, these sequences converge to h respectively H , h; H 2 M0 : Since the limit of a sequence of holomorphic functions is unique, lim S hn D S h D H;

n!1

and since both h and S h are in M0 , h 2 L. Further, by (2) in Lemma 7, f D lim hn gn D hg; n!1

where h 2 L and g 2 M0 : That is, L D LM0 , which completes the proof. Acknowledgements The authors thank the anonymous referee for the careful review and the several suggestions which improved the presentation of the article.

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The Corona Problem in Several Complex Variables Steven G. Krantz

Abstract We survey and describe some results about the corona problem in several complex variables. We indicate connections with classical results, and point in some new directions as well. Keywords Several complex variables • Corona problem • Maximal ideal space Subject Classifications: 32A10, 32A35, 32A38, 32A60

1 Introduction Banach algebras were invented by I. M. Gelfand in his thesis in 1936. In 1941, S. Kakutani first formulated the corona problem: Are the point evaluation functionals dense in the maximal ideal space of H 1 .D/? The popular Bezout formulation of the corona problem is this: Suppose that f1 ; f2 ; : : : ; fk are bounded holomorphic functions on the disc D that satisfy jf1 ./j C jf2 ./j C C jfk ./j > ı > 0 for some positive ı. Then do there exists bounded, holomorphic g1 ; g2 ; : : : ; gk such that f1 ./ g1 ./ C f2 ./ g2 ./ C C fk ./ gk ./ 1 ‹

The equivalence of these two formulations of the corona is explored in [KRA2, Ch. 11].

S.G. Krantz () Department of Mathematics, Washington University in St. Louis, St. Louis, MO 63130, USA e-mail: [email protected] R.G. Douglas et al. (eds.), The Corona Problem, Fields Institute Communications 72, DOI 10.1007/978-1-4939-1255-1__6, © Springer Science+Business Media New York 2014

107

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S.G. Krantz

It was not until 1962 that Lennart Carleson [CAR] was able to solve this problem in the affirmative. His result was very important, but his proof techniques have proved to be even more influential. His idea of the Carleson measure, for example, has become an important artifact of harmonic analysis and function theory. In the late 1960s and early 1970s, when the function theory of several complex variables was becoming more concrete, and starting to resemble one complex variables in certain key respects (as with explicit integral formulae), people started thinking about the corona problem in the several complex variables context. The first notable contribution was from N. Sibony, who produced in his thesis [SIB1] a domain in Cn which is pseudoconvex, but on which the corona problem is not solvable. This clever and dramatic example spawned a small industry, as we shall describe below. It should be noted that this first example of Sibony does not have smooth boundary. Later, Sibony was able to modify his first example to produce a smoothly bounded, pseudoconvex domain in C3 [SIB2] on which the corona fails. This example is notable because the boundary is strongly pseudoconvex at every boundary point except one. A few years after that, Fornæss and Sibony [FSIB] produced an example in C2 of a smoothly bounded, pseudoconvex domain on which the corona fails. Again, every boundary point except one is strongly pseudoconvex. It might also be noted that Stout [STO] and Alling [ALL] proved the corona theorem for a finitely connected Riemann surface. As a counterpoint, Brian Cole (see Chap. 4 of [GAM]) gave an example of an infinitely connected Riemann surface on which the corona theorem fails. It is possible to “fatten up” Cole’s example to get a counterexample in several complex variables. Our goal in this paper is not to give a comprehensive treatment of all the work on the corona problem in several complex variables in the last forty years. Instead we want to give a detailed taste of the subject, and to point to some directions for further reading. We hope that the interested reader will get a start in exploring the many avenues that are open for further investigation.

2 The First Sibony Example In the rest of this paper, D denotes the unit disc in C, B denotes the unit ball in Cn , and D n denotes the unit polydisc in Cn . We use the terminology domain to denote a connected, open set. If W CN is a domain, then let int W denote its interior. Now Sibony’s theorem (see [SIB1]) is this:

The Corona Problem in Several Complex Variables

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Theorem 1. There is a Runge,1 pseudoconvex domain U C2 such that (a) int U D U D 2 , U ¤ D 2 ; (b) All functions in H 1 .U / analytically continue to H 1 .D 2 /. What must be understood here is that, because U is pseudoconvex, it is a domain of holomorphy (see [KRA1]). So there is a holomorphic function on U that does not analytically continue to any larger domain. But Sibony shows us that any bounded holomorphic function on U does analytically continue to a strictly larger domain. Proof of the Theorem. Set U D f.z; w/ 2 C2 W jzj < 1; jwj exp V .z/ < 1g ; where V .z/ exp.'.z// and './

X j

ˇ ˇ ˇ pj ˇ ˇ: j log ˇˇ 2 ˇ

Here fj g is a summable sequence of positive real numbers. The fpj g are a countable set of points in the disc D with no interior accumulation point, but such that every boundary point of D is a nontangential accumulation point of the pj . Clearly the function ', being the sum of subharmonic functions, is subharmonic. Likewise V , being the exponential of a subharmonic function, is subharmonic. Observe that (a) V is subharmonic; (b) 0 < V .z/ 1 for all z 2 D; (c) V is continuous. This last property holds because fpj g is discrete and V takes the value 0 only at the pj . We see that U is a proper subset of D 2 —clearly there are points of the form .0; ˛/ in D 2 n U . Now let g 2 H 1 .U / with kgk1 1. We expand g in a power series in w and write X h .z/w : (?) g.z; w/ D 0

1 A Runge domain is one on which any holomorphic function can be approximated, uniformly on compact sets, by polynomials.

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Of course h .z/ D

1 @ g .z; 0/ ; Š @w

so that h is holomorphic. Thus, by the Cauchy formula, with 0 < r < 1 and r very near 1, Z 2 g.z; r exp.V .z//e i / 1 i r exp.V .z//e i d 2 i 0 r C1 exp.. C 1/V .z//e i.C1/ Z 2 1 g.z; r exp.V .z/e i / D d : 2 0 r exp.V .z//e i

h .z/ D

Since kgk1 1, we see that jh .z/j r exp.V .z// : Since r is arbitrarily close to 1, we have jh .z/j exp.V .z// ;

8z 2 D :

(*)

Hence, for each fixed , h is a bounded, holomorphic function on the disc D. Also line (*) tells us that jh .pj /j 1 for each ; j . It follows that jh j clearly has at most unimodular nontangential boundary limits at every boundary point of D. Therefore jh j, being the modulus of a bounded holomorphic function, is bounded by 1 on all of D. As a result, the series (?) converges on all of D 2 . We see, using the open mapping principle, that in fact the extension gO of g to D 2 has the same sup norm. We may suppose, just by choosing none of the pj to be the origin, that V .0/ ¤ 0. Thus there exists an ˛ 2 D such that .0; ˛/ 62 U . So there is a ı > 0 so that jzj C jw ˛j dist ..z; w/; .0; ˛// > ı > 0 for all .z; w/ 2 U . Seeking a contradiction, we suppose that there exists g1 ; g2 2 H 1 .U / such that z g1 .z; w/ C .w ˛/ g2 .z; w/ D 1 : Then the g1 ; g2 analytically continue to D 2 (by Sibony’s theorem). So we can evaluate the last equation at the point .z; w/ D .0; ˛/. The result is 0 C 0 D 1; and that is a contradiction. So the corona problem cannot be solved on U .


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Another way to look at this matter is as follows. The point evaluation at .0; ˛/ is certainly a multiplicative linear functional on the H 1 .U /. But it is not in the weak closure of the multiplicative linear functionals coming from points of U (since .0; ˛/ is spaced away from U ). Therefore the corona cannot be solved on U . See [KRA2] for a thorough discussion and background of the corona problem. See [[JAP] for a treatment of ideas related to Sibony’s theorem.

3 The Smooth Example of Sibony Now we provide some consideration of Sibony’s smooth example in C3 . [The smooth example of Fornæss and Sibony in C2 —see[FSIB]—is quite technical, and we shall not treat it here.] See [SIB2] for the details. Let us write Sibony’s domain U from his first theorem as U D [1 j D1 Dj ; where D1 D2 and each Dj is polynomially convex. Since U is Runge, this construction is clear. Lemma 1. Let j be positive real numbers with j ! 0. Set ˚j .z; w/ D . j z; j w; j / : Define Xn D ˚n .Dn / and set XD

[1 nD1 Xn

[ f0g :

Then X is a compact, polynomially convex subset of C3 . Proof. Let .z1 ; z2 ; z3 / be coordinates in C3 . Let A be the set consisting of f j g [ f0g. O its polynomial envelope, is contained in C2 A. Clearly X is compact and X, Each Dj is polynomially convex, and one checks that X is polynomially convex by utilizing polynomials of the form p.z3 /q.z1 ; z2 /. Lemma 2. There exists a function ' 2 C 1 .C3 / which is plurisubharmonic and so that (i) ' 0 and ' 1 .0/ D X . (ii) ' is flat on X and '.z/ kzk2 for z large. This lemma was proved by David Catlin in his thesis—see also [CAT]. It is valid for any compact, polynomially convex set Y Cn . We write X D \p ˝p , where each ˝p is smooth and strongly pseudoconvex. We assume that the ˝p are decreasing and each is polynomially convex.

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For each index p, there is a function p 2 C 1 .C3 /, with p 0, p1 .0/ D ˝p ,and p plurisubharmonic in C3 . We set '1 D

X

˛p p :

p

Here f˛p g is a sequence of positive numbers chosen so that '1 is a C 1 function. One can add to '1 a function that is convex in kzk2 and is equal to 0 on D 3 (the unit polydisc). Also we add on .Im z3 /2 . We take ' to equal '1 . For any c > 0, set ˝c D f.z1 ; z2 ; z3 / W Im z3 C c'.z1 ; z2 ; z3 / < 0g : Then ˝c is bounded and, for almost every c, it has smooth boundary (use Sard’s lemma). We may suppose that this is the case for c D 1. Further suppose that ˝1 D 3 . Note that ˝1 \ fIm z3 D 0g D X and ˝1 fIm z3 < 0g. The functions z1 =z3 and z2 =z3 are bounded by 2 in a neighborhood of X n f0g. Thus there exists a sequence fıj g, ıj > 0, such that z1 =z3 and z2 =z3 are bounded by 2 on ˝1 \ Œ[n1 fjz3 n j ın g : Set Vn D f.z1 ; z2 ; z3 / W jz3 n j ın g : We may suppose that ın 0 such that jf1 j C jf2 j ı on ˝ \ .[n Vn /, possibly by decreasing the ın . Let f3 .z3 / be a holomorphic function on fIm z3 < 0g, bounded by 1, continuous on fIm z3 0g n f0g, zero on f n g, and such that jf3 .z3 /j ı outside [1 nD1 Vn . It suffices to set f3 .z3 / D

1 Y z3 n n nD1

2

:

One has outside [n Vn that log jf3 j D

X n

ˇz ˇ X ˇ 3 nˇ n log ˇ n log ın : ˇ 2 n

By taking n sufficiently small, we may check that X

n log ın log ı :

In conclusion, the functions f1 , f2 , f3 are bounded and continuous on ˝ n f0g and satisfy jf1 j C jf2 j C jf3 j ı on ˝1 . Now we obtain the domain ˝ by perturbing ˝1 , and then we show that g1 , g2 , g3 solving the corona problem do not exist. Let an be a sequence of strictly positive numbers with limit 0. There is a sequence ˇn > 0, such that

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(i) For n sufficiently large, Xn n .0; 0; iˇn / ˝1 ; (ii) For j D 1; 2; 3; : : : and .z1 ; z2 ; z3 / 2 Xn , one has jfj ..z1 ; z2 ; n iˇn / fj .z1 ; z2 ; n /j an :

()

Condition (i) is true because is flat on X ; and condition (ii) is true because fj is continuous on ˝ n f0g. Let 2 C 1 .R/ be such that 1. is convex and increasing; 2. .t / D 0 for t 0 and .2 n2 / < 1=.2ˇn /. Set ˝ D f.z1 ; z2 ; z3 / W Im z3 C '.z/ C .jz1 j2 C jz2 j2 / < 0g : By multiplying with a constant near 1, we obtain a domain with smooth boundary. By construction, '.z/ .Im z3 /2 is plurisubharmonic on ˝ and strictly plurisubharmonic except at 0. For n sufficiently large, Xn iˇn ˝ : Now we show that there do not exist functions g1 , g2 , g3 , gj 2 H 1 .˝ for all j , such that X fj gj 1 : (**) j

Suppose that such gj do exist. For .z; w/ 2 Dn , set n .z; w/

D . n z; n w; n iˇ/ :

Inequality () entails that the functions ffj ı n g converge uniformly on compact subsets of U to z; w ˛, and 0 (there are three sequences, for j D 1; 2; 3). Set gjn D gj ı n . The sequence fgjn gn is bounded so one can elicit a subsequence converging to fhj g, j D 1; 2; 3. These are holomorphic and bounded in U . Passing to the limit in equation (**), after having composed with n , we see that zh1 .z; w/ C .w ˛/h2 .z; w/ D 1 in U . We know that this is impossible.


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4 Ideas of Gunnar Berg In his thesis [BER], Gunnar Berg capitalized on the work of Sibony (described above) and explored H 1 domains of holomorphy, H 1 convexity, and connections with completeness of the Carathéodory metric. In a later work [KRA3], Krantz built on both Berg’s and Sibony’s ideas. We describe some of these here. Definition 1. Let ˝ be a domain (a connected open set) in Cn . 1. We say that ˝ is HLp if there is a holomorphic f 2 Lp .˝/ such that f cannot be analytically continued to any larger domain. 2. We say that ˝ is ELp if there is a strictly larger domain ˝O so that every O holomorphic Lp function f on ˝ does analytically continue to ˝. Matters in the complex plane (complex dimension one) are fairly straightforward. We briefly treat them now. First consider the planar domain D 0 f 2 C W 0 < jj < 1g : This is the punctured disc. (1) If f is holomorphic and bounded on D 0 , then f analytically continues to D D f 2 C W jj < 1g. This is just the Riemann removable singularities theorem. (2) In fact, if f holomorphic and Lp on D 0 for p 2 then f analytically continues to all of D. This phenomenon occurs mainly because 1= 62 Lp for p 2. If p < 2, then g./ D 1= is an Lp holomorphic function on D 0 that does not analytically continue to D. Now we have: Proposition 1. If ˝ C is bounded and is the interior of its closure, then ˝ is of type HLp for 1 p 1. Proof. Let fpj g be a countable, dense subset of c ˝. For each j , the function 'j ./ D

1 pj

is holomorphic and bounded on ˝ and does not analytically continue past pj . Let O.˝/ denote the ring of holomorphic functions on ˝. For each j , let dj be an open disc centered at pj which intersects ˝ and is nearly as small as possible. Consider the linear mapping Ij W O.˝ [ dj / \ Lp .˝ [ dj / ! O.˝/ \ Lp .˝/ given by restriction. Each of the indicated spaces is equipped with the Lp norm, so it is a Banach space. Notice that, because of the existence of 'j , we know that Ij is

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not surjective. Thus the open mapping principle implies that the image Mj of Ij is of first category in O.˝/ \ Lp .˝/. Thus, by the Baire category theorem, M D [j Mj is of first category in O.˝/ \ Lp .˝/. So the set of holomorphic functions on ˝ that can be analytically continued past some pj is of first category. The conclusion then is that the set of holomorphic functions that cannot be analytically continued across @˝ is dense in O.˝/ \ Lp .˝/. We note that the zeros of a holomorphic function of one complex variable are always isolated, while the zeros of a holomorphic function of several complex variables are never isolated. Nonetheless, one can take advantage of pseudoconvexity in several complex variables to imitate Berg’s argument in some contexts. We give now some sample results, without proofs (see [KRA3] for the details). Proposition 2. If ˝j C, j D 1; 2; : : : ; k, are each domains which are the interior of their closures, then ˝ D ˝1 ˝2 ˝k if of type HLp for 1 p 1. Proof. Just take the product of functions from the preceding result. Proposition 3. Let ˝ Cn be bounded and convex. Let 1 p 1. Then ˝ is a domain of type HLp . Proposition 4. Let ˝ Cn be bounded and strongly pseudoconvex with C 2 boundary. Let 1 p 1. Then ˝ is of type HLp . Proposition 5. Let ˝ be a smoothly bounded domain of finite type in C2 (see [KRA1] for this concept). Let 1 p 1. Then ˝ is of type HLp . Proposition 6. Let ˝ Cn be a bounded, pseudoconvex domain with a Stein neighborhood basis. Then ˝ is of type HLp , 1 p 1. Corollary 1. If ˝ is of type ELp , then ˝ does not have a Stein neighborhood basis. Proof. This is just the contrapositive of the preceding result.

Proposition 7. Let ˝ be of finite type in Cn with not necessarily C 1 boundary. Assume that the @ problem satisfies uniform estimates on ˝. Then ˝ is of type HL1 .


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5 The Koszul Complex and Applications The Koszul complex was originally developed for calculating the cohomology of certain Lie algebras. Thanks to the insight of Lars Hörmander and others, it has now become a powerful tool in mathematical analysis. Let f1 , f2 , . . . , fk be corona data on a domain ˝. This means that there is a ı > 0 so that k X

jfj .z/j ı

j D1

for all z 2 ˝. Set

ı Uj D z 2 ˝ W jfj .z/j > 2k

:

Then the Uj are open sets and [j Uj ˝. Let f'j g be a partition of unity subordinate to the covering fUj g. Set X 'j C vj i fi ; gj D fj iD1 k

where vij D vj i . Then, by a simple algebra calculation, X

fj gj 1 :

j

We seek vij so that each gj is bounded and holomorphic. Thus what we require is that 0 D @gj D

X @'j C @vj i fi : fj i

As a result, we need to solve X

@vj i f i D

i

@'j fj

or X i

@.fi vj i / D

@'j : fj

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One can show that @'j =fj is a Carleson measure (see [HOR] for the provenance of these ideas). Thus we need to be able to solve @g D ; for a @-closed (0,1) form with Carleson measure coefficients, and obtain a solution g that is bounded. This is essentially what Tom Wolff accomplished in his celebrated proof of the corona theorem—see [KRA2, Ch. 11]. Varopoulos [VAR] showed that we cannot accomplish the same thing on the unit ball in Cn for n 2. That is to say, he proved that you could get BMO estimates for the @ problem with Carleson measure data, but not uniform estimates. He did this by proving a remarkable new characterization of BMO using balyage. We shall describe some of Varopoulos’s results here. One of Varopoulos’s main theorems is as follows. Note that we use Mp;q to 1 denote .p; q/ forms with measure coefficients and Cp;q to denote .p; q/ forms with 1 C coefficients. Theorem 2. Let ˝ Cn be strongly pseudoconvex and smoothly bounded. (i) Let 2 Mp;q .int ˝/ satisfy a Carleson condition and @ D 0 on ˝. Then there is a g 2 BMOp;q1 .@˝/ such that Z Z g^ D ^ @˝

˝

1 for all 2 Cnp;nq .˝/ such that @ D 0 in a neighborhood of ˝. (ii) Conversely let g 2 BMOp:q1 .@˝/. Then there is a 2 Mp;q .˝/ (a form with Carleson measure coefficients in ˝) and an f 2 L1 .@˝/ such that

Z

Z g^ @˝

for all

Z

D

^

C

˝

1 2 Cnp;nq .˝/ that satisfies @

f ^ @˝

D 0 in a neighborhood of ˝.

This is in effect a global formulation of the solution of the @b problem with Carleson data and a BMO solution. Theorem 3. Let f1 ; f2 2 H 1 .B/ be corona data. Then there are two holomorphic function '1 and '2 on B that satisfy f1 '1 C f2 '2 1 ; sup k'1 .r/kBMO.@B/ < 1 ; r

sup k'2 .r/kBMO.@B/ < 1 : r

The same result holds on strongly pseudoconvex domains.


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We note that this result provides BMO solutions to the corona problem. But it is only valid for two pieces of data. More recently, Costea, Sawyer, and Wick [CSW] have extended the result to k pieces of data, or even infinitely many pieces of data. The main message here is that, in one complex variable, you can solve the @problem with corona data and get an L1 solution. But in several complex variables you cannot. This certainly does not mean that the corona problem cannot be solved in several complex variables; but it cripples one of our most important approaches to the problem. Next we present a theorem that contains the real-variable essence of Varopoulos’s work: Theorem 4. Let f 2 BMO.RN / with compact support. Then there is a function N C1 F 2 C 1 .RC / such that (i) limy!0C F .x; y/ f .x/ 2 L1 .RN /. (ii) The measure 0

N ˇ X ˇ @F ˇ @ jrF j dxdy ˇ @x j D1

j

1 ˇ ˇ ˇ ˇ ˇ @F ˇ ˇ C ˇ ˇA dxdy ˇ ˇ @y ˇ

is Carleson. (iii) There exists a function g 2 L1 .RN / such that sup jF .x; y/j g.x/ : y>0

(iv) jrF j D O.1=y/ as y ! C1. This is a version of the result that expresses a BMO function as the balyage of a function on the upper halfspace.

6 Factorization of Functions In this section we discuss work of Coifman/Rochberg/Weiss and Krantz/Li about factorization of holomorphic functions. These ideas are inspired by classical results that are rooted in the theory of Blaschke products (see [KRA1, Ch. 8] for some of the details of that theory). Let us begin with some background information. Let X be a set and a measure on X . We say that .X; / is a space of homogeneous type2 if, for each x 2 X and each r > 0, there is a ball B.x; r/ X such that

2 Do not confuse this idea with the notion of “homogeneous space.” The latter is an idea from harmonic analysis (see [HEL]) that has no bearing on the present discussion.

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(a) 0 < .B.x; r// < 1 8x; r; (b) .B.x; 2r// C.B.x; r/ for some C > 0 and all x; r; (c) If B.x; r/ \ B.y; s/ ¤ ; and r s, then B.x; C r/ B.y; s/. Here C is as in part (b). When there is a quasimetric d present then part (c) simply follows from the triangle inequality. In practice, property (c) is the most interesting and the trickiest to verify. If f is a locally integrable function on X and B.x; r/ X , we set Z 1 fB.x;r/ f .t / d.t / : .B.x; r// B.x;r/ In other words, fB.x;r/ is the average of f on B.x; r/. On a space .X; / of homogeneous type, we define BMO.X / D f 2 L1loc W sup x;r

1 .B.x; r/

Z B.x;r/

jf .t / fB.x;r/ j d.t / kf k < 1 :

These are the functions of bounded mean oscillation. Also we take Z 1 1 VMO.X /D f 2 Lloc W lim sup jf .t /fB.x;r/ j d.t /D0 : r0 !0 x;rr0 .B.x; r/ B.x;r/ These are the functions of vanishing mean oscillation. The space BMO was first created by F. John and L. Nirenberg in the context of partial differential equations [JON]. It has become important in harmonic analysis because it is the dual of the real variable Hardy space (see below). The space VMO was created by D. Sarason [SAR] because it is the predual of the real variable Hardy space. Now we define the atomic Hardy space Hat1 by way of atoms.3 A function a on X is called an atom if it is -measurable and if (a) supp a B.x; r/, some x; r; 1 (b) ja.t /j .B.x;r// for all t 2 B.x; r/; R (c) a.x/ d.x/ D 0 : We say that f 2 Hat1 H 1 if and only if f D

X

j aj

j

for some fj g 2 `1 and some atoms aj . 3 These spaces were originally defined by Stein and Weiss [STW] in terms of partial differential equations. The atomic approach, which is more recent (and which was first conceived by C. Fefferman in a conversation at tea), turns out to be more flexible and is well adapted to the theory of spaces of homogeneous type.


121

Recall from measure theory that, if 1 p 1, then the dual exponent is p 0 p=.p 1/. Fact. A function f is in L1 .X; / if and only if there exist f1 2 Lp and f2 2 Lp such that f D f1 f2

0

(*)

with kf kL1 D kf1 kLp kf2 kLp0 : Now we know that .H 1 / D BMO—see [[FST]. Se we might ask whether there 0 is a factorization similar to (*) with the roles of Lp and Lp played by H 1 and BMO. The answer is “yes,” as Krantz and Li proved (see [KRL1]): Theorem 5. Let .X; / be a space of homogeneous type, and assume that .X / < 1. If f 2 L1 .X; /, then there exist f1 2 BMO and f2 2 Hat1 such that f D f1 f2 and kf kL1 kf1 kBMO kf2 kHat1 : Furthermore, if b 2 L1 , then bf2 2 Hat1 . This latter says that f2 2 L log L. This new type of result encouraged us to explore other types of factorization theorems. Serendipitously, some of these turned out to be connected with the corona problem. Now let ˝ Cn be bounded and pseudoconvex. Let H p .˝/ be the holomorphic Hardy space, defined as in [STE] or [KRA1]. Of course a function f in H p .˝/ has a boundary limit functions in Lp .@˝/—see [KRA1, Ch. 8]. When ˝ D D, the unit disc in C, then f 2 H 1 .D/ implies that f D f1 f2 , where f1 ; f2 2 H 2 .D/. This is because f has a factorization as f D B F , where B is a Blaschke product that contains all the zeros of f , and F is zero-free and has the same norm as f . So we may write

p p f DB F D B F F f1 f2 : Such a result is definitely not true in higher dimensions—see, for instance, [RUS]. However, Coifman/Rochberg/Weiss in 1976 (see [CRW]) showed that, if f 2 H 1 .B/, then f D

X

fj gj ;

j

where fj ; gj 2 H 2 . This result was generalized by Krantz/Li [KRL2] to strongly pseudoconvex domains in Cn .

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Fix a domain ˝ Cn with C 2 boundary. Now let L 2 .@˝/ D ff W f 2 L2 .@˝/ H 2 .@˝/g : These are complex conjugates of functions that are orthogonal to H 2 . For 1 p < 1, let

L p .@˝/ D closure of Lp .@˝/ \ L 2 .@˝/ in Lp .@˝/ : We can identify L 1 .@˝/ with the complex conjugates of functions in L1 .@˝/=H 1 .˝/. We further set H01 .D/ D those elements of H 1 .D/ that vanish at 0 : This space can be identified with the complex conjugates of harmonic extensions of functions in L 1 .T/. [Clearly, if f 2 L 1 , then f ? H 2 so f is perpendicular to the constant 1. Hence it has mean value 0, so vanishes at the origin.] This last fact played an important role in Tom Wolff’s proof of the corona theorem (see [KRA2, Ch. 11]). A natural question to ask at this point is: Given f 2 L 1 , can we decompose f as an infinite sum of products of functions in H 2 times functions in L 2 ? If this is true, then it in fact leads to a solution of the corona problem (see [LI] and [AND]). However, it is false in several complex variables. This negative result is related to the fact that f 2 L 1 .@˝/ does not imply that f 2 Hat1 .@˝/. A related question is: Can one decompose f 2 Hat1 .@˝/ \ L 1 .@˝/ as a finite sum of products of elements of H 2 .˝/ and L 2 .@˝/? The answer to this question is “yes.” We conclude this discussion with one final decomposition theorem. Theorem 6. Let 1 < p < 1. Let ˝ be a smoothly bounded, strongly pseudoconvex domain (or else the polydisc). Let f 2 L 1 .@˝/ \ Hat1 .@˝/. Then there are a sequence fj g of numbers, a sequence of functions ffj g in H p .˝/, and a sequence 0 of functions fgj g in L p .@˝/ such that X

jj j kf kH 1

j

kfj kp kgj kp0 Cp for all j ; and f D

1 X j D1

j fj gj :


123

Conversely, any such sum lies in L 1 .@˝/ \ Hat1 .@˝/ : All the ideas discussed above have applications to the corona problems of one and several complex variables, as we shall see momentarily. In what follows, let us say that a domain ˝ is regular if one of the following holds: (a) ˝ is strongly pseudoconvex; (b) ˝ is of finite type in C2 ; (c) ˝ is of finite type and convex in Cn . Of course (b) is an allusion to work of Kohn [KOH] and (c) is a reference to work of McNeal [MCN]. The next results are due to Krantz and Li [KRL2]. Theorem 7. Let ˝ be a regular domain in Cn . Let f1 , f2 be corona data satisfying 0 < ı 2 jf1 j2 C jf2 j2 1 : Then there exist g1 ; g2 2 H 1 such that f1 g1 C f2 g2 1 and kg1 kH 1 C kg2 kH 1 C if and only if Sf W L 1 .˝/2 ! L 1 .˝/ is onto, where Sf .u/ D f1 u1 C f2 u2 for u D .u1 ; u2 / 2 L 1 .˝/2 L 1 .˝/ L 1 .˝/. Consider the equation f1 .z/'1 .z/ C f2 .z/'2 .z/ 1

()

on ˝. Let g1 D '1 f2 a and g2 D '2 C f1 a, where we seek a “good” function a so that g1 and g2 so defined are holomorphic and bounded. If we can find such an a, then g1 , g2 solve the corona.

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Proposition 8. If '1 , '2 solve (), then (i) '2 @'1 '1 @'2 is @-closed. (ii) The functions g1 , g2 as defined above are both holomorphic on ˝ if and only if a is a solution of the equation @a D '2 @'1 '1 @'2 :

()

Proof. This is a straightforward calculation. Theorem 8. Let ˝ be a bounded domain in Cn with C 2 boundary (or else the polydisc). Let f1 , f2 be corona data on ˝. If the corona problem is solved by g1 , g2 so that jg.z/jjf .z/j1 1 C.ı/ L .@˝/ (where jgj2 D jg1 j2 C jg2 j2 and jf j2 D jf1 j2 C jf2 j2 ), and if a is a solution of the equation @a D '2 @'1 '1 @'2 ; then kakL1 .@˝/=H 1 .˝/ C.ı/ : Theorem 9. Let f1 , f2 be corona data. Then (i) Suppose that ˝ is a regular domain and that () has a solution a 2 BMO.@˝/. Then Sf W H p .˝/2 ! H p .˝/ is onto for all 1 < p < 1. (ii) If ˝ is such that Theorem 8 holds, and if Sf W H p .˝/2 ! H p .˝/ is onto for some 1 < p < 1, then () has a solution a 2 BMO.@˝/. Remark. Let ˝ D B, the unit ball in Cn , n > 1. Then () having a solution a with a 2 BMO.@B/ does not imply that () has a solution a1 2 L1 . But for the disc D C, this implication is actually true!!

7 Concluding Remarks The simple truth is that our understanding of the corona problem in several complex variables is meager. While there are no known domains in the complex plane for which the corona problem is known to fail, there are also no domains in multidimensional complex space for which the corona problem is known to hold.


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There are a number of partial results in norms other than H 1 . These include L (see [AMA1]) and, most notably, BMO. There are various functional analysis versions of the corona problem. The Toeplitz operator version (see [BRS, AMA2]) is of particular interest. There are also some connections with geometry and other parts of mathematics. We cannot explore those here. Clearly there is a great deal of work remaining to be done in the several complex variables setting. Perhaps a reformulation, or an adjustment, of the problem is in order. This will be the subject of future work. p

References [ALL] N. L. Alling, A proof of the corona conjecture for finite open Riemann surfaces, Bull. Amer. Math. Soc. 70(1964), 110–112 [AMA1] E. Amar, On the corona problem, Jour. Geom. Analysis 1(1991), 291–305. [AMA2] E. Amar, On the Toeplitz corona problem, Publ. Mat. 47(2003), 489–496. [AND] M. Andersson, The H 2 corona problem and @b in weakly pseudoconvex domains, Trans. Amer. Math. Soc. 342(1994), 241–255. [BRS] J. A. Ball, L. Rodman, and I. M. Spitkovsky, Toeplitz corona problem for algebras of almost periodic functions. Toeplitz matrices and singular integral equations, (Pobershau, 2001), 25–37, Oper. Theory Adv. Appl., 135, Birkhäuser, Basel, 2002. [BER] G. Berg, Bounded holomorphic functions of several variables, Ark. Mat. 20(1982), 249–270. [CAR] L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76(1962), 547–559. [CAT] D. Catlin, Boundary behavior of holomorphic functions on weeakly pseudoconvex domains, Princeton University thesis, 1978. [CRW] R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103(1976), 611–635. [CSW] S. Costea, E. Sawyer, and B. Wick, BMO estimates for the H 1 .Bn / Corona problem, J. Funct. Anal. 258(2010), 3818–3840. [[FST] C. Fefferman and E. M. Stein, H p spaces of several variables, Acta Math. 129(1972), 137–193. [FSIB] J. E. Fornæss and N. Sibony, Smooth pseudoconvex domains in C2 for which the corona theorem and Lp estimates for @ fail, Complex Analysis and Geometry, 209–222, Univ. Ser. Math., Plenum, New York, 1993. [GAM] T. W. Gamelin, Uniform Algebras and Jensen Measures, LMS Lecture Notes Series 32, Cambridge University Press, Cambridge, 1978. [HEL] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. [HOR] L. Hörmander, Generators for some rings of analytic functions, Bull. Amer. Math. Soc. 73(1967), 943–949. [[JAP] M. Jarnicki and P. Pflug, Extension of Holomorphic Functions, de Gruyter, Berlin, 2000. [JON] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure and Appl. Math. 14(1961), 415–426. [KOH] J. J. Kohn, Boundary behavior of @ on weakly pseudoconvex manifolds of dimension two, J. Diff. Geom. 6(1972), 523–542. [KRA1] S. G. Krantz, Function Theory of Several Complex Variables, 2nd ed., American Mathematical Society, Providence, RI, 2001.

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[KRA2] S. G. Krantz, Cornerstones of Geometric Function Theory: Explorations in Complex Analysis, Birkhäuser Publishing, Boston, 2006. [KRA3] S. G. Krantz, Int. J. Math. Math. Sci. 2010, Art. ID 648597, 18 pp. [KRL1] S. G. Krantz and S.-Y. Li, Factorization of functions in subspaces of L1 and applications to the corona problem, Indiana Univ. Math. J. 45(1996), 83–102. [KRL2] S. G. Krantz and S.-Y. Li, On decomposition theorems for Hardy spaces on domains in Cn and applications, Jour. Four. Anal. and Applics. 2(1995), 65–107. [LI] S.-Y. Li, Corona problem of several complex variables, The Madison Symposium on Complex Analysis (Madison, WI, 1991), 307–328, Contemp. Math., 137, Amer. Math. Soc., Providence, RI, 1992. [MCN] J. McNeal, Convex domains of finite type, J. Funct. Anal. 108(1992), 361–373. [RUS] L. Rubel and A. Shields, The failure of interior-exterior factorization in the polydisc and the ball, Tôhoku Math. J. 24(1972), 409–413. [SAR] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207(1975), 391–405. [SIB1] N. Sibony, Prolongement analytique des fonctions holomorphes bornées (French), C. R. Acad. Sci. Paris Ser. A–B 275(1972), A973–A976. [SIB2] N. Sibony, Probléme de la couronne pour des domaines pseudoconvexes bord lisse (French) [The corona problem for pseudoconvex domains with smooth boundary], Ann. of Math. (2) 126(1987), 675–682. [STE] E. M. Stein, The Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton University Press, Princeton, NJ, 1972. [STW] E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables, I. The theory of H p -spaces, Acta Math. 103(1960), 25–62. [STO] E. L. Stout, Bounded holomorphic functions on finite Reimann surfaces, Trans. Amer. Math. Soc. 120(1965), 255–285. [VAR] N. Th. Varopoulos, BMO functions and the @-equation, Pacific J. Math. 71(1977), 221–273.

Corona-Type Theorems and Division in Some Function Algebras on Planar Domains Raymond Mortini and Rudolf Rupp

Abstract Let A be an algebra of bounded smooth functions on the interior of a compact set in P the plane. We study the following problem: if f; f1 ; : : : ; fn 2 A n satisfy jf j jf j, does there exist gj 2 A and a constant N 2 N such Pn j D1 j N that f D j D1 gj fj ? A prominent role in our proofs is played by a new space, C@;1 .K/, which we call the algebra of @-smooth functions. In the case n D 1, a complete solution is given for the algebras Am .K/ of functions holomorphic in K ı and whose first m-derivatives extend continuously to K ı . This necessitates the introduction of a special class of compacta, the so-called locally L-connected sets. We also present another constructive proof of the Nullstellensatz for A.K/, that is only based on elementary @-calculus and Wolff’s method. Keywords Algebras of analytic functions • Ideals • Corona-type theorems • Nullstellensatz • Division in algebras of smooth functions Subject Classifications: Primary 46J10; Secondary 46J15, 46J20, 30H50

R. Mortini () Département de Mathématiques et Institut Élie Cartan de Lorraine, Université de Lorraine, UMR 7502, Ile du Saulcy, 57045 Metz, France e-mail: [email protected] R. Rupp Fakultät Allgemeinwissenschaften, TH-Nürnberg, Kesslerplatz 12, 90489 Nürnberg, Germany e-mail: [email protected] R.G. Douglas et al. (eds.), The Corona Problem, Fields Institute Communications 72, DOI 10.1007/978-1-4939-1255-1__7, © Springer Science+Business Media New York 2014

127

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R. Mortini and R. Rupp

1 Introduction Our paper is motivated by the following problem. Problem 1. Let A.K/ be the algebra of all complex-valued functions that are continuous on the compact set K C and holomorphic in the interior K ı of K. Given f; f1 ; : : : ; fn 2 A.K/ satisfying jf j

n X

jfj j;

(1)

j D1

does there exist a power N such that f N belongs to the ideal IA.K/ .f1 ; : : : ; fn / generated by the fj in A.K/? In view of Wolff’s result for the algebra of bounded holomorphic functions in the open unit disk D (see [2]), the expected value for N seems to be 3. We are unable to confirm this; our intention therefore is to present sufficient conditions that guarantee the existence of such a constant that may depend on the n-tuple .f1 ; : :P : ; fn /. If f is zero-free on K then, by the classical Nullstellensatz for A.K/, f D nj D1 fj gj for some gj 2 A.K/, whenever the fj satisfy (1). We present a constructive proof of this assertion by using @-calculus. A different constructive proof, indirectly based on properties of the Pompeiu operator ZZ Pf .z/ WD K

f ./ d ./ z

and elementary approximation theory, was recently given by the authors in [6]. To achieve our goals, we consider various algebras of smooth functions on K. Here is our setting. First we present some notation and introduce, apart from A.K/ that was defined above, the spaces we are dealing with. As usual, if fx and fy are the partial derivatives of f , then @f D .1=2/.fx ify / and @f D .1=2/.fx C ify / denote the Wirtinger derivatives of f . Let K be either R or C. Definition 2. (1) C.K; K/ is the set of all continuous, K-valued functions on K. If K D C, then we also write C.K/ instead of C.K; C/. (2) C m .K/ is the set of all f 2 C.K/ that are m-times continuously differentiable on K ı and such that each of the partial derivatives up to the order m extends continuously to K. (3) Am .K/ WD A.K/ \ C m .K/. Of course Am .K/ is the set of all functions f 2 A.K/ such that f .j / jK ı extends continuously to K for j D 1; : : : ; m. Finally, we introduce the following

Corona-Type Theorems and Division in Some Function Algebras on Planar Domains

129

space of @-smooth functions. This space will play an important role when solving @-equations. (4) C@;1 .K/ is the set of all f 2 C.K/ continuously differentiable in K ı with @f 2 C 1 .K ı / and such that @f admits a continuous extension to K. The set of continuous (respectively infinitely often differentiable) functions on C with compact support is denoted by Cc .C/ (respectively Cc1 .C/). If f 2 C.K; K/, then jjf jj1 D supz2K jf .z/j and Z.f / D fz 2 K W f .z/ D 0g is the zero set of f . If A is a commutative unital algebra with unit element denoted by 1, then the ideal generated by f1 ; : : : ; fn 2 A is denoted by IA .f1 ; : : : ; fn /; that is IA .f1 ; : : : ; fn / D

X n

gj fj W gj 2 A :

j D1

2 The Algebra of @-Smooth Functions If we endow C@;1 .K/ with the pointwise operations .C; ; s /, then it becomes an algebra; just note that for f; g 2 C@;1 .K/, @.fg/ D .@f /g C f @g. The most important subalgebra of C@;1 .K/ is A.K/. It is obvious that C@;1 .K/ contains all the polynomials in the real variables x and y, or what is equivalent, all polynomials in z and z. In particular C@;1 .K/ is uniformly dense in C.K/. Note that the class of real-valued functions u in C@;1 .K/ coincides with C 2 .K ı ; R/ \ C 1 .K; R/, since 2@f D ux C i uy 2 C 1 .K ı / \ C.K/. We point out that for f 2 C@;1 .K/, the function @f W K ı ! C may not even be bounded. Consider p for example on the unit disk D the standard holomorphic branch of the function 1 z. Here 1 @f .z/ D f 0 .z/ D .1 z/1=2 ; 2 which is unbounded. Hence C@;1 .D/ n C 1 .D/ 6D ; and C 1 .D/ n C@;1 .D/ 6D ;: On the other hand, C@;1 .K/ is strictly bigger than C 2 .K/. The example above also shows that if f 2 C@;1 .K/, then f may not belong to C@;1 .K/ (note that @ f D @f ).

130


Question. Is the restriction algebra C@;1 .K/jK ı a subalgebra of C 2 .K ı /? Note that we only assume that @f D

ux vy 2

Ci

uy Cvx 2

is continuously differentiable in K ı .

A useful algebraic property is that C@;1 .K/ is inversionally closed; this means that if f 2 C@;1 .K/ has no zeros on K, then 1=f 2 C@;1 .K/. P A generalization to n-tuples (Dsolution to the Bézout equation nj D1 xj fj D 1) is given by the following theorem. Theorem 3. Suppose that the functions P f1 ; : : : ; fn 2 C@;1 .K/ have no common zero on K. Then the Bézout equation nj D1 xj fj D 1 admits a solution .x1 ; : : : ; xn / in C@;1 .K/. We present two proofs. Proof. (1) Let fj : 2 kD1 jfk j

qj WD Pn

P Then qj 2 C.K/ and nj D1 qj fj D 1. By Weierstrass’ approximation theorem choose a polynomial pj .z; z/ such that on K jpj qj j 2

n X

!1 jjfk jj1

:

kD1

Then ˇ ˇ ˇ ˇ ˇ n ˇ ˇX ˇ n ˇ ˇ n ˇ ˇ ˇX ˇX 1 1 ˇ ˇ ˇ ˇ ˇ pj fj ˇ qj fj ˇ ˇ .pj qj /fj ˇˇ 1 D : 2 2 ˇj D1 ˇ ˇj D1 ˇ j D1 Note that we get that

Pn

j D1

pj fj 2 C@;1 .K/. Because C@;1 .K/ is inversionally closed, pj 2 C@;1 .K/: kD1 pk fk

x j D Pn

P Since nj D1 xj fj D 1, we see that .x1 ; : : : ; xn / is the desired solution to the Bézout equation in C@;1 .K/. (2) This is similar to [1, Exercice 25, p. 139]. P Let fj be continuously extended to an open neighborhood U of K so that nj D1 jfj j ı > 0 on U . Let Ej D fz 2 U W jfj .z/j > 0g. Since the functions fj have no common zeros on U , S n 1 j D1 Ej D U . Let f˛j W j D 1; : : : ; ng be a Cc -partition of unity subordinate to the open covering fE1 ; : : : ; En g of K; that is,


˛j 2 Cc1 .C/;

0 ˛j 1;

supp ˛j Ej ;

n X

131

˛j D 1 on K

j D1

(see [9, p. 162]). Then xj WD ˛j fj1 2 C@;1 .K/ and n X

xj fj D

j D1

n X

˛j D 1:

j D1

Next we deal with the generalized Bézout equation

Pn

j D1

xj fj D f .

Proposition 4. LetT f; fj 2 C@;1 .K/ and suppose that f vanishes in a neighborhood (within K) of nj D1 Z.fj /. Then f 2 IC@;1 .K/ .f1 ; : : : ; fn /. T Proof. Let V C be an open set satisfying nj D1 Z.fj / V \ K Z.f /. If P S D K n V , then infS nj D1 jfj j ı > 0. As usual we extend the functions fj to continuous functions in Cc .C/ and denote these extensions S by the same symbol. Let Uj D fz 2 C W jfj j > 0g, j D 1; : : : ; n. Then S nj D1 Uj . Let f˛j W j D 1; : : : ; ng be a Cc1 -partition of unity subordinate to the open covering fU1 ; : : : ; Un g of S . Then xj WD ˛j fj1 2 C@;1 .K/ and n X

xj fj D

j D1

n X

˛j D 1 on S :

j D1

Noticing that f 0 on K n S , we get n X

.f xj /fj D f on K;

j D1

where f xj 2 C@;1 .K/.

3 A New Proof of the Nullstellensatz for A.K / Here we give yet another elementary proof of the “Corona Theorem” (or Nullstellensatz) for the algebra A.K/ (see [6] for the preceding one). We use Wolff’s @-method (see for example [1, p. 130 and p. 139]). Our main tool will be the following wellknown theorem (see [1, 8], and [6]). Here denotes the 2-dimensional Lebesgue measure. Theorem 5. Let K C be compact, f 2 C.K/ \ C 1 .K ı / and let u.z/ D

1

ZZ K

f .w/ d .w/: wz

132


Then O 1. u 2 C.C/, 2. u 2 C 1 .K ı / and holomorphic outside K, 3. @u D f on K ı . TheoremP 6. Let K C be a compact set and let fj 2 A.K/. Then the Bézout equation nj D1 hj fj D 1 admits a solution .h1 ; : : : ; hn / in A.K/ if and only if the functions fj have no common zero on K. P Proof. Assume that nj D1 jfj j ı > 0 on K. Applying Theorem 3, there is a P solution .x1 ; : : : ; xn / 2 C@;1 .K/n C.K/n to nj D1 xj fj D 1. Now we use Wolff’s method to solve a system of @-equations. Consider f D .f1 ; : : : ; fn / as a P row matrix; its transpose is denoted by ft . Let jfj2 D nj D1 jfj j2 ; that is jfj2 D fft . The Bézout equation now reads as xft D 1. It is well-known (see for instance [7, p. 227]) that any other solution u 2 C.K/n to the Bézout equation uft D 1 is given by ut D xt C H ft ; or equivalently u D x fH; where H is an n n antisymmetric matrix over C.K/; that is H t D H . Let

t 1 F D @xt f @xt f : jfj2 Since x 2 C@;1 .K/n , we see that F is an antisymmetric matrix over C.K/\C 1 .K ı /. Thus, by Theorem 5, the system @H D F admits a matrix solution H over C.K/ \ C 1 .K ı /. Note that H can be chosen to be antisymmetric, too. It is now easy to check that on K ı , @u D 0. In fact t 1 @u D @x f @H D @x f f @x @xt f jfj2 @.f xt / f @.x ft /t f .f @xt / f D D D D0 jfj2 jfj2 jfj2 Thus u D x fH 2 A.K/. Hence u is the solution to the Bézout equation in A.K/.


133

4 The Principal Ideal Case In this section we consider the following division problems: let A be one of the algebras C m .K/; Am .K/ and C@;1 .K/. Determine the best constant N 2 N such that jf j jgj implies f N 2 IA .g/; that is g divides f N in A. Note that this is the (n D 1)-case of Problem 1. We need to introduce a special class of compacta.

4.1 Locally L-connectedness Definition 7. A compact set K C satisfying K D K ı is said to be locally Lconnected if for every z0 2 @K there is an open neighborhood U of z0 in K and a constant L > 0 such that every point z 2 U \ K ı can be joined with z0 by a piecewise C 1 -path z entirely contained in K ı (except for the end-point z0 ) and such that L.z / L jz z0 j; where L.z / denotes the length of the path z . As examples we mention closed disks, compact convex sets with interior, and finite unions of these sets. Counterexamples, for instance, are comb domains, spirals having infinite length and certain disjoint unions of infinitely many disks (see below Figs. 1, 2, 3). Definition 8. A topological space X is said to be locally path-connected if for every x 2 X and every neighborhood U X of x there exists a neighborhood V of x such that V U and for any pair .u; v/ of points in V there is a path from u to v lying in U .1 Proposition 9. Every locally L-connected compactum is locally path-connected.

Fig. 1 The comb domain

1

In the “usual” definition the path is assumed to lie in V ; these two definitions coincide

134


Fig. 2 The spiral domain

Fig. 3 A non locally L-connected compactum

Proof. Let z0 2 @K D K ı n K ı . Given the open disk D.z0 ; r/, we choose the open neighborhood U of z0 in K and L D L.z0 / > 0 according to the definition of local L-connectivity. In particular, every point z 2 D.z0 ; r/\U \K ı can be joined with z0 by a curve z;z0 contained in K ı [fz0 g. Note that for z 2 U \K ı , z;z0 D.z0 ; Ljz z0 j/. Hence, if we choose 0 < r 0 < r=2L so small that D.z0 ; r 0 / \ K U , then every point z 2 D.z0 ; r 0 / \ K ı can be joined with z0 by a path z;z0 contained in K \ D.z0 ; Ljz z0 j/ K \ D.z0 ; r/: We still need to show that every point z1 2 @K \ D.z0 ; r 0 / can be joined with z0 by a path contained in K \ D.z0 ; r/. By the same argument as above, there is a disk D.z1 ; r 00 / D.z0 ; r 0 / such that every point z 2 D.z1 ; r 00 / \ K ı can be joined with z1 by a curve z;z1 contained in K \ D.z1 ; L.z1 /r 00 / K \ D.z0 ; r/: 1 Thus the concatenation of the inverse path z;z with z;z0 joins z1 with z0 within 1 D.z0 ; r/ \ K. Hence every point w 2 D.z0 ; r 0 / \ K can be joined to z0 by a path contained in D.z0 ; r/ \ K. If z0 2 K ı , then the assertion is trivial.


135

We conclude that for z0 2 K and D.z0 ; r/ we find a neighborhood V of z0 in K such that any two points u; v 2 V can be joined by a path (passing through z0 ) that stays in D.z0 ; r/. Hence K is locally path-connected. We mention that there exist locally connected, path-connected continua K with K ı D K that are not locally L-connected. For example, let K be the inner spiral . C 1/1 r./ 1 , 1. See the following figure. Here the point 0 2 @K, but cannot be joined with any other point z 2 K by a rectifiable path. We can also take a spiral consisting of “thick” half-circles with radii 1=n, n D 1; 2; : : : .

4.2 The Taylor formula on the boundary We need the following Taylor formula for functions in Am .K/. Proposition 10. (1) Let K be a locally L-connected compactum. Given f 2 Am .K/ and z0 2 K ı n K ı D @K, we denote the continuous extension of f .j / to z0 by the symbol f .j / .z0 /, .j D 0; 1; : : : ; m/. Let 1 0 1 .m/ f .z0 / .z z0 / C C f .z0 / .z z0 /m 1Š mŠ

Pm .z; z0 / D f .z0 / C

be the m-th Taylor polynomial of f at z0 . Then f .z/ D Pm .z; z0 / C Rm .z/; .j /

where Rm 2 Am .K/, Rm .z0 / D 0 for j D 0; 1; : : : ; m and .j / .z/ D O.z z0 /mj as z ! z0 : Rm

Moreover, if m 1,

136


lim f 0 .z/ D

z!z0 z2K ı

f .z/ f .z0 / : z z0

lim

z!z0 z2Knfz0 g

(2)

(2) If K is not locally L-connected, then the equality (2) does not hold in general. Proof. (1) We first show equality (2). Set ` WD z!z lim f 0 .z/. Let U K be the 0 z2K ı

neighborhood of z0 and L > 0 the associated constant given by the definition 7 of locally L-connectedness. Given " > 0, choose ı > 0 so small that jf 0 ./ `j < "=L for j z0 j < ı, 2 K ı . For z 2 D.z0 ; ı=L/ \ U \ K ı , let z;z0 be a path in K ı [ fz0 g joining z with z0 and with length L.z;z0 / L jz z0 j. Note that z;z0 D.z0 ; L jz z0 j/ D.z0 ; ı/: Then, by integrating along z;z0 from z to a point zQ0 close to z0 and passing to the limit, we obtain Z f .z/ f .z0 / D f 0 ./ d : z;z0

Hence ˇ ˇ Z ˇ f .z/ f .z0 / ˇ 1 ˇ ˇ ` jf 0 ./ `j jd j ˇ ˇ jz z j z z0 0 z;z0

" 1 L.z;z0 / ": jz z0 j L

This confirms (2). Since f 2 Am .K/, we obviously have Rm 2 Am .K/. Moreover, by taking .j / derivatives in K ı and extending them to the boundary, we see that Rm .z0 / D .m/ .j / .f Pm / .z0 / D 0 for j D 0; 1; : : : ; m. Moreover, Rm .z/ D O.1/ for z ! z0 . As in the proof of (2), for z 2 K ı \ U , Z .m1/ .z/ D Rm

z;z0

.m/ Rm ./ d :

Hence, .m1/ .m/ jRm .z/j Ljz z0 j max jRm ./j D O.z z0 /: 2z;z0


Also,

137

Z .m2/ .z/ D Rm

Z

z;z0

.m1/ Rm ./ d .m1/

D

. z0 / z;z0

Rm ./ d ; z0

and so .m1/

.m2/ jRm .z/j

jRm ./j max 2z;z0 j z0 j

Z j z0 j jd j z;z0

O.1/L.z;z0 /2 D O.z z0 /2 : Using backward induction, we obtain the assertions that for the indeces j D m; m 1; : : : ; 1; 0 .j / .z/ D O.z z0 /mj Rm

as z ! z0 . (2) Let K be the union of the disk D0 D fz W jz C 1j 1g and the disks Dn D fz W jz 1=nj 1=n3 g; n D 3; 4; : : : : Note that Dn \ Dm D ; for n 6D m, n; m 3, and that K D K ı , but that K is not locally path-connected (at the origin). Let f be defined as follows: f D 0 p on D0 and f D 1= n on Dn . Then f is continuous on K, holomorphic on K ı and f 0 0 on K ı . It is obvious that f 0 admits a continuous extension to K, namely by the constant 0. However, p f .1=n/ f .0/ D n ! 1: 1=n 0 Thus the continuous extension of f 0 to 0 is distinct from the limit of the associated differential quotient: 0 6D

lim

z!0;z2K ı Re z>0

f .z/ f .0/ D 1: z0

Corollary 11. Let f 2 A1 .K/, where K is a locally L-connected compactum. Then for every z0 2 K ı n K ı there exists F 2 A.K/ with F .z0 / D f 0 .z0 /, respectively H 2 A1 .K/ satisfying H.z0 / D H 0 .z0 / D 0 and H.z/ D O.z z0 /, such that f .z/ D f .z0 / C .z z0 /f 0 .z0 / C H.z/ D f .z0 / C .z z0 /F .z/

(3) (4)

138


Remark. The function F in the representation f .z/ D f .z0 / C .z z0 /F .z/ for f 2 A1 .K/ may not belong to A1 .K/ itself. In fact, let f .z/ D .1 z/3 S.z/, z 2 D, where

1Cz S.z/ D exp 1z is the atomic inner function. Then f 2 A1 .D/ and f .z/ D .1 z/F .z/ with F .z/ D .1 z/2 S.z/. But F … A1 .D/.

4.3 Division in A m .K / We first consider the cases m D 0; 1. Theorem 12. (a) If f; g 2 C.K/ satisfy jf j jgj, then f 2 2 IC.K/ .g/. (b) If f; g 2 C 1 .K/ satisfy jf j jgj and if Z.g/ has no cluster points in K ı , then f 3 2 IC 1 .K/ .g/. (c) If f; g 2 A.K/ satisfy jf j jgj, then f 2 2 IA.K/ .g/. (d) If f; g 2 A1 .K/ satisfy jf j jgj, then f 3 2 IA1 .K/ .g/. (e) If f; g 2 A1 .K/ satisfy jf j jgj, then f 2 2 IA1 .K/ .g/ whenever K is locally L-connected. (f) If f; g 2 C@;1 .K/ satisfy jf j jgj, and if Z.g/ has no cluster points in K ı , then f 4 2 IC@;1 .K/ .g/. The powers 2,3,2,3,2,4 (within N) are optimal. Proof. (a) Since the quotient f =g is bounded on K n Z.g/, we just have to put h D f 2 =g on K n Z.g/ and h D 0 on Z.g/ in order to see that f 2 D gh, where h 2 C.K/. In fact, if z 2 @Z.g/, then g.z/ D 0 and so jf j jgj implies f .z/ D 0, too. Hence, for any sequence zn in K n Z.g/ with zn ! z0 we see that f 2 =g.zn / ! 0. Thus h 2 C.K/ and so f 2 2 IC.K/ .g/. (b) Let h D f 3 =g on K n Z.g/ and h D 0 on Z.g/. Then by a), h 2 C.K/ and h 2 C 1 .K n Z.g//. Let D be one of the derivatives d=dx or d=dy. Then Dh D

3gf 2 .Df / f 3 .Dg/ on K ı n Z.g/: g2

Because f; g 2 C 1 .K/, Df jK ı and DgjK ı extend continuously to K ı . So DhjK ı nZ.g/ extends continuously to K ı n Z.g/. Moreover, the assumption jf j jgj implies that jDhj jgj on K ı n Z.g/. Hence, DhjK ı nZ.g/ extends continuously to K ı \ Z.g/ (with value 0). By combining both facts, we conclude that DhjK ı nZ.g/ extends continuously to K ı . Tietze’s theorem now yields the extension of DhjK ı to a continuous function on K.


139

If a 2 Z.g/ \ K ı is an isolated point then, by the mean value theorem, Dh.a/ exists and coincides with the continuous extension of DhjKnZ.g/ at a. Thus h 2 C 1 .K ı /. Putting it all together, we conclude that h 2 C 1 .K/. Hence f 3 D gh 2 IC 1 .K/ .g/. We remark that if Z.g/ \ K ı is not discrete, then we are not always able to conclude that h is differentiable at points in K ı n Z.g/ \ Z.g/ \ K ı .2 (c) If f; g 2 A.K/, then jf j jgj implies that every zero of g is a zero of f . By Riemann’s singularity theorem, the boundedness of the quotient f =g around isolated zeros of g within K ı implies that f =g, and hence f 2 =g, are holomorphic. If g is constantly zero on a component ˝0 of K ı , then f 0 on ˝0 , too. So we have just to define f 2 =g D 0 there. Using a) and the fact that @.Z.g/ı / @K, we conclude that ( h.z/ D

f 2 .z/ g.z/

0

if z 2 K n Z.g/ if z 2 Z.g/

belongs to A.K/. Hence f 2 2 IA.K/ .g/. (d) Let S D K n Z.g/ı . Since the zeros of the holomorphic function g do not accumulate at any points in S ı , we deduce from b) that on S , f 3 D kg for some k 2 C 1 .S /. Moreover, k D 0 and Dk D 0 on Z.g/ \ S . If we let h D k on S and h D 0 on Z.g/, then h 2 C.K/. Let Kı D

[

0

!

˝n [ @

n

[

1 ˝j0 A

j

where the ˝n are those components of K ı containing only isolated zeros of g and where the ˝j0 are those components of K ı where g vanishes identically. Note that ˝n and ˝j0 are open sets. Since the quotient f 3 =g extends holomorphically at every isolated zero of g, we conclude that h is holomorphic on each of these components. Hence h 2 A.K/. Moreover, 0 @K D @S [ @ @

[

1 ˝j0 A :3

j

Note that the possibility of a continuous extension of the partial derivatives to E D Z.g/ does not mean that the function itself is differentiable at the points in E. See Proposition 10(2) or just consider the Cantor function C associated with the Cantor set E in Œ0; 1 . Here the derivative of C is zero at every point in R n E, C.0/ D 0, C.1/ D 1, C increasing, but C itself does not belong to C 1 .R/. 3 Actually we have @.Z.g/ı / @S D @K 2

140


S Since k 2 C 1 .S / and Dh D 0 on j ˝j0 , we deduce that Dh has a continuous extension to @K. Consequently, h 2 A1 .K/. (e) Let f; g 2 A1 .K/. Define h by ( h.z/ D

f 2 .z/ g.z/

if z 2 K n Z.g/

0

if z 2 Z.g/.

Then, by c), h 2 A.K/. Since K is locally L-connected, and so K D K ı , we may apply Corollary 11. Hence, for every z0 2 @K there exists F; G 2 A.K/ such that f .z/ D f .z0 / C .z z0 /F .z/ and g.z/ D g.z0 / C .z z0 /G.z/. In particular, if g.z0 / D 0, jf j jgj implies that f .z0 / D 0 and jF j jGj. If additionally g 0 .z0 / D 0, then G.z0 / D F .z0 / D 0; hence f 0 .z0 / D 0, too. Now, for z 2 K ı n Z.g/, @

f 2 .z/ g.z/

2f .z/g.z/f 0 .z/ f 2 .z/g 0 .z/ g 2 .z/

f .z/ 2 0 f .z/ 0 g .z/: f .z/ D2 g.z/ g.z/

D

(5) (6)

Case 1. Let z0 2 K n Z.g/ \ Z.g/. This means that z0 is in the boundary of Z.g/ with respect to the topological space K. Note also that by our global assumption, z0 2 @K. 1.1 If g 0 .z0 / D f 0 .z0 / D 0, then, ˇ the boundedness of the quotient 2by (5), f .z/ .f =g/jK ı nZ.g/ implies that @ g.z/ ˇK ı nZ.g/ admits a continuous extension to z0 . 1.2 If g 0 .z0 / 6D 0, then G.z0 / 6D 0. Hence f .z/ .z z0 /F .z/ F .z/ D D ; g.z/ .z z0 /G.z/ G.z/ is continuous at z0 . Thus, by (5), the continuity of f 0 and g 0 implies that

f 2 .z/ ˇ ˇ ı @ K nZ.g/ g.z/

admits a continuous extension to z0 . Case 2. Let z0 2 int Z.g/ \ @K, the interior being taken in the topological space K. Then there is an open set U in C containing z0 such that U \ K Z.g/. Since K D K ı , V WD U \ K ı 6D ;. Moreover g 0 on V . Note that by the definition of h, .f 2 =g/jK ı nZ.g/ has a continuous extension to K with


141

values 0 on Z.g/. Thus the derivative of h is zero on V , and the derivative of the null-function hjV has a continuous extension to z0 with value 0. Combining both cases, we conclude that h 2 A1 .K/ and so f 2 2 IA1 .K/ .g/. (f) Let h D f 4 =g on K n Z.g/ and h D 0 on Z.g/. Using b), we know that h D f .f 3 =g/jK ı nZ.g/ admits an extension to a function in C 1 .K ı /. (Here we have used that the zeros of g are isolated in K ı ). Next we have to consider the second-order derivatives. Since on K ı n Z.g/ @h D

4g .@f /f 3 f 4 @g g2

(7)

we obtain @.@h/ D

g 2 4.@g/.@f /f 3 C4g.@@f /f 3 C12g.@f /f 2 .@f /4f 3 .@f /.@g/f 4 .@@g/ 8g 2 .@f /f 3 .@g/C2f 4 g.@g/.@g/ g4

and @.@h/ D

g 2 4.@g/.@f /f 3 C 4g.@ @f /f 3 C 12g.@f /2 f 2 4f 3 .@f /.@g/ f 4 .@ @g/ 8g 2 .@f /f 3 .@g/ C 2f 4 g.@g/2 g4

:

Using that jf j jgj we conclude from (7) that on K ı n Z.g/, j@hj 4j@f j jgj2 C j@gj jgj2 C jgj2 : Hence @hjK ı nZ.g/ admits a continuous extension to K ı with value 0 on Z.g/ \ K ı . Using Tietze’s theorem we get a continuous extension to K, too. To show that @h is in C 1 .K ı /, we use the formulas above to conclude that for every z0 2 K ı n Z.g/ \ K ı there is a small neighborhood U of z0 such that on U n Z.g/ maxfj@.@h/j; j@.@h/jg C jgj: Thus @.@h/jK ı nZ.g/ and @.@h/jK ı nZ.g/ admit continuous extensions to K ı with value 0. Since the zeros of g within K ı are isolated, we deduce that @hjK ı nZ.g/ belongs to C 1 .K ı /. Thus h 2 C@;1 .K/ and so f 4 2 IC@;1 .K/ .g/. To show the optimality of the powers, let K be the closed unit disk D and consider for (a) and (c) the functions f .z/ D .1 z/S.z/ and g.z/ D 1 z where

1Cz S.z/ D exp 1z

142


Fig. 4 The sectors

is the atomic inner function. Then on D, f =g D S and jf j jgj. But S does not admit a continuous extension to D. For (b), let f .z/ D z and g.z/ D z. Then the @-derivative of f 2 =g outside 0 does not admit a continuous extension to 0; in fact, @.f 2 .z/=g.z// D @.z2 =z/ D z2 =z2 , a function that is discontinuous at 0. For (e) let f .z/ D .1 z/3 S.z/ and g.z/ D .1 z/3 . Then f; g 2 A1 .D/, jf j jgj but f =g … A1 .D/. For (f), let f .z/ D z and g.z/ D z. Then the @-derivative of @.f 3 =g/ outside 0 does not admit a continuous extension to 0; in fact, @.f 3 .z/=g.z// D @.z3 =z/ D z3 =z2 : Hence @[email protected] 3 .z/=g.z/// D @.z3 =z2 / D 3z2 =z2 : For (d) we necessarily have to consider a compactum that is not locally Lconnected. Moreover, as the proof of e) shows, the extensions of f 0 jK ı nZ.g/ and g 0 jK ı nZ.g/ to a point z0 2 @K \ Z.g/ cannot both be zero. As an example we take the following compact set K (see Fig. 4): K D f0g [

1 [

z2CW

nD1

1 22nC1

1 jzj 2n W j arg zj =4 : 2

Let Cn be the upper right corner of the sector Sn WD z 2 C W

1 22nC1

jzj

1 W j arg zj =4 : 22n


143

Define the functions f and g by f .z/ D z and ( g.z/ D

Cn

if z 2 Sn

0

if z D 0:

Then f and g belong to A1 .K/ and jf j jgj; note that for z 2 Sn jf .z/j max juj D 22n D jCn j D jg.z/j: u2Sn

Since g 0 0 on K ı , we obtain for every z 2 K, z 6D 0: d .z/ WD dz

f 2 .z/ g.z/

D 2f 0 .z/f .z/

z 1 D2 : g.z/ g.z/

If zn D Cn D rn e i=4 , then .zn / D 2e 2i=4 ; but if zn D C n , then zn 2 K and .zn / D 2. Thus lim .z/ does not exist. Hence f 2 =g … A1 .K/. z!0 z2Knf0g

In order to study division in the algebras C m .K/ and Am .K/, m 3, we use a variant of the Faà di Bruno formula, given in [4]. Theorem 13. Let M j D fk D .k1 ; : : : ; kj / 2 .N /j ; k1 k2 kj 1g be the set of ordered multi-indices in N D f1; 2; : : : g. Then for every f; g 2 C n .R/ 0

.f ı g/.n/ .x/ D

n X

1 X B C f .j / .g.x// @ Ckn g .k/ .x/A ;

j D1

(8)

k2M j jkjDn

where

Ckn D Y

! n k

:

n.k; i /Š

i

Here g

.k/

Dg

.k1 / .k2 /

g

g

.kj /

,

n k

! n is the multinomial coefficient defined by D k

nŠ , where jkj WD k1 C C kj D n, and n.k; i / is the number of times k1 Šk2 Š : : : kj Š the integer i appears in the j -tuple k (i 2 N and k 2 .N /j ).

144


Theorem 14. Let f; g 2 Am .K/ and suppose that jf j jgj. Then, for every n 2 N with 0 n m, the following estimate holds on K ı n Z.g/:

f mC2 g

.n/ C jgjmC1n :

Proof. Let I.z/ WD 1=z and h.z/ WD 1=g.z/. Then h D I ı g. Also, I .j / .z/ D j Š.1/j

1 zj C1 :

By Theorem 13, for 1 m, 1

0 h./ D .I ı g/./ D

X

B X .k/ C .I .j / ı g/ B Ck g C A @

j D1

k2M j jkjD

0 X

D

j Š.1/j

j D1

1 g j C1

1

B X .k/ C B Ck g C @ A: k2M j jkjD

We may assume that jjgjj1 1. Since the derivatives of g are bounded up to the order m, we conclude that on K ı n Z.g/ jh

./

ˇ ˇC1 ˇ1ˇ j C ˇˇ ˇˇ : g

By Leibniz’s formula .f

mC2

h/

.n/

! n X n .n`/ mC2 .`/ D .f / : h `

(9)

`D0

Next we have to estimate the derivatives of f mC2 . Let p.z/ D zmC2 . Using a second time Theorem 13, we obtain for 1 ` m, 1 0 ` X X C B .f mC2 /.`/ D .p ı f /.`/ D .p .j / ı f / @ Ck` f .k/ A : j D1

k2M j jkjD`

Since .zmC2 /.j / D .m C 2/ : : : .m C 2 j C 1/ zmC2j ;


145

we estimate as follows (note that jf j jgj 1): ` ˇ mC2 .`/ ˇ X ˇ.f / ˇ Cj jf jmC2j CQ jf jmC2` : j D1

In view of the assumption jf j jgj, equation (9) then yields ˇ ˇ !ˇ ˇ n ˇ f mC2 .n/ ˇ X ˇ 1 ˇn`C1 n ˇ ˇ ˇ ˇ jf jmC2` ˇ CQ ˇ ˇ ˇ g ` ˇg ˇ `D0

! n X n CQ jgjmC1n D jgjmC1n : ` jf jjgj

`D0

A similar proof applied to the mixed partial derivatives of f; g 2 C m .K/ yields an analogous result. Theorem 15. Let f; g 2 C m .K/ and suppose that jf j jgj. Then, for every n 2 N with 0 n m, the following estimate holds on K ı n Z.g/: Dn

f mC2 g

C jgjmC1n ;

where Dn D .@x/j1 .@y/j2 with j1 C j2 D n. Theorem 16. Let K C be a compact set. Then the following assertions hold: (a) If f; g 2 Am .K/ satisfy jf j jgj, then f mC2 2 IAm .K/ .g/. (b) If f; g 2 C m .K/ satisfy jf j jgj and if Z.g/ has no cluster points in K ı , then f mC2 2 IC m .K/ .g/. Proof. (a) Due to holomorphy, the quotient f mC2 =g is holomorphic at every isolated zero z0 of g in K ı , since m.f; z0 / m.g; z0 /, where m.f; z0 / denotes the multiplicity of the zero z0 . Let ( h.z/ D

f mC2 .z/ g.z/

if z 2 K n Z.g/

0

if z 2 Z.g/.

Since h 0 on Z.g/ı , the fact that @Z.g/ı @K implies that h is holomorphic on

K ı D.K n Z.g/ı /ı [ Z.g/ı : (* was shown in [5, p. 2219]). We may assume that jgj 1 on K. By Theorem 14,

146


.j / j f mC2 =g j jgj on K ı n Z.g/ for every 0 j m. Since f; g 2 A.K/m , the derivatives (a priori defined only on K ı ) are continuously extendable to K. We denote them by the usual symbol f .j / etc. Thus jh.j / j jgj on K n Z.g/ and so h.j / jKnZ.g/ admits a continuous extension to K. (b) Similar proof, since we assumed that the zeros of f and g are isolated.

5 The f N -Problem in A.K / Here we present some sufficient conditions on the generators fj that guarantee that P N jgj N j D1 jfj j implies that g 2 IA.K/ .f1 ; : : : ; fn / for some N 2 N. 1 ı Lemma 17. Let A D C 1 .K/or C.K/ \ C .K /. If g; fj 2 A satisfy jgj jfj, 4 2 where f D .f1 ; : : : ; fn /, then g =jfj jKnZ.jfj2 / admits an extension to an element in A whenever jfj has only isolated zeros in K ı . The power 4 is best possible (within N). In particular, g 4 2 IA .f1 ; : : : ; fn /.

Proof. To show that 4 is best possible, consider the function f .z/ D z and g.z/ D z. Then z3 =jzj2 D z2 =z is not differentiable at 0 (see the example in Theorem 12(b)). Now if D D @ or @, then4 on K ı n Z.jfj2 / D

g4 jfj2

4jfj2 g 3 Dg g 4 .f Df C Df f/ jfj4

g 2 g 4 f Df C Df f : D 4g Dg jfj jfj

D

(10) (11)

If jf.z/j D 0, then g.z/ D 0 and so the boundedness of the term g=jfj yields the g4 with value 0. Since the zeros of jfj are isolated continuous extensions of D jfj 2 ı within K , we get from the mean value theorem of the differential calculus, that both partial derivatives exist at those zeros and coincide with these extensions. Hence .g 4 =jfj2 /jKnZ.f/ admits the desired extension. By a similar proof we have 1 ı 2 Lemma 18. Let A D C 1 .K/or C.K/ \ C .K /. If g; fj 2 A satisfy jgj jfj, 7 2 where f D .f1 ; : : : ; fn /, then g =jfj jKnZ.jfj2 / admits an extension to an element in A whenever jfj has only isolated zeros in K ı . In particular, g 7 2 IA .f1 ; : : : ; fn /.

We don’t know whether the power 7 is optimal. Now we use again the convenient matricial notation from section 3.

4

Here a b means the scalar product of the row-vectors a and b.


147

Proposition 19.PSuppose that f D .f1 ; : : : ; fn / 2 A.K/n and that g 2 A.K/ satisfies jgj nj D1 jfj j. If there is a solution x in C@;1 .K/ to xft D g, then there exists u D .u1 ; : : : ; un / 2 A.K/n such that uft D g 5 whenever jfj has only isolated zeros in K ı and there exists v D .v1 ; : : : ; vn / 2 A.K/n such that vft D g 6 whenever f 2 A.K/n is arbitrary. Proof. We first suppose that Z.jfj/ does not admit a cluster point within K ı . As in Theorem 6, we consider on K n Z.jfj/ the matrix F D

t 1 @xt f @xt f : jfj2

Using the facts that x 2 C@;1 .K/n and g 4 = jfj2 2 C.K/ \ C 1 .K ı / we conclude from Lemma 17 that F g 4 extends to an antisymmetric matrix over C.K/\C 1 .K ı /. Thus, by Theorem 5, the system @H D F g 4 admits a matrix solution H over C.K/ \ C 1 .K ı /. Note that H can be chosen to be antisymmetric, too. Now let u D g 4 x fH: n Then u 2 C.K/n \ C 1 .K ı / . Moreover, on K ı n Z.jfj/, @u D 0 because t g4 @u D g 4 @x f @H D g 4 @x f f @x @xt f jfj2 t .f @xt / f .@g/ f 4 @.f x / f Dg Dg D g4 D 0: 2 2 jfj jfj jfj2 4

Since it is assumed that every point in Z.jfj/\K ı is an isolated point, the continuity of @u implies that @u D 0 on K ı . Hence u 2 A.K/n . By antisymmetry,5 fH ft D 0 and so uft D g 4 xft fH ft D g 4 g D g 5 : Now let f 2 A.K/n be arbitrary. If jfj is identically zero, then nothing is to prove. Hence we may assume that S WD K n Z.jfj/ı 6D ;. By the first case, there is u 2 A.S /n such that uft D g 5 on S . Since @.Z.jf j/ı / @K, we again have S ı [ Z.jf j/ı D K ı :

5

fH ft D .fH ft /t D fH t ft D fH ft

148


Hence the vector-valued function ( v.z/ D

g.z/u.z/

if z 2 S

0

if z 2 K n S

is well defined, continuous on K and each of its coordinates is holomorphic on K ı . It easily follows that vft D g 6 . According to [3], an ideal I in a uniform algebra A is said to have T the Forelli property (say I 2 F ), if there exists f 2 I with Z.f / D Z.I / WD h2I Z.h/. It is obvious that in C.K/ every finitely generated ideal has the Forelli-property; just consider the function f D

n X

jfj j D 2

j D1

n X

f j fj 2 IC.K/ .f1 ; : : : ; fn /:

j D1

It was shown in [3] that there exist finitely generated ideals in the disk-algebra that do not have this property. A natural question, therefore, is whether C@;1 .K/ has the Forelli-property. A sufficient condition for I D I.f1 ; : : : ; fn / to belong to F is that there exists h 2 I such that6 jhj

n X

jfj j2 :

j D1

Theorem 20. Let f D .f1 ; : : : ; fn / 2 A.K/n and suppose that jfj has only isolated zeros in K ı . We assume that there are hj 2 C@;1 .K/ such that n n ˇ X ˇX ˇ hj fj ˇ jfj j2 : j D1

j D1

Pn 12 Then for every g 2 A.K/ satisfying jgj 2 j D1 jfj j we have g IA.K/ .f1 ; : : : ; fn /. Pn Pn 2 2 Proof. Let h D j D1 hj fj . Then jgj n j D1 jfj j n jhj. By Proposi8 tion 12, there is k 2 C@;1 .K/ such that g D kh. Hence g8 D

n X

.khj /fj 2 IC@;1 .K/ .f1 ; : : : ; fn /:

j D1

p Since jgj n jfj implies that g 4 =jfj2 2 C.K/ \ C 1 .K ı / (Proposition 17), we obtain in a similar manner as in Proposition 19 that g 12 2 IA.K/ .f1 ; : : : ; fn / (just consider the data @H D F g 4 , x D .kh1 ; : : : ; khn /, u D g 4 x fH , and use the fact that xft D g 8 in order to get uft D g 12 ). 6

Instead of 2 we may of course take any power ˛ > 0.


149

6 The f N -Problem in C.K / and C 1 .K / Pn 2 Proposition 21. Let h; fj 2 C.K/ satisfy jhj 2 j D1 jfj j. Then h IC.K/ .f1 ; : : : ; fn /. Within N, the constant 2 is best possible. T Proof. Consider on K n nkD1 Z.fk / the functions hf j qj D Pn : 2 kD1 jfk j Then, by Cauchy-Schwarz, Pn kD1

jqj j Hence qj is bounded on K n and that

Tn kD1

jfk j

2

jfj2

n:

Z.fk /. It is then clear that gj WD hqj 2 C.K/

n X

gj fj D h2 :

j D1

By Theorem 12(a), the power n D 2 is best possible. Pn 1 Proposition 22. Let h; fj 2 C .K/ satisfy jhj j D1 jfj j. If we suppose that Tn ı 3 j D1 Z.fj / \ K is discrete, then h 2 IC 1 .K/ .f1 ; : : : ; fn /. Within N, the constant 3 is best possible. Proof. The example h.z/ T D z and f .z/ D z in Proposition 12 (b) shows that 3 is best possible. On K n nj D1 Z.fj /, let f j h3 qj D Pn : 2 kD1 jfk j We claim thatT qj has an extension to a function in C 1 .K/. Let f D .f1 ; : : : ; fn /. Since outside nj D1 Z.fj / jfjjh3 j jqj j D jfj2

jhj jhj2 C jfj2 ; jfj

we immediately see that qj can be continuously extended to K with value 0 on T Z.jfj/. Now, if D D @ or @, then on K ı n nj D1 Z.fj / D K ı n Z.jfj/

150


Dqj D

jfj2 .Df j /h3 C 3f j .Dh/h2 f j h3 .fDf C fDf/ jfj4

:

Using that the derivatives are continuous and that maxfjhj; jfj jg jfj, we obtain constants Cj such that jDqj j

jhj jfj

2

jhj jDf j j jhj C 3jfj jDhj jfj

2 C

jfj jhj3 jfj.jDfj C jDfj/ jfj4

C1 jfj C C2 jfj C jfj .jDfj C jDfj/ C3 jfj: Thus the partial derivatives admit a continuous extension to K. Since at an isolated zero of jfj the partial derivatives of qj exist by the mean value theorem in differential calculus, we are done.

7 Questions In this final section we would like to ask several questions and present a series of problems. P (1) Let f; f1 ; : : : ; fn 2 A.D/ satisfy jf j nj D1 jfj j. Is f 3 2 IA.D/ .f1 ; : : : ; fn /? (2) Is there a simple example of a triple .f; f1 ; f2 / in A.D/ such that jf j jf1 j C jf2 j, but for which f 2 … IA.D/ .f1 ; f2 /? (3) Is there a simple example of a function f , continuous on D, such that the Pompeiu-integral ZZ Pf .z/ D

D

f ./ d ./ z

is not continuously differentiable in D? Let M and S be two specific classes of functions on the unit disk. For example M and S coincide with Cb .D/, the set of all bounded, continuous and complex-valued 1 functions on D; Cb1 .D/ D Cb .D/ \ C 1 .D/; or Cbb .D/, the set of all C 1 -functions u on D for which u and ru are bounded. Give necessary and sufficient conditions on f 2 M such that the @-equation @u D f admits a solution u 2 S : (4) (5) (6) (7) (8)

f f f f f

2 Cb1 .D/, u 2 Cb1 .D/; 2 Cb1 .D/, u 2 C.D/ \ C 1 .D/; 1 2 Cb1 .D/, u 2 C.D/ \ Cbb .D/; 1 1 2 C .D/, u 2 C .D/; 2 C 1 .D/, u 2 Cb1 .D/;


(9) (10) (11) (12) (13) (14)

f f f f f f

151

2 C 1 .D/, u 2 C.D/ \ C 1 .D/; 1 2 C 1 .D/, u 2 C.D/ \ Cbb .D/; 1 2 C.D/, u 2 C .D/; 2 C.D/, u 2 Cb1 .D/; 2 C.D/, u 2 C.D/ \ C 1 .D/; 1 2 C.D/, u 2 C.D/ \ Cbb .D/.

References 1. M. Andersson, Topics in Complex Analysis, Springer, New York 1997. 2. J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. 3. R. Mortini, The Forelli problem concerning ideals in the disk algebra A.D/, Proc. Amer. Math. Soc. 95 (1985), 261–264. 4. R. Mortini, The Faà di Bruno formula revisited, Elem. Math. 68 (2013), 33–38. 5. R. Mortini, R. Rupp, The Bass stable rank for the real Banach algebra A.K/sy m , J. Funct. Anal. 261 (2011), 2214–2237. 6. R. Mortini, R. Rupp, A solution to the Bézout equation in A.K/ without Gelfand theory. Archiv Math. 99 (2012), 49–59. 7. R. Mortini, B. Wick, Simultaneous stabilization in AR .D/, Studia Math. 191 (2009), 223–235. 8. R. Narasimhan Complex variables in one variable, Birkhäuser, Boston, 1985. 9. W. Rudin Functional Analysis, Second Edition, MacGraw-Hill, 1991.

The Ring of Real-Valued Multivariate Polynomials: An Analyst’s Perspective Raymond Mortini and Rudolf Rupp

Abstract In this survey we determine an explicit set of generators of the maximal ideals in the ring RŒx1 ; : : : ; xn of polynomials in n variables with real coefficients and give an easy analytic proof of the Bass-Vasershtein theorem on the Bass stable rank of RŒx1 ; : : : ; xn . The ingredients of the proof stem from different publications by Coquand, Lombardi, Estes and Ohm. We conclude with a calculation of the topological stable rank of RŒx1 ; : : : ; xn , which seems to be unknown so far. Keywords Ring of real polynomials • Bass stable rank • Topological stable rank • Prime ideals • Krull dimension Subject Classifications: Primary 46E25; Secondary 13M10, 26C99

1 Introduction In his seminal paper [14, Theorem 8], that paved the way to all future investigations of the Bass stable rank for function algebras, L. Vasershtein deduced from a Theorem of H. Bass [2] (see below) that the Bass stable rank of the ring of real polynomials in n variables is n C 1. Since for an analyst Bass’ fundamental paper is very hard to understand it is desirable to develop an analytic proof of Bass’ important result that can easily be read. This was done in a paper by Estes and Ohm [4]. The whole depends on the determination of the Krull dimension of R. Mortini () Département de Mathématiques et Institut Élie Cartan de Lorraine, Université de Lorraine, UMR 7502,l Ile du Saulcy, 57045 Metz, France e-mail: [email protected] R. Rupp Fakultät Allgemeinwissenschaften, TH-Nürnberg, Kesslerplatz 12, 90489 Nürnberg, Germany e-mail: [email protected] R.G. Douglas et al. (eds.), The Corona Problem, Fields Institute Communications 72, DOI 10.1007/978-1-4939-1255-1__8, © Springer Science+Business Media New York 2014

153

154


RŒx1 ; : : : ; xn , the known proofs prior to 2005 were rather involved. But also here, a nice elementary proof had been developed around 2005 by Coquand and Lombardi [3]. Their short proof depends on the standard algebraic tool of “localization of rings”. We shall replace this by a direct construction of a chain of prime ideals of length n and obtain in this way an entirely analytic proof of the Bass-Vasershtein Theorem. The only tool used in the proof will now be Zorn’s Lemma. In our survey we present all these proofs so that it will be entirely self-contained; it will no longer be necessary to look up half a dozen papers in order to admire this nice result by Bass and Vasershtein. We conclude the paper with a result we could not trace in the literature: the determination of the topological stable rank of RŒx1 ; : : : ; xn : every .n C 1/-tuple of real-valued polynomials can be uniformly approximated on Rn be invertible .n C 1/-tuples in RŒx1 ; : : : ; xn . This survey forms part of an ongoing textbook project on stable ranks of function algebras, due to be finished only in a couple of years from now (now D 2013). Therefore we decided to make this chapter already available to the mathematical community (mainly for readers of this Proceedings and for master students interested in function theory and function algebras).

2 The Maximal Ideals of RŒx1 ; : : : ; xn Associated with RŒx1 ; : : : ; xn is the following algebra of real-symmetric polynomials: Csym Œz1 ; : : : ; zn n o D f 2 CŒz1 ; : : : ; zn W f .z1 ; : : : ; zn / D f .z1 ; : : : ; zn / 8.z1 ; : : : ; zn / 2 Cn : For shortness we write z for the n-tuples .z1 ; : : : ; zn / and z for .z1 ; : : : ; zn /. Lemma 1. Csym Œz1 ; : : : ; zn is a real algebra of complex-valued polynomials that is real-isomorphic to RŒx1 ; : : : ; xn . Proof. It is easy to see that Csym Œz is a real algebra. Let W Csym Œz ! RŒx be the restriction map p 7! pjRn . Note that is well defined, since the coefficients of a polynomial in Csym Œz are real; in fact if p.z/ D

X

an zn 2 Csym Œz ;

n2I

then p.z/ D

X n2I

an zn D

X n2I

an zn :

The Ring of Real-Valued Multivariate Polynomials: An Analyst’s Perspective

155

The uniqueness of the coefficients implies that an D an . Hence an 2 R. The rest is clear. For a 2 Cn let Ma WD fp 2 CŒz1 ; : : : ; zn W p.a/ D 0g: By Hilbert’s Nullstellensatz (see for instance [9] or [8]) an ideal in CŒz1 ; : : : ; zn is maximal if and only if it has the form Ma for some a 2 Cn . This will be used in the following to determine the class of maximal ideals in Csym Œz1 ; : : : ; zn . Theorem 2. The class of maximal ideals of Csym Œz1 ; : : : ; zn coincides with the class of ideals of the form Sa WD Ma \ Ma \ Csym Œz1 ; : : : ; zn ; where a 2 Cn . The set fa; ag is uniquely determined for a given maximal ideal. Proof. We first note that Ma \ Ma \ Csym Œz D Ma \ Csym Œz , because for every polynomial p in Csym Œz it holds that p.a/ D 0 if and only if p.a/ D 0. Next we show that the ideals Sa are maximal. So suppose that f 2 Csym Œz does not vanish at a. Then f f .a/ f f .a/ D f 2 2Re f .a/ f C jf .a/j2 2 Sa and f f .a/ f f .a/ f .f .a/ C f .a// f : 1D 2 jf .a/j jf .a/j2

Hence the ideal, ICsym Œz .Sa ; f /, generated by Sa and f is the whole algebra and so Sa is maximal. We note that in the case where f .a/ is real, we simply could argue as follows, since the constant functions z 7! f .a/ and z 7! 1=f .a/ then belong to Csym Œz : 1D

f f f .a/ C 2 ICsym Œz .Sa ; f /: f .a/ f .a/

It remains to show that every maximal ideal M in Csym Œz coincides with Sa for some a 2 Cn . Suppose, to the contrary, that M is not contained in any ideal of the form Sa . Hence, for every a 2 Cn , there is pa 2 M such that pa .a/ 6D 0. By Hilbert’s Nullstellensatz, the ideal generated by the set S D fpa W a 2 Cn g in CŒz coincides with CŒz . Hence there are qj 2 CŒz and finitely many aj 2 Cn , .j D 1; : : : ; N /, such that

156

R. Mortini and R. Rupp N X

qj paj D 1:

j D1

Now, by taking complex conjugates, and using the fact that paj 2 M Csym Œz , we get 1D

N X

qj .z/ paj .z/ D

j D1

X

qj .z/ paj .z/:

j D1

Hence, with qj .z/ D

1 qj .z/ C qj .z/ ; 2

we conclude that N X

qj paj D 1:

j D1

Since qj 2 Csym Œz and paj 2 M we obtain the contradiction that 1 2 M . Thus M Sa for some a 2 Cn . The maximality of M now implies that M D Sa . Finally we show the uniqueness of fa; ag. So suppose that b 62 fa; ag. Case 1 There is an index i0 such that bi0 … fai0 ; ai0 g. Then the polynomial p, given by p.z1 ; : : : ; zn / D .zi0 ai0 /.zi0 ai0 / belongs to Csym Œz1 ; : : : ; zn , vanishes at a, but not at b (if ai0 2 R, then it suffices to take p.z1 ; : : : ; zn / D zi0 ai0 ). Case 2 There are two indices i0 and i1 such that ai … R, bi0 D ai0 and bi1 D ai1 . Then the polynomial q given by q.z1 ; : : : ; zn / D .zi0 ai0 / C .zi1 ai1 / .zi0 ai0 / C .zi1 ai1 / belongs to Csym Œz1 ; : : : ; zn , vanishes at a, but not at b. Hence, in both cases, p 2 Sa n Sb . There are no other cases left. Pk Theorem 3. The Bézout equation j D1 qj pj D 1 admits a solution in the ring R D Csym Œz1 ; : : : ; zn or RŒx1 ; : : : ; xn if and only if the polynomials pj do not have a common zero in Cn .


157

Proof. By the identity P theorem for holomorphic functions of several complex variables, the condition kj D1 qj pj D 1 on Rn implies that the same equality holds on Cn . Hence the given polynomials pj do not have a common zero in Cn . Conversely, if the pj do not have a common zero in Cn then, by Theorem 2, the ideal generated by the pj in R cannot be a proper ideal. Hence there are qj 2 R P such that kj D1 qj pj D 1. We shall now determine an explicit class of generators for the maximal ideals in RŒx1 ; : : : ; xn . Recall that by Hilbert’s Nullstellensatz the maximal ideals in CŒz1 ; : : : ; zn are generated by n polynomials of the form z1 a1 ; : : : ; zn an , where a WD .a1 ; : : : ; an / 2 Cn . The situation for the real algebra RŒx1 ; : : : ; xn is quite different. Here are some examples that will reflect the general situation dealt with below. We identify RŒx1 ; : : : ; xn with Csym Œz1 ; : : : ; zn . The proof of the assertions is left as an exercise to the reader. Example 4. (1) Let 2 C n R and rj 2 R, j D 1; 2; : : : ; n 1. Then the ideal generated by xn2 .2 Re / xn C j j2 and xj rj , .j D 1; : : : ; n 1/, is maximal in RŒx1 ; : : : ; xn . It corresponds to the ideal S.r1 ;:::;rn1 ; / . (2) The ideal IRŒx;y .1 C x 2 ; 1 C y 2 / generated by 1 C x 2 and 1 C y 2 is not maximal. (4) The ideal M WD IRŒx;y .1Cx 2 ; 1Cy 2 ; 1Cxy; xy/ is maximal and corresponds to S.i;i/ . (5) The following representations hold: M D IRŒx;y .1 C x 2 ; 1 C y 2 ; x y/ D IRŒx;y .1 C x 2 ; 1 C y 2 ; 1 C xy/ D IRŒx;y .1 C xy; x y/: Theorem 5. Modulo a re-enumeration of the indices, the maximal ideals M of R WD RŒx1 ; : : : ; xn are generated by polynomials of the form pj WD xj rj ; .j D 1; : : : ; k/; 2 .2Re akCj / xkCj C jakCj j2 ; .j D 1; : : : ; m/ pkCj WD xkCj

.rj 2 R, aj 2 C n R, k C m D n/, and 2nk 2 multilinear polynomials qj in RŒxkC1 ; : : : ; xn vanishing at akC1 ; : : : ; an . More precisely, we have M D

n X j D1

pj .xj /R C

2nk X2 j D1

qj .xkC1 ; : : : ; xn / R:

158


Proof. Since RŒx1 ; : : : ; xn is isomorphic to S WD Csym Œz1 ; : : : ; zn , it suffices to show that every maximal ideal Sa in S is generated by polynomials of the desired type (Theorem 2). Fix a 2 Cn . We may assume that a D .r1 ; : : : ; rk ; akC1 ; : : : ; akCm /; with k C m D n, where rj 2 R and akC1 ; : : : ; akCm 2 C n R. Note that k or m may be 0. Let f 2 Sa and z D .z1 ; : : : ; zn /. By the Euclidean division procedure f .z/ D

n X

pj .z/qj .z/ C r.zkC1 ; : : : ; zkCm /;

j D1

where degzj r < 2 for k C 1 j k C m D n. Hence r is a multilinear polynomial of the form X j jm 1 cj zkC1 zkCm ; r.zkC1 ; : : : ; zkCm / D j

j D .j1 ; : : : ; jm /, j` 2 f0; 1g, cj 2 R. Moreover, r.akC1 ; : : : ; akCm / D f .a/ D 0. Now the real vector-space, V , of all multilinear real-symmetric polynomials in m variables has the algebraic dimension 2m . Hence, the subspace V of all p 2 V with p.akC1 ; : : : ; akCm / D p.akC1 ; : : : ; akCm / D 0 has dimension 2m 2. Let fq1 ; : : : ; q2m 2 g be a basis of V . Then f 2

n X

.pj R/ C

j D1

m 2 2X

.qj R/:

j D1

We shall now unveil an explicit basis for V whenever aj D i for every j . Lemma 6. Let i D .i; : : : ; i / 2 Cm . Then a (vector-space) basis of 8 9 < = X j1 jm V D f .z1 ; : : : ; zm / D cj zkC1 zkCm ; j` 2 f0; 1g; cj 2 R; f .i/ D 0 : ; j1 ;:::;jm

is given by x1 xj 1 < j m 1 C xj1 xj2 1 j1 < j2 m x1 C xj1 xj2 xj3 1 j1 < j2 < j3 m


1

4 Y

159

xj` 1 j1 < < j4 m

`D1

x1

5 Y

xj` 1 j1 < < j5 m

`D1

:::::: :::::: The last element has exactly one of the following forms: 8 Q ˆ x1 m ˆ j D1 xj ˆ ˆ < 1 C Qm x j D1 j Qm ˆ x1 C j D1 xj ˆ ˆ ˆ Q : 1 m j D1 xj

if m 1 mod 4 if m 2 mod 4 if m 3 mod 4 if m 0 mod 4

Proof. All the polynomials above vanish at i (this is the reason for their cyclic behaviour mod 4). Moreover, they are linear independent and there are exactly 2m 2 of them. Note that the second summand has the form "m ; x1"1 x2"2 xm

the monomials 1 D

Qm

j D1

xj0 and x1 D x1

"j 2 f0; 1g; Qm

j D2

xj0 being excluded.

We conclude this section with the final form of the generators of the maximal ideals in RŒx1 ; : : : ; xn . Theorem 7. Let m C k D n, m 2, and a W D.i; : : : ; i; rmC1 ; : : : ; rmCk / 2 Cm Rk . The maximal ideal Sa of RŒx1 ; : : : ; xn is generated by the 2m 2 multilinear polynomials in Lemma 6 and the polynomials pmCj WD xmCj rmCj ; .j D 1; : : : ; k/: Proof. Using Theorem 5, it suffices to show that the quadratic polynomials 1 C xj2 , j D 1; : : : ; m, belong to the ideal generated by the 2m 2 multilinear polynomials in Lemma 6. This is clear, however, in view of the following relations: 1 C xj2 D .x1 xj /xj C .1 C x1 xj / for j D 2; : : : ; m; 1 C x12 D .x1 xm /x1 C .1 C x1 xm /:

The general case of an arbitrary maximal ideal Sa is easily deduced by using the transformation

160


.z1 ; : : : ; zm / D

z1 ˛1 zm ˛m ; :::; ˇ1 ˇm

of Cm onto Cm , whenever a D .˛1 C iˇ1 ; : : : ; ˛m C iˇm ; rmC1 ; : : : ; rmCk / 2 Cm Rk Cn ; with ˇj 6D 0 for j D 1; : : : ; m. Using more algebraic methods, it can be shown, that every maximal ideal in R WD F Œx1 ; : : : ; xn is generated by n elements (see [8, p. 20]), where F is a field. Finally, let us mention that R is a Noetherian ring (this means that every ideal in R is finitely generated; this is Hilbert’s basis theorem).

3 The Bass Stable Rank of RŒx1 ; : : : ; xn Definition 8. Let R be a commutative unital ring with identity element 1. We assume that 1 6D 0, that is R is not the trivial ring f0g. (1) If aj 2 R, .j D 1; : : : ; n/, then IR .a1 ; : : : ; an / WD

8 n <X :

xj aj W xj 2 R

j D1

9 = ;

is the ideal generated by the aj in R. (2) An n-tuple .f1 ; : : : ; fn / 2 Rn is said to be invertible (or P unimodular), if there exists .x1 ; : : : ; xn / 2 Rn such that the Bézout equation nj D1 xj fj D 1 is satisfied. The set of all invertible n-tuples is denoted by Un .R/. Note that U1 .R/ D R1 . An .nC1/-tuple .f1 ; : : : ; fn ; g/ 2 UnC1 .R/ is called reducible if there exists .a1 ; : : : ; an / 2 Rn such that .f1 C a1 g; : : : ; fn C an g/ 2 Un .R/. (3) The Bass stable rank of R, denoted by bsr R, is the smallest integer n such that every element in UnC1 .R/ is reducible. If no such n exists, then bsr R D 1. Note that if bsr R D n, n < 1, and m n, then every invertible .m C 1/-tuple .f; g/ 2 RmC1 is reducible [14, Theorem 1]. Let N D f0; 1; 2; : : : g. For the sequel, we make the convention that the symbol denotes strict inclusion. Definition 9. Let R be a commutative unital ring, R 6D f0g. (1) A chain C D fI0 ; I1 ; : : : ; In g of ideals in R is said to have length n (n 2 N), if I0 I1 In ;


161

the inclusions being strict. We also call C an n-chain. Note the length of a chain C counts the number of strict inclusions between the members of C and not the cardinal of C. (2) The Krull dimension, Krd R, of R is defined to be the supremum of the lengths of all increasing chains P0 Pn of prime ideals in R. Note that Krd R 2 f0; 1; : : : ; 1g. Here is now Vasershtein’s theorem (and its proof sketched in [14]). Theorem 10 (Vasershtein). bsr RŒx1 ; : : : ; xn D n C 1: Proof. We first show that bsr RŒx1 ; : : : ; xn nC1. Consider the invertible .nC1/tuple 0 @x 1 ; : : : ; x n ; 1

n X

1 xj2 A

j D1

in RŒx1 ; : : : ; xn . This tuple cannot be reducible in RŒx1 ; : : : ; xn C.Rn ; R/, since otherwise the n-tuple .x1 ; : : : ; xn /, restricted to the unit sphere @B in Rn , would have a zero-free extension e to the unit ball B, where e is given by 0 @x1 C u1 1

n X

! xj2 ; : : : ; xn C un 1

j D1

n X

!1 xj2 A

j D1

for some uj 2 RŒx1 ; : : : ; xn . This means that e does not take the value .0; : : : ; 0/ on B. This contradicts Brouwer’s result that the identity map .x1 ; : : : ; xn / W @B ! @B defined on the boundary of the closed unit ball B in Rn does not admit a zero-free continuous extension to B. Next we prove that bsr RŒx1 ; : : : ; xn n C 1. This follows from a combination of Theorem 27 below, telling us that the Bass stable rank of a Noetherian ring with Krull dimension n is less than or equal to n C 1, and Theorem 20, according to which the Krull dimension of RŒx1 ; : : : ; xn is n. In the next section we shall now present analytic proofs of both Theorems mentioned above. They were given by Estes and Ohm for Theorem 27 ([4]) and Coquand and Lombardi for Theorem 20 ([3]).

162


Remark 11. Concerning the polynomial ring CŒz1 ; : : : ; zn , to the best of our knowledge, the exact value of the Bass stable rank for CŒz1 ; : : : ; zn is not yet known. Only estimates are available: bsr CŒz1 ; : : : ; zn n C 1; (follows as in the proof for RŒx1 ; : : : ; xn because the Krull dimension of CŒz1 ; : : : ; zn is also n by Theorem 20) and bsr CŒz1 ; : : : ; zn

jnk 2

C1

(see [5, 6]).

4 The Krull Dimension of RŒx1 ; : : : ; xn We begin with our own proof of the Coquand-Lombardi result concerning an elementary characterization of the Krull dimension of a commutative unital ring. We avoid the algebraic tool of considering localized rings and explicitly construct (with the help of Zorn’s Lemma) chains of prime ideals having the correct length. Our tool will be the following standard result, which we would like to present, too. Lemma 12 (Krull). Let R be a commutative unital ring, R 6D f0g, and S a multiplicatively closed set in R with 1 2 S . Suppose that I is an ideal in R with I \ S D ;. Then there exists a prime ideal P with I P such that P \ S D ;. Proof. Let V be the set of all ideals J with I J and J \ S D ;. Then V 6D ; because I 2 V , and S V is partially ordered by set inclusion. If C is any increasing chain in V then J 2C J obviously is an ideal belonging to V . Hence, by Zorn’s Lemma, V admits a maximal element P . Since 1 2 S and S \ P D ;, we obtain that P R, the inclusion being strict. Moreover, I P . We claim that P is a prime ideal. For f; g 2 R, let fg 2 P and suppose that neither f nor g belongs to P . Since P is maximal in V , the ideals P C fR and P C gR meet S . Hence there exist s; s 0 2 S , p; p 0 2 P , and r; r 0 2 R such that s D p C rf and s 0 D p 0 C r 0 g. Multiplying both terms, we obtain ss 0 D pp 0 C .r 0 g/p C .rf /p 0 C rr 0 .fg/ 2 P: Since S is multiplicatively closed, ss 0 2 S . Hence P \ S 6D ;; a contradiction. We conclude that P is prime.


163

For a; x 2 R and n 2 N, let La;n;x .y/ D an .y C ax/. If aj ; xj 2 R and nj 2 N are given, then we abbreviate Lj .y/ WD Laj ;nj ;xj .y/. Lemma 13. Let Q0 and Q1 be ideals in a commutative unital ring R with Q0 Q1 and let a 2 Q1 n Q0 . Suppose that Q0 is prime and that for r; x 2 R and n 2 N we have an .r C ax/ 2 Q0 : Then r 2 Q1 . Proof. Because a … Q0 , the primeness of Q0 implies that r C ax 2 Q0 Q1 . Since a 2 Q1 , we conclude that r 2 Q1 . Theorem 14 (Coquand-Lombardi). Let R be a commutative unital ring, R 6D f0g. For N 2 N, the following assertions are equivalent: (1) The Krull dimension of R is at most N . (2) For all .a0 ; : : : ; aN / 2 RN C1 there exists .x0 ; : : : ; xN / 2 RN C1 and .n0 ; : : : ; nN / 2 NN C1 such that a0n0

a1n1

nN aN .1

C aN xN / C

! C a0 x0

D 0;

(1)

in other words L0 ı ı LN .1/ D 0:

(2)

Proof. We show the contraposition of the assertion. :.1/ H) :.2/ Assume that the Krull dimension of R is at least N C 1. Then R admits a strictly increasing (N C 1)-chain P0 PN PN C1 of prime ideals Pj . Choose aj 2 Pj C1 n Pj , j D 0; : : : ; N . If we suppose, contrariwise, that (2) holds, then a0n0 L1 ı ı LN .1/ C a0 x0 D 0 2 P0 : Hence, by Lemma 13, L1 ı ı LN .1/ 2 P1 . Now L1 ı ı LN .1/ D a1n1 L2 ı ı LN .1/ C a1 x1 2 P1 : Hence, by Lemma 13, L2 ı ı LN .1/ 2 P2 . Continuing in this way, we deduce that

164

R. Mortini and R. Rupp nN aN .1 C aN xN / D LN .1/ 2 PN :

Hence, by Lemma 13, 1 2 PN C1 ; a contradiction. :.2/ H) :.1/ Suppose that there exists a D .a0 ; : : : ; aN / 2 RN C1 such that for all x 2 RN C1 and n 2 NN C1 one has L0 ı ı LN .1/ 6D 0: For j D 0; : : : ; N , let o n Sj D Lj ı ı LN .1/ W x 2 RN j C1 ; n 2 NN j C1 : Then SN SN 1 S0 , or, what is the same, S0c S1c SNc where S c denotes the complement of S . Note that f1; aj ; : : : ; aN g Sj . We claim that Sj is multiplicatively closed. This follows by an inductive argument on N j . If j D 0, then nN SN D faN .1 C aN xN / W xN 2 R; nN 2 Ng

is easily seen to be multiplicatively closed. If for some j , SN j is multiplicatively closed, then we use that n .j C1/ L LN .j C1/ ı LN j ı ı LN .1/ D aNN.j ı ı L .1/ C a x N j N N .j C1/ N .j C1/ C1/ and observe that an .s C ax/ am .s 0 C ax 0 / D anCm .ss 0 C ax 00 / where x 00 D sx 0 C xs 0 C axx 0 . Step 1 Looking at the zero ideal I WD f0g, and noticing that by assumption 0 … S0 , Krull’s Lemma 12 tells us that there exists a prime ideal P0 with P0 \ S0 D ;, or in other words, P0 S0c . Now a0 2 S0 implies that a0 … P0 . We claim that P0 IR .P0 ; a0 / S1c :

(3)

In fact, if the second inclusion does not hold, then there is p0 2 P0 and x 2 R such that p0 C xa0 D L1 ı ı LN .1/ 2 S1 S0 :


165

Hence p0 D L1 ı ı LN .1/ xa0 D a00 L1 ı ı LN .1/ xa0 2 S0 : Thus p0 2 P0 \ S0 ; a contradiction to the choice of P0 . Hence the inclusions (3) hold. Step 2 Now we apply Krull’s Lemma again to get a prime ideal P1 with P0 IR .P0 ; a0 / P1 S1c : Observe that a1 … P1 because a1 2 S1 . We claim that P1 IR .P1 ; a1 / S2c :

(4)

In fact, if the second inclusion does not hold, then there is p1 2 P1 and x 2 R such that p1 C xa1 D L2 ı ı LN .1/ 2 S2 S1 : Hence p1 D L2 ı ı LN .1/ xa1 D a10 L2 ı ı LN .1/ xa1 2 S1 : Thus p1 2 P1 \ S1 ; a contradiction to the choice of P1 . Hence the inclusions (4) hold. Step N Continuing in this way, we get a chain of prime ideals Pj , j D 0; : : : ; N , with P0 P1 PN SNc : Observe that aN … PN because aN 2 SN . Hence PN is a proper ideal. Therefore, PN is contained in a maximal ideal M . We claim that M \ SN 6D ;. In fact, if aN 2 M , then we are done. If aN … M , then IR .aN ; M / D R. In other words, there is x 2 R and m 2 M such that aN x C m D 1; that is 1 C aN x 2 M . By the definition of SN , 1 C aN x 2 SN . Hence 1 C aN x 2 M \ SN . Thus M \ SN 6D ;. Hence PN M , the inclusion being strict, and so we have found a chain of prime ideals of length N C 1: P0 P1 PN M:

166


We conclude that the Krull dimension of R is at least N C 1.

The proof that the Krull dimension of RŒx1 ; : : : ; xn is n now works as in Coquand and Lombardi’s paper: Proposition 15 (Coquand-Lombardi). Let F be a field and R 6D f0g a commutative unital algebra over F . If any .nC1/-tuple .f0 ; : : : ; fn / 2 RnC1 is algebraically dependent over F , that is, if there is a non-zero polynomial Q 2 F Œy0 ; : : : ; yn such that Q.f0 ; : : : ; fn / D 0, then the Krull dimension of R is at most n. Proof. Let Q.f0 ; : : : ; fn / D 0 for some non-zero polynomial Q 2 F Œy0 ; : : : ; yn . We assume that the monomials are ordered lexicographically with respect to the powers .i0 ; i1 ; : : : ; in / 2 NnC1 . This means that .i0 ; i1 ; : : : ; in / .j0 ; j1 ; : : : ; jn / if either i0 < j0 or if there is m 2 f0; : : : ; n 1g such that i D j for all with 0 m and imC1 < jmC1 . Let ai0 ;:::;in f0i0 f1i1 : : : fnin be the “first” monomial appearing in the relation above (here the coefficient ai0 ;:::;in belongs to F and .i0 ; : : : ; in / 2 NnC1 ). Without loss of generality we may assume that the coefficient of this monomial is 1. Then Q.f0 ; : : : ; fn / can be written as in1 in in1 1Cin 1Cin1 fn C f0i0 : : : fn1 fn Rn C f0i0 : : : fn1 Rn1 C : : : Q D f0i0 : : : fn1

C f0i0 f11Ci1 R1 C f01Ci0 R0 where Rj belongs to F Œfj ; fj C1 ; : : : ; fn , j D 0; 1; : : : ; n. Hence Q has been written in the form given by equation 1 (with aj WD fj and xj WD Rj ), that is f0i0

f1i1

fnin .1

C fn Rn / C

! C f0 R0

D 0:

We conclude from Theorem 14, that the Krull dimension of R is at most n.

In order to show that RŒx1 ; : : : ; xn satisfies the assumptions of Proposition 15 and to deduce its Krull dimension, we need some additional information. Proposition 16. Let R D F Œx1 ; : : : ; xn . Then the ideals IR .x1 /, IR .x1 ; x2 /, . . . , IR .x1 ; : : : ; xj / with 1 j n are prime ideals. Proof. We may assume that 1 j < n, since the ideals IR .x1 ; : : : ; xn / are maximal, hence prime. First we observe that F Œx1 ; : : : ; xn D F Œx Œx1 ; : : : ; xj , the polynomial ring with indeterminates x1 ; : : : ; xj and coefficients from the ring F Œx , where x D .xj C1 ; : : : ; xn /. Then every f 2 R can uniquely be written as f .x1 ; : : : ; xn / D where a`1 ;:::;`j 2 F Œx .

X

`

a`1 ;:::`j x1`1 : : : xj j ;


167

Let us now consider the surjective ring-homomorphism 8 ˆ A.L/ for some L (depending on n). In fact, looking upon B.L/ and A.L/ as polynomials in L, we have that deg B D n C 1 and deg A D n. Thus, for large L, we obtain that B.L/ > A.L/. Thus the cardinal of set S is strictly bigger than the dimension of the vector space V it belongs to. Hence S is a linear dependent set in V . In other words, there is a non-trivial linear combination of the elements from S that is identically zero. This implies that there is a non-zero polynomial P 2 F Œy1 ; : : : ; ynC1 of degree at most L such that P .p1 ; : : : ; pnC1 / D 0.


169

Theorem 20. If F is a field then the Krull dimension of F Œx1 ; : : : ; xn is n. Proof. By Perron’s Theorem 19, R WD F Œx1 ; : : : ; xn satisfies the assumption of Proposition 15. Hence the Krull dimension of R is less than or equal to n. By Proposition 16, we have a chain of prime ideals f0g IR .x1 / IR .x1 ; x2 / IR .x1 ; : : : ; xn /: Since f0g is a prime ideal too, this chain has length n. Thus the Krull dimension of R is n.

5 Anderson’s Approach to Noether’s Minimal Prime Theorem To prove Bass’ Theorem along the lines developed by Estes and Ohm [4], we need to collect in the following classical Theorem by E. Noether some information on the abundance of minimal prime ideals in Noetherian rings. We present a recent proof developed by D. Anderson [1]. Theorem 21. Let R be a Noetherian ring. Then the system, Pmi n , of prime ideals containing a given proper ideal I R and that are minimal (with respect to set inclusion) is a non-empty finite set. Proof. Let PI be the set of all prime ideals containing I . Then PI 6D ;, because there exists (using Zorn’s Lemma) a maximal ideal M containing I . Obviously M is prime. A second use of Zorn’s Lemma shows that PI also admits minimal elements. Hence Pmi n 6D ;.2 Next we consider the set J of all ideals of the form P1 ˇ ˇ Pn WD

m nX

f1;j : : : fn;j W fk;j 2 Pk ; m 2 N ;

j D1

where Pk 2 Pmi n , n 2 N. Since R is a Noetherian ring, every ideal in J is finitely generated. Case 1. If for some J WD P1 ˇ ˇ Pn0 2 J we have J I , then P1 Pn0 I P for every P 2 Pmi n . Hence, the primeness of P implies that there is i0 2 f1; : : : ; n0 g depending on P , such that Pi0 P (for if this is not the case, there exists for every j 2 f1; : : : ; n0 g an element fj 2 Pj n P with f1 fn 2 I P , contradicting the primeness of P ). Since P is minimal, P D Pi0 . Hence Pmi n D fP1 ; : : : ; Pn0 g and we are done. For later purposes we note that, by the same reason, if I and P are ideals, P prime and I P , then there exist minimal prime ideals Pmi n with I Pmi n P . 2

170


Case 2. Let us suppose that J 6 I for every J 2 J. The aim is to show that this case does not occur. Consider the set L WD fL R; L ideal, I L, J 6 L for each J 2 Jg. Then L 6D ; because, by assumption, L WD I 2 L . Moreover all ideals in L are proper, because J R for all J 2 J. With respect to set inclusion, L is partially ordered. We claim that L admits a maximal element M and M is automatically prime:

(5)

Suppose for the moment that this has been verified. Then, using Zorn’s Lemma, there exists a minimal prime ideal P over I (hence P 2 J) with I P M . This is a contradiction, though, to the fact that M 2 L . We conclude that this second case cannot occur. Hence, in view of the first case, Pmi n is finite. Let us verify the two assertions in (5). ToS this end, let fL W 2 g be an increasing chain in L . Let us show that L WD L 2 L . (i) I L is obviously satisfied and L is an ideal because the chain is increasing. (ii) Let J 2 J. Note that J is finitely generated, say J D IR .f1 ; : : : ; fd /, and that J 6 S L for any . If we suppose (in view of achieving a contradiction) that J 2 L D L, then fj 2 Lj for some j 2 , .j D 1; : : : ; d /. The monotonicity of the chain implies that there exists 0 such that fj 2 L0 for all j D 1; : : : ; d . Thus J L0 , a contradiction to the hypothesis that L0 2 L . Thus we have shown that L is an inductive set and so, by Zorn’s Lemma, L admits a maximal element M . In particular, I M . We claim that M is prime. To see this, we first observe that M is proper since otherwise J M D R for every J 2 J. Now let f; g 2 R with fg 2 M . Suppose, to the contrary, that f … M and g … M . Since M is a maximal element in L , and I M , there is Jf 2 J and Jg 2 J such that Jf IR Œf; M and Jg IR .g; M /: By the definition of J, there exists Pi and Pk in Pmi n with I Pi Jf and I Pk Jg . We claim that Pi Pk IR .fg; M / M: In fact, if pi 2 Pi and pk 2 Pk , then there are xi ; xk 2 R and mi ; mk 2 M such that pi pk D .xi f C mi /.xk g C mk / D xi xk .fg/ C mi .xk g/ C mk .xi f / C mi mk 2 M:


171

Because M is an ideal, Pi ˇ Pk M . Thus we have found an element in J that is contained in M . Since M 2 L , this is a contradiction. This proves that M is prime.

6 The Estes-Ohm Approach Here we present the approach to Bass’ Theorem given by Estes and Ohm [4]. Lemma 22. Let R be a commutative ring and I0 ; I1 ; I2 three ideals with I0 I1 [ I2 . Then I0 I1 or I0 I2 . Proof. Suppose that neither I0 I1 nor I0 I2 . Then there are a1 2 I0 n I2 I1 and a2 2 I0 nI1 I2 . Since I0 is an ideal, s WD a1 Ca2 2 I0 I1 [I2 . Without loss of generality we may assume that s 2 I1 . Then a2 D s a1 2 I1 ; a contradiction. We conclude that I0 I1 or I0 I2 . Remark 23. The assertion above does not hold (in general) for unions of three (or more) ideals. In fact, let R be a finite ring such that not all of its maximal ideals are principal. S Let I R be a non-principal maximal ideal. Then N D card I 4 and I D N j D1 Rxj . But of course, Ij WD Rxj does not contain I . A specific example is, for instance, the quotient ring R D Z2 Œx; y =M where M is the ideal generated by x 2 , xy and y 2 . When denoting the equivalence class of u 2 Z2 Œx; y by uQ , we have Q 1; Q x; R D f0; Q y; Q xQ C y; Q 1Q C xQ C yg; Q Q x; Q y/, Q which coincides with the set f0; Q y; Q xQ C yg. Q and as I we may take I D IR .x; If one stipulates, however, that the ideals Ij are prime, then one obtains the following well-known result. For the readers’ convenience we present its proof, too. Lemma 24. Let R be a commutative ring and I; P1 ; : : : ; Pm ideals in R such that each Pj is prime and m [

I

Pj :

j D1

Then I Pj0 for some j0 2 f1; : : : ; mg. Proof. If m D 1, then nothing has to be shown. If m D 2, then the assertion holds by Lemma 22. So we may assume that m 3. For j 2 f1; : : : ; mg, let Qj D

[ k6Dj

Pk :

172


We claim that there is j such that I Qj . Suppose, to the contrary, that for every j , I 6 Qj . Then we may choose for j 2 f1; : : : ; mg elements aj 2 I n Qj . Note that I n Qj Pj . Let Y ak : bj D k6Dj

S Then bj 2 I and b1 C C bm 2 I m kD1 Pk . Hence there is j 2 f1; : : : ; mg such that b C C b 2 P . Since b 2 Pj for every k 6D j , we deduce that 1 m j k P b 2 P . Therefore b 2 P . Since P k j j j j is prime, there is k 6D j such that k6Dj ak 2 Pj Qk . This is a contradiction to the choice of the elements a1 ; : : : ; am . We S conclude that I k6Dj Pk for some j . Now we proceed by backwards induction to reduce the number of prime ideals up to the case m D 2. That case, though, is handled by Lemma 22. Lemma 25. Let Pj be prime ideals in a commutative unital ring R with Pj 6 Pk for j 6D k, j; k 2 f1; : : : ; mg. Let a; r 2 R. Then there exists b 2 R such that for all j 2 f1; : : : ; mg: if a … Pj then r C a b … Pj :

(6)

S T any element not in m Proof. If r 2 m j D1 Pj , then we may choose for b j D1 Pj . If Qm Sm r … j D1 Pj , then we take bj 2 Pj and let b D j D1 bj . In the remaining cases, S T P . modulo a re-enumeration, we may assume that r 2 sj D1 Pj but r … m Ssj DsC1 j j D 6 k implies that P 6 P By Lemma 24, the hypothesis Pj 6 Pk for j kD1 k for S each j 2 fs C 1; : : : ; mg. Let bj 2 Pj n skD1 Pk , j s C 1, and let b D bsC1 bm : Tm

Then b 2 j DsC1 Pj . But b … Pk for 1 k s, because otherwise the primeness of Pk implies that one of the factors bj with s C 1 j m belongs to Pk , a contradiction to the choice of bj . Fix j0 2 f1; : : : ; mg and assume that a … Pj0 . We claim that r C ab … Pj0 . In fact, assuming the contrary, let u WD r C ab 2 Pj0 . If 1 j0 s, then r 2 Pj0 , hence ab D u r 2 Pj0 . The assumption on a and the primeness of Pj0 imply that b 2 Pj0 , a contradiction. If s C 1 j0 m, then b 2 Pj0 . Hence r D u ab 2 Pj0 ; this is a contradiction to the choice of r. Lemma 26. Let R be a Noetherian ring. Given a; aj 2 R, j D 0; 1; : : : ; s, there exists bj 2 R, j D 1; : : : ; s, such that for i D 1; 2; : : : ; s, the (finite) set, Pi1 , of minimal prime ideals P containing IR .a0 ; a1 C b1 a; : : : ; ai1 C bi1 a/3

3

If i D 1, then we consider IR .a0 /.


173

has the following property: if a … P , where P 2 Pi1 , then ai C bi a … P :

(7)

Proof. To prove the assertions via induction, we will use several times Lemma 25. i D 1: Recall that P0 is the class of all minimal prime ideals P with IR .a0 / P . By Theorem 21, P0 is finite; say P0 D fP1 ; : : : ; Pt g. To apply Lemma 25, we let r D a1 . This gives b1 2 R such that for all 2 f1; : : : ; t g a10 WD a1 C b1 a … P whenever a … P . Thus (7) is satisfied. Now let us suppose that b1 ; : : : ; bi have been constructed and that (7) is satisfied for some i 2 f1; : : : ; s 1g. Let aj0 WD aj C bj a, where j 2 f1; : : : ; i g. i ! i C 1: By definition, Pi is the class of all minimal prime ideals P with IR .a0 ; a10 ; : : : ; ai0 / P: By Theorem 21, Pi is finite. To apply Lemma 25, we let r D aiC1 . This gives biC1 2 R such that for all P 2 Pi : 0 WD aiC1 C biC1 a … P whenever a … P . aiC1

Thus (7) is satisfied for i C 1.

Theorem 27 (Bass). Let R be a Noetherian ring with Krull dimension less than or equal to n. Then bsr R n C 1. Proof. This is the proof given by Estes and Ohm [4]. Let a WD .a1 ; : : : ; anC1 ; a/ 2 UnC2 .R/. We have to show that a is reducible. Choose a0 D 0. Associate with aj the elements bj coming from Lemma 26, j D 1; : : : ; n C 1, and let aj0 D aj C bj a. For i D 1; : : : ; n C 1, consider the ideals Ii WD IR .a0 ; a10 ; : : : ; ai0 /; and let Pi be the (finite) set of minimal prime ideals P with Ii P . Note that the reducibility of a is a consequence to the assertion that InC1 D R. Suppose, to the contrary, that InC1 is a proper ideal. We claim that a … P for every P 2 PnC1

(8)

In fact, if we suppose that a 2 P for some P 2 PnC1 , then the invertibility of a implies that

174

R. Mortini and R. Rupp 0 P IR .a10 ; : : : ; anC1 ; a/ D IR .a1 ; : : : ; anC1 ; a/ D R;

a contradiction to the fact that prime ideals are proper ideals. Hence assertion (8) holds. Consider now the following chain of ideals: 0 IR .a0 / IR .a0 ; a10 / IR .a0 ; a10 ; : : : ; an0 / IR .a0 ; a10 ; : : : ; anC1 / D InC1 R:

We shall construct a chain of prime ideals that has length n C 1 (in other words n C 2 elements) which will yield a contradiction to the assumption that the Krull dimension of R is less than or equal to n. Let PnC1 2 PnC1 . Choose a minimal prime ideal Pn 2 Pn with IR .a0 ; a10 ; : : : ; an0 / Pn PnC1 : Backwards induction yields minimal prime ideals Pi with IR .a0 ; a10 ; : : : ; ai0 / Pi PiC1 PnC1 (i D 1; : : : ; n), and finally a minimal prime ideal P0 with IR .a0 / P0 P1 : We claim that all the inclusions in the chain P0 P1 Pn PnC1 are strict. To do so, we use Lemma 26. Fix i 2 f1; : : : ; n C 1g and consider 0 Ii1 D IR .a0 ; a10 ; : : : ; ai1 /, with the convention that I0 D IR .a0 /. We first observe that a … Pi1 , since otherwise a 2 PnC1 , a contradiction to (8). Hence, by Lemma 26, ai0 … Pi1 . But by construction, ai0 2 Pi . Thus Pi1 Pi , the inclusion being strict. Hence, under the assumption that InC1 is proper, we have shown that the Krull dimension of R is at least nC1. This contradicts the hypothesis. Consequently, InC1 D R. Thus, as already mentioned, a 2 UnC2 .R/ is reducible. Hence bsr R n C 1. To conclude this section, let us mention the following generalization of Bass’ Theorem given by R. Heitmann [7]. It shows that the Noetherian condition can be dropped. Theorem 28. Let R be a commutative unital ring with Krull dimension n. If R is an integral domain, then bsr R n C 1. If, on the other hand, R has zero-divisors, then bsr R n C 2.


175

7 The Topological Stable Rank of RŒx1 ; : : : ; xn Definition 29. Let R be a ring endowed with a topology T (we do not assume that the topology is compatible with the algebraic operations C and ). The topological stable rank, tsrT R, of .R; T / is the least integer n for which Un .R/ is dense in Rn , or infinite if no such n exists. If the ring R is endowed with two topologies T1 and T2 such that T1 is weaker than T2 , then tsrT1 R tsrT2 R: For the ring of polynomials, we work with the topology of uniform convergence. Theorem 30. The topological stable rank of RŒx1 ; : : : ; xn is n C 1. Proof. We first prove that tsr RŒx1 ; : : : ; xn n C 1:

(9)

Let p WD .p1 ; : : : ; pnC1 / be an .n C 1/-tuple in RŒx1 ; : : : ; xn . Now we look upon p as being an .n C 1/-tuple in CŒz1 ; : : : ; zn . Choose, according to Perron’s Theorem 19, a non-zero polynomial P over C with n C 1 indeterminates such that P .p1 ; : : : ; pnC1 / D 0: Then P , looked upon as a polynomial function, vanishes identically on the image p.Cn /. But P cannot vanish identically on the ball B.0; "/ WD fx D .x1 ; : : : ; xnC1 / 2 RnC1 W jjxjj2 "g; since otherwise P would be the zero-polynomial (just consider the partial derivatives at the origin). Since " > 0 can be chosen arbitrarily small, we obtain a null-sequence ."k / in RnC1 such that "k … p.Cn /. Hence the .n C 1/-tuple p "k is invertible in C.Cn ; C/. From Theorem 3 we deduce that p "k is in UnC1 .RŒx1 ; : : : ; xn /. Since p "k uniformly approximates p, we are able to conclude that tsr RŒx1 ; : : : ; xn n C 1. Next we show that tsr RŒx1 ; : : : ; xn n C 1. Consider the identity map x D .x1 ; : : : ; xn / of Rn onto Rn . Note that x 2 RŒx1 ; : : : ; xn RŒx1 ; : : : ; xn : ƒ‚ … „ ntimes

Suppose that there exist invertible n-tuples in RŒx1 ; : : : ; xn that uniformly approximate x. That is, for every " > 0 there is f D .f1 ; : : : ; fn / 2 Un .RŒx1 ; : : : ; xn / such

176


that jxj fj j < " for j D 1; : : : ; n. This implies of course that xj fj is a constant (we keep this generality though, since it also works for non-polynomial rings). In particular, this inequality then holds on the unit sphere Sn1 . Let A be the Banach algebra A D C.Sn1 ; R/ of all continuous, real-valued functions on Sn1 , endowed with the supremum norm. We obviously have that f 2 Un .A/. By a classical theorem in the theory of Banach algebras ([10] or [13]) there exists a matrix H 2 Mn .A/ with .x1 ; : : : ; xn /t D .exp H / .f1 ; : : : ; fn /t whenever " is chosen small enough. Extending the entries of H with the help of Tietze’s Theorem to continuous functions on Rn we obtain a zero-free extension of xj@Bn to Bn . As above (see the proof of Theorem 10), this contradicts Brouwer’s fixed point theorem. Acknowledgements We thank Leonhard Frerick for providing us the reference [12]; Witold Jarnicki for the proof of Theorem 19; Amol Sasane for the reference [1] as well as Peter Pflug and Thomas Schick for valuable comments yielding to a proof of tsr RŒx1 ; : : : ; xn n C 1 in Theorem 30.

References 1. D. Anderson, A note on minimal prime ideals, Proc. Amer. Math. Soc. 122 (1994), 13–14. 2. H. Bass, K-theory and stable algebra, Publications Mathématiques de L’I.H.É.S., 22 (1964), 5–60. 3. T. Coquand, H. Lombardi, A short proof for the Krull dimension of a polynomial ring, Amer. Math. Monthly 112 (2005), 826–829. 4. D. Estes, J. Ohm, Stable range in commutative rings, J. of Algebra 7 (1967), 343–362. 5. M.R. Gabel, Lower bounds on the stable range of polynomial rings, Pacific J. Math. 61 (1975), 117–120. 6. M. R. Gabel, A.V. Geramita, Stable range for matrices, J. Pure Appl. Algebra 5 (1974), 97–112. 7. R. Heitmann, Generating ideals in Prüfer domains, Pacific J. Math. 62 (1976), 117–126. 8. I. Kaplansky, Commutative rings, Allyn and Bacon, Boston 1970. 9. S. Lang, Algebra Springer, New York, 2002. 10. T.W. Palmer, Banach algebras and the general theory of -algebras, Vol 1, Cambridge Univ. Press, London, 1994. 11. O. Perron, Algebra I. Die Grundlagen, 2. Auflage, W. de Gruyter u. Co, Berlin, 1932. 12. A. Płoski, Algebraic dependence of polynomials after O. Perron and some applications,in: Computational commutative and non-commutative algebraic geometry,Proceedings of the NATO Advanced Research Workshop, Chisinau, Republic of Moldova, June 6–11, 2004. Ed.: Cojocaru, Svetlana et al., 167–174 (2005) 13. R. Rupp. Zerofree extension of continuous functions on a compact Hausdorff space, Topology Appl. 93 (1999), 65–71. 14. L. Vasershtein, Stable rank of rings and dimensionality of topological spaces, Funct. Anal. Appl. 5 (1971), 102–110; translation from Funkts. Anal. Prilozh. 5 (1971), No.2, 17–27.

Structure in the Spectra of Some Multiplier Algebras Richard Rochberg

Abstract We examine the spectra of the multiplier algebras of the Dirichlet space and of the generalized Dirichlet spaces (spaces between the Hardy space and the Dirichlet space). For the generalized Dirichlet spaces we show that the spectrum contains nontrivial analytic disks. For the Dirichlet space we are able to show that the spectrum contains nontrivial Gleason parts. Keywords Dirichlet space • Multiplier algebra • Spectrum • Analytic disk • Gleason parts Subject Classifications: 46E22, 47B32, 46J20

1 Introduction We begin with informal comments. Precise definitions and statements are given in the later sections. Active study of the fine structure of the maximal ideal space of H 1 began in the 1950’s and still continues. The early story is in Hoffman’s book [Ho1], a description of what was known in 1980 is in the book of Garnett [Ga], and a sample of recent work is [II]. The functions in H 1 are the multipliers (bounded pointwise multiplication operators) of the classical Hardy space H 2 . In recent years we have learned that viewing H 1 as the multiplier algebra of such a Hilbert space can lead to valuable new insights [AM]. Hence it seems natural to revisit the classical results about H 1

R. Rochberg () Department of Mathematics, Washington University, St. Louis, MO 63130, USA e-mail: [email protected] R.G. Douglas et al. (eds.), The Corona Problem, Fields Institute Communications 72, DOI 10.1007/978-1-4939-1255-1__9, © Springer Science+Business Media New York 2014

177

178

R. Rochberg

and see what we can say about similar questions for multiplier algebras of spaces such as the Dirichlet space, which share certain crucial technical properties with the Hardy space. That is what we do here. We have two groups of results. The first group, in Sect. 4, concerns the spectra of the multiplier algebras of Dirichlet type spaces D˛ ; 0 < ˛ < 1. We find that the spectra of those algebras have a structural property in common with that of H 1 . Namely, for 0 < ˛ < 1, every point in the spectrum which is in the closure of an interpolating sequence is the center of an analytic disk. We show this by showing that, for this range of ˛, every interpolating sequence for D˛ is the zero set of a Blaschke product in the multiplier algebra. With that result in hand, the argument used in showing that the maximal ideal space of H 1 contains analytic structure (e.g. [Ga, Sec. V3]) can be used. The second group of results, in Sect. 5, concern the classical Dirichlet space, D1 . Our results there are less incisive. This is perhaps foretold by the fact that neither the Dirichlet space nor its multiplier algebra contain any Blaschke products. To elaborate on this distinction we study a metric on the disk which is associated to the Dirichlet space. An incompatibility between that metric and the pseudohyperbolic metric on the disk precludes adapting the arguments which are successful for the Hardy space to the Dirichlet space. On the other hand the spectrum of the multiplier algebra of the Dirichlet space has a very rich structure. We show that every Gleason part that meets the closure of an interpolating sequence contains infinitely many points. Unfortunately, however, we do not have a further description for those parts. The ordering of the presentation is based on familiarity, and it is opposite the order of the sophistication of the techniques used. The results in Sect. 4 are adaptations to Dirichlet type spaces of a well known result in [Sc], and, just as in [Sc], Blaschke products play a crucial role. On the other hand the results in Sect. 5 do not use Blaschke products; and those results apply, essentially as stated, to a large class of spaces, including both the Dirichlet type spaces of Sect. 4 and some spaces of functions of several variables. The author is not aware of other research on the specific topics considered here. However there is a continuous flow of work which investigates generalizations of the classical results about the spectrum of H 1 . Instances that seem to resonate with this work include [CP1, CP2, CGJ], and [Gam]. Thanks to Orr Shalit for an improvement in a proof, and for catching an error.

2 Background In this section we present definitions and notation, and collect some facts we will use later.

Structure in the Spectra of Some Multiplier Algebras

179

2.1 Spaces of Holomorphic Functions 2.1.1

Spaces

For 0 ˛ 1, let D˛ be the Hilbert space P of functions which are holomorphic on the unit disk, and for which, with f D an zn , kf k2˛ D

X

Z Z .n C 1/˛ jan j2 kf k2H 2 C

ˇ 0 ˇ2 ˇf ˇ .1 jzj2 /1˛ dA < 1:

D

Each space is given the natural inner product. The space D0 is the Hardy space H 2 I D1 is the classical Dirichlet space.

2.1.2

Kernels

Each of the D˛ is a reproducing kernel Hilbert space. That is, for each ˛; 0 ˛ 1, and each z 2 D there is a kernel function k˛;z 2 D˛ with the reproducing property that, 8f 2 D˛ ; hf; k˛;z i˛ D f .z/. (Below, when it seems safe and convenient to do so, we will suppress ˛ in the notation.) For 0 ˛ < 1 these kernels are given by k˛;z .w/ D

1 .1 zNw/1˛

:

(1)

When ˛ D 1 we have k1;z .w/ D

1 1 : log zNw 1 zNw

(2)

bb

We will denote the normalized kernel functions by k˛;z ; k˛;z D k˛;z = kk˛;z k˛ I and we will use analogous notation for the normalized kernel functions of other Hilbert spaces.

2.1.3

Multipliers and Carleson Measures

A holomorphic function m, defined on D, is said to be a multiplier for D˛ ; m 2 M˛ , if Mm , the operator of multiplication by m, is a bounded operator on D˛ . Pointwise multiplication of multipliers, or what is the same, composition of multiplication operators, gives a product on M˛ . With that product and normed by the operator norm, M˛ is a commutative Banach algebra with unit. The function 1 is a unit vector in each D˛ , and hence, for each ˛; m 2 M˛ is contractively contained in D˛ .

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Understanding multipliers for these spaces is intimately related to understanding Carleson measures and the functions which generate Carleson measures. A nonnegative measure supported on the disk, D, is a Carleson measure for D˛ ; 2 CM .D˛ /, if there is a constant C./2 so that 8f 2 D˛ Z Z D

jf j2 d C./2 kf k2˛ :

For each ˛; 0 ˛ 1; 1 2 D˛ and hence any Carleson measure, 2 CM .D˛ /, is a finite measure. A special role is played by the subspace BMO˛ of functions in D˛ which generate Carleson measures in a particular way: o n ˇ ˇ2 BMO˛ D f 2 Hol .D/ W ˇf 0 ˇ .1 jzj2 /1˛ dxdy 2 CM .D˛ / : The choice of notation reflects the fact that for ˛ D 0 we recapture the classical space of holomorphic functions of bounded mean oscillation, BMO0 D BMO \ Hol .D/. The multipliers are the bounded functions in this space. Proposition 1 ([W]). For 0 ˛ 1; M˛ D BMO˛ \ H 1 . In the classical case, ˛ D 0, but in that case only, H 1 BMO˛ , in which case the result simplifies to the more familiar M0 D H 1 .

2.1.4

The Complete Nevanlinna Pick Property

Some reproducing kernel Hilbert spaces (or their reproducing kernels) have (are) complete Nevanlinna Pick kernels (CNPK) and some of the consequences of having this property play a crucial role below. However we will not introduce that property here, nor review its consequences; for that we refer to [AM, S]. Here we just state that the spaces we consider have that property and note the consequences we will use. For D 1; 2; : : :, let B be the unit ball in C . Let H be the Hilbert space of holomorphic functions on B which is determined by specifying the reproducing kernels 8z; w 2 B I kH ;z .w/ D .1 hw; ziC /1 :

(3)

For D 1 we let H be the Hilbert space of holomorphic functions on B1 , the unit ball of `2 .C/, which is determined by reproducing kernels 8z; w 2 B1 I kH1 ;z .w/ D .1 hw; zi`2 .C/ /1 :

(4)

The following result, which is far from the traditional definition of a CNPK, is a property we will use.


181

Proposition 2 ([AM, Thm. 8.2]). Suppose H is a reproducing kernel Hilbert space of functions on a set X , and suppose H has an irreducible reproducing kernel kx .y/ D k.x; y/. H has a CNPK if and only if there is an injection W X ! B (with D 1 allowed) and a nonvanishing scalar function ˇ defined on X such that ˝ ˛ 1 8x; x 0 2 X; k(x,x’) D ˇ.x/ˇ.x 0 / 1 .x 0 /; .x/ `2

(5)

D ˇ.x/ˇ.x 0 /kH ;z ..x 0 /; .x// Proposition 3 ([AM, Cor. 7.41]). For 0 ˛ 1, the space D˛ has an irreducible reproducing kernel and has a CNPK. Proposition 4 ([ARSW, Prop 12]). Suppose H is a reproducing kernel Hilbert space on the set X and that it has a CNPK. Let M.H / be its multiplier algebra. Given x; x 0 2 X ˚

sup Re m.x/I m 2 M.H /; kmkM.H /

D 1; m.x / D 0 D 0

r

ˇD Eˇ2 ˇ ˇ 1 ˇ kbx ; kc (6) x0 ˇ

That the left hand side is no larger does not require a CNPK. However having such a kernel insures that there is a close relationship between the extremal problem (6) in M.H / and an extremal problem in H [AM, Prop. 9.3.1]. With that fact in hand the proof proceeds by solving the easy Hilbert space extremal problem and explicit computation.

2.1.5

Gramm Matrices and Domination

For fixed ˛, for any fzn g D Z D, we define the Gramm matrix of Z; Gr .Z/ to be the matrix E1 D Gr .Z/ D k˛;zI ; k˛;zj : (7)

b b

i;j D1

This, of course, relates to the space D˛ I but we will use the same notation for other spaces, clarifying the particulars as necessary. We will also be interested in the matrix obtained by making these entries positive. Set

b b

Eˇ1 ˇD ˇ ˇ jGrj .Z/ D ˇ k˛;zI ; k˛;zj ˇ

i;j D1

:

(8)

It is elementary that if the matrix jGrj .Z/ is bounded then so is the matrix Gr .Z/. We will be interested in the converse statement. That is, we say that a reproducing kernel Hilbert space of holomorphic functions, H , on a space X satisfies the domination relationship (DOM) if, given Z X , and using Gramm matrices based on the inner product of H , we have, uniformly in Z,

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If Gr .Z/ is bounded then jGrj .Z/ is bounded.

(DOM)

When explicit formulas are available for the kernel functions this property may be easy to verify. That is done, for instance, in [B] to show that, for 0 < ˛ 1; D˛ , satisfies (DOM). On the other hand, the Hardy space, D0 , does not satisfy (DOM).

2.1.6

Metrics and the Strengthened Triangle Inequality

Suppose H is a reproducing kernel Hilbert space of holomorphic functions on D, with n o kernel functions fkz g ; z 2 D, and normalized reproducing kernel functions kOz . There is an associated metric, ıH .; /, defined on D by the formula r ıH .z; w/ D

ˇD Eˇ2 ˇ ˇ 1 ˇ kbz ; kbw ˇ :

(9)

For more about such metrics see [ARSW]. If H D D˛ then we will write ı˛ rather than ıD˛ . These metrics are generalizations of the pseudohyperbolic metric ; in fact, ı0 .z; w/ D .z; w/ D jz wj = j1 zNwj. The metric ıH ; < 1, which is defined by the formula (9) with H D H , is called the pseudohyperbolic metric on the ball B . It is discussed in [DW] We will say a metric ı on a space X satisfies the strengthened triangle inequality, (STI), if 8x; y; z 2 X ı.x; y/ C ı.y; z/ jı.x; y/ ı.y; z/j ı.x; z/ : 1 ı.x; y/ı.y; z/ 1 C ı.x; y/ı.y; z/

(STI)

The pseudohyperbolic metric, ı0 , satisfies (STI) [Ga, Lemma 1.4]. More generally we have the following Proposition 5. For 0 ˛ 1, the metric ı˛ satisfies (STI). Proof. Suppose two metrics ı and ı 0 are related as follows. There is map of the metric space X on which ı is defined to the metric space XO on which ıO is defined O .x 0 //. In such a case, tracking through such that 8x:x 0 2 X; ı.x; x 0 / D ı..x/; O the definitions is enough to see that if ı satisfies (STI) then so does ı. It is shown in [DW] that, for < 1, the pseudohyperbolic metric ıH , on the ball B , satisfies (STI). We noted in Proposition 3 that the spaces D˛ have a CNPK. We now invoke Theorem 2 and obtain, for some 1, a map ˛ W D ! B such that (5) holds and hence, for all z; z0 2 D ı˛ .z; z0 / D ıH .˛ .z/; ˛ .z0 //:


183

If we knew < 1 we would be done. However, in fact, when applying Theorem 2 to a D˛ ; 0 < ˛ 1, the most natural construction of ˛ gives a map into the infinite dimensional ball. We know of no obstacle to extending the results of [DW] to D 1, but we avoid that issue with the following observations. Any finite set of points in B1 determine a finite dimensional subspace of C1 which, when intersected with B1 , produces a finite dimensional ball Bn . Associated with that Bn is the Hilbert space Hn with the inner product given by (3). Furthermore, the natural inclusion of this H into H1 , which has the inner product (4), is isometric. Thus, computation of inner products for the three points involved in any verification of (STI) reduces to evaluating inner products in an H3 (as opposed to an H1 / where, as we noted, (STI) does hold. Having just described a large class of H for which the (STI) holds (and that proof applies unchanged to a much larger class of Hilbert spaces than D˛ ; 0 ˛ 1/ we should note that the property is not ubiquitous. Suppose H is the classical Bergman space, D1 in our notation. Consider, for small t > 0, the three points .x; y; z/ D .t; 0; t /. For these points the claim in (STI) is that the right hand quantity, R.t /, dominates the expression in the center, C.t /I C.t / R.t /, where v u u C.t / D t1

.1 t 2 /2 .1 C t 2 /2

!2 ;

q 2 1 .1 t 2 /2 R.t / D

2 : q 1C 1 .1 t 2 /2 However the Taylor expansions at t D 0, for t > 0 (so that

p

t 2 D t / are

8 p 3 2t C O.t 5 /; 2

p 9 p 3 R.t / 2 2t 2t C O.t 5 /; 2 p C.t / 2 2t

and we see that C.t / R.t / fails.

2.2 Interpolating Sequences A sequence fzn g D Z D D D˛ is said to be an interpolating sequence, Z 2 IS D IS˛ , if every bounded function on Z is the restriction to Z of a function in M˛ . For the case of interest to us (as opposed to the situation for general multiplier algebras) we have a useful description of interpolating sequences.

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The following description of interpolating sequences is classical for the Hardy space [Ga]. The results for D˛ ; 0 < ˛ 1, is given a short, elegant proof in [B] where one can also find references to earlier work. Recall that Carleson measures are defined in Sect. 2.1.3. Proposition 6 ([S, Theorem 5.31]). For 0 ˛ 1, a sequence fzn g D Z D D D˛ is an interpolating sequence, Z 2 IS˛ , if and only if both 1. (Separation Condition) 9ı > 08zn ; zm 2 Z; zn ¤ zm ı˛ .zn ; zm / > ıI

(Sep)

and 2. (Carleson Measure Condition) the measure z associated to the sequence Z is a D˛ Carleson measure: X k˛;zn .zn /1 ızn 2 CM.D˛ /: (CM) Z D zn 2Z

The condition (Sep) is independent of ˛ for 0 ˛ < 1. More precisely, starting from (1) and (2), direct computation gives that for 0 ˛ < 1 1˛ 1 ı˛2 D 1 ı02 : (10) Hence, if for some ˇ; 0 ˇ < 1, a set of values of ıˇ is bounded away from zero, or bounded away from one, then the same holds for the corresponding set of values of ı˛ for every other value of ˛ in the range. However, the equivalence is not uniform in ˛ and the statements with ˛ D 1 are not equivalent to the others. The metric ı1 on the disk is fundamentally different from the others. Because M˛ M0 D H 1 we have IS˛ IS0 . Hence the following result which holds for all Z 2 IS0 holds, in particular, for Z 2 IS˛ . For fzn g D Z 2 IS0 , let BZ be the associated Blaschke product BZ .z/ D

Y jzn j zn z z 1 zNn z z 2Z n n

(with the factor taken to be z if zn D 0:/. Define Y ˇ ˇ ı0 .zn ; zk /: .BZ / D inf.1 jzj2 / ˇBZ0 .zn /ˇ D inf n

n

(11)

k¤n

(The IS0 condition is more than enough to insure convergence.) The following result actually holds for all sequences in IS0 . Proposition 7 ([Ga, Chpt. VII]). For Z 2 IS˛ ; 0 ˛ 1, we have .BZ / > 0. Remark 1. We are not claiming BZ 2 M˛ .


185

2.3 The Spectrum, Gleason Parts, Analytic Discs We now introduce the spectrum. We will be brief and informal. Further details and proofs can be found in [Ho1] or [Ga]. Associated with any commutative Banach algebra A, in our case A D M˛ , is a compact Hausdorff space X.A/, its spectrum or maximal ideal space. We write X˛ for X.M˛ /. The spectrum is the set of nonzero continuous multiplicative linear functionals on A and is topologized is such a way that for any a 2 A the map of X.A/ to C which takes x 2 X.A/ to x.a/ is continuous on X.A/. This function on X.A/ is called the Gelfand transform of a and is denoted by a. O The map of a to aO is a Banach algebra contraction of A into the C.X.A// the algebra of continuous functions on X.A/ normed by the supremum norm. Given 2 D, the evaluation map e on M˛ , defined by e .m/ D m./, is a nonzero multiplicative linear functional on M˛ . Our earlier observation that M˛ is contractively contained in D˛ is enough to insure that e is continuous, hence e 2 X˛ . Thus, the map ˚ defined by ˚./ D e is a one to one map of D into X˛ . Unpacking the definitions we see that m O .˚.// D m.e O / D m./. ˚ Taking note of this we will abuse notation and denote e by and denote the set e W 2 D X˛ by D, or, if useful, by D˛ . We will be interested in analytic disks in X˛ . An analytic disk in X˛ is nonconstant map ˚ of D into X˛ such that 8m 2 M˛ ; m O .˚.z// is a holomorphic function of z for z 2 D. The point ˚.0/ 2 X˛ is called the center of the disk. The maps ˚ of the previous paragraph which take D to D˛ are obvious examples. A basic problem is to find the others, if there are any. A fundamental first step in that study was the introduction of Gleason parts [Gl]. To define them we first define the Gleason distance between and in X˛ by ˚ ı˛ .; / D sup jm ./j W m 2 M˛ ; kmkM˛ D 1; m./ D 0 : Using Proposition 4 one checks that this agrees with the earlier usage, ı˛ .z; w/, for z; w 2 D D D˛ X˛ . Proposition 8. For 0 ˛ 1; ı˛ is a distance function on M˛ . That distance satisfies (STI) and hence the relation ı˛ .; / < 1 is an equivalence relation on M˛ . Proof. The case ˛ D 0 goes back to Gleason. His work, and other early work using these ideas, was done in the context of uniform algebras, algebras in which, for any element m; m2 D kmk2 . Among the algebras M˛ , only M0 D H 1 is a uniform algebra, however we can still adapt the classical arguments to our situation. To do that we need the following fact. If m 2 M˛ ; kmkM˛ D 1, and if L is a conformal automorphism of the disk, then the composite, L ı m, satisfies L ı m 2 M˛ ; kL ı mkM˛ 1. For ˛ D 0 we have M0 D H 1 and this is immediate; for general ˛ we need to invoke von Neumann’s inequality. Once that is settled the proofs in ([Gl]) can be used to prove the proposition.

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We now define the Gleason parts of M˛ to be the equivalence classes into which M˛ is decomposed under the equivalence relation ı˛ .; / < 1. We denote the part containing by P./. Parts were introduced because they are the carriers of whatever analytic structure there may be in the spectrum. In particular we have the following proposition which is a direct consequence of the definitions and the Schwarz lemma. Proposition 9 ([Gl]). Suppose there is an analytic disk in X˛ given by the map ˚ of D into X˛ . Then the range of ˚ is contained in a single part. That is, ˚ .D/ P.˚.0//. If we can find a map ˚ such as the one in the proposition which maps D univalently onto all of the part P, then we say that P is an analytic disk. For instance, D˛ is an analytic disk in X˛ .

3 The Structure of X0 D X.H 1 / A goal of this study was to obtain for X˛ ; 0 < ˛ 1, results analogous to the classical results for X0 which we now recall. Proposition 10 ([Sc]). Suppose Z 2 IS0 . Any m 2 ZN X0 is the center of an analytic disk in X0 . This was substantially refined by Hoffman [Ho2] Theorem 1. Suppose m 2 X0 and let P.s/ be the part containing s. Then 1. Either P.s/ D s, or P.s/ is an analytic disk. N 2. The second case occurs exactly when there is a Z 2 IS0 with s 2 Z.

4 Dirichlet Type Spaces The following result for D˛ ; 0 < ˛ < 1, is an analog of what is true for the Hardy space. The proof for the Hardy space only needs that M0 D H 1 . The proof here requires an additional step because the multiplier algebra M˛ is more subtle. The following theorem is a consequence of Proposition 6 and Verbitski˘ı’s description of inner functions in M˛ [V, Theorem 1]. Here we provide a different proof which has different potential for generalization. Recall that we are writing BZ for the Blaschke product with zero set Z. Theorem 2. Suppose 0 < ˛ < 1 and fzn g D Z 2 IS˛ , then BZ 2 M˛ . Proof. Note that by Proposition 6 we have


Z D

X

k˛;zn .zn /1 ızn D

187

X .1 jzn j2 /1˛ ızn 2 CM.D˛ /:

(12)

zn 2Z

P Ever Carleson measure is a finite measure, .1 jzn j2 /1˛ < 1. It was shown by Carleson [C] that this insures BZ 2 D˛ . However, by Proposition 1, to show that BZ 2 BMO˛ we must further show that ˇ ˇ2 (13) B D ˇBZ0 .z/ˇ .1 jzj2 /1˛ dxdy 2 CM .D˛ / : We now finish the proof by showing that (12) implies (13). The general idea is that B is a spread out version of Z and the particular type of spreading involved preserves membership in CM .D˛ /. The version of that general scheme which we will use is Lemma 6 of [RW]. To specialize that lemma to our needs, we set the parameters ˛; ˇ; b in that lemma to the values ˛=2; 1 ˛; 0. Lemma 1 ([RW] Lemma 6). Suppose v is a function on D such that v D jv.z/j2 .1 jzj2 /1˛ dxdy 2 CM .D˛ / : Set Z Z .T v/ .w/ D D

v.z/ j1 zNwj2

dxdy:

Then T v D jT v.z/j2 .1 jzj2 /1˛ dxdy 2 CM .D˛ / : Proof. In using this Lemma to show that (12) implies (13) we will make certain approximations which we will discuss at the end of the proof. n o Pick and fix a small positive ". For zk 2 Z let Dk;" D W jzk j ".1 jzj2 / and let jDk;" j denote its Euclidean area, "2 .1 jzj2 /2 . Let k;" be the characteristic function of Dk;" . By choosing " sufficiently small, and taking note of (10) which lets us compare ı˛ and ı0 , we can insure that the Dk;" are disjoint and contained in D. Our first approximation is Z D

X .1 jzn j2 /1˛ ızn

X k;" .z/ dxdy .1 jzj2 /1˛ jDk;" j !2 X k;" .z/ D .1 jzj2 /1˛ dxdy 1=2 jDk;" j

D jv.z/j2 .1 jzj2 /1˛ dxdy

(14)

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The last measure is in the form required for use in the lemma. We compute Z Z

X k;" .z/

1

T v.w/ D

!

dxdy j1 zNwj2 jDk;" j1=2 X Z Z jDk;" j1=2 k;" .z/ dxdy D j1 zNwj2 jDk;" j X Z Z .1 jzk j2 / k;" .z/ dxdy C."/ j1 zNk wj2 jDk;" j C."/

X .1 jzk j2 / j1 zNk wj2

:

We have BZ .z/ D

Y jzn j zn z z 1 zNn z z 2Z n

(15)

n

and hence ˇ X ˇ 0 ˇB .w/ˇ Z

.1 jzj2 / j1 zNk wj2

!

number of modulus less than one :

ˇ ˇ Hence ˇBZ0 .w/ˇ C T v.w/. By the lemma T v 2 CM .D˛ /. Hence, by the pointwise majorization of the densities which we just obtained, B 2 CM .D˛ /. We made several approximations. We wanted to know that a large multiple of T v is a pointwise upper bound for jB 0 j, and we don’t care if the constants in the comparison depend on ". In estimating T v we used that .1 jzj2 / C."/ jDk;" j1=2 and for z 2 Dk;" ; w 2 D we have j1 zNwj j1 zNk wj. These are easy to check and compatible with our goal of showing that a lower bound for T v dominates jB 0 j. In our approximation for Z we had also made some straightforward estimates which, again, cause no problems. However the passage from the first line, the definition of Z to the second, in which the ızn are replaced by jDk;" j1 k;" .z/dxdy, needs some attention. What we actually want is that, given that Z 2 CM .D˛ / then the new measure is in CM .D˛ / with norm control. In making the estimate we replaced each point mass with a measure of the same total mass, and within a small pseudohyperbolic disk centered at the location of the point mass. It is easy to see, using the characterization of Carleson measures using tree models, for instance Theorem 1 of [AR] or Theorem 5.31 of [S], that such a change replaces a Carleson measure with another Carleson measure. With this result in hand we can follow the classical proof.


189

Theorem 3. For 0 < ˛ < 1, every s 2 X˛ that is in the closure of an interpolating sequence is the center of an analytic disk. Proof. Suppose Z 2 IS˛ and s 2 ZN X˛ . If s 2 D˛ the conclusion is immediate and we now consider the other case. Let BZ .z/ be the Blaschke product (15). For each n let Ln be the automorphism of the disk Ln ./ D

C zn ; 1 C zNz

˚ ˚ and note that Ln .0/ D zn . Hence there is a subnet of maps, L with L .0/ converging to s. ˚ We now regard L as a net of maps of the open unit disk into the compact Hausdorff space X˛ . As such, it has a limit map L and for any f 2 M˛ ; 2 D we have fO.L.// D lim f .L .//: We have that fO ı L is the pointwise limit of a net of uniformly bounded analytic functions. Hence it must be analytic (although possibly constant), and its derivative must be the pointwise limit of the derivatives of functions in the net. It is now that we use Theorem 2, which allows us to make the choice f D BZ . With D 0 we have 0 .BO Z ı L/0 .0/ D lim. BZ ı L .0/ D lim BZ0 .L .0//L0 .0/ ˇ ˇ2 D lim BZ0 .z /.1 ˇz ˇ / We now use Proposition 7 to conclude that this derivative is nonzero, and hence that L is not a constant map. A great deal of Hoffman’s refined work in [Ho2] relies on technical factorization theorems for Blaschke products. Many of those details hold as stated or with minor modifications, for Blaschke products in M˛ for ˛ > 0. Hence it is tempting to speculate that a full analog of Theorem 1 could be obtained. However we do not see a clear path forward. For instance, in Theorem 3.4 of [Ho2] Hoffman establishes a converse to the previous theorem; every point in X0 that is the center of an analytic disk is in the closure of an interpolating sequence. In his proof he first finds a useful function f 2 M0 D H 1 and then uses the fact that the Blaschke product, Bf , cf to a function with the same zeros as f is also in M0 and hence has an extension B 1 on X0 . Because M0 D H this is automatic; however, for M˛ ; ˛ > 0, the situation is unclear and seems interesting.

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5 The Dirichlet Space 5.1 A No-go Theorem In proving the previous theorem we constructed a map L of D into X˛ and then studied BO ı L for an appropriately chosen Blaschke product B. However M1 D1 and D1 contains no nontrivial Blaschke products [Ro] and thus the previous argument can not be repeated in this context. In fact there is a more fundamental obstruction to any argument of that type; that is an incompatibility between the pseudohyperbolic metric, D ı0 and the metric ı1 . Lemma 2. Suppose f˛n g ; fˇn g D and 9C; 8n; ı0 .˛n ; ˇn / < C < 1. If j˛n j ! 1I then ı1 .˛n ; ˇn / ! 0. Theorem 4. Suppose fL˛ g are maps of D to D and jLn .0/j ! 1. If L is map of D into X1 obtained as a limit of some fL˛ g then L is a constant map. First we prove the theorem assuming the lemma. Proof. By passing to a subnet we may suppose lim Ln D L. Suppose there are ˛; ˇ 2 D such that L.˛/ D a; L.ˇ/ D b with a; b 2 X1 X D and a ¤ b. Because a and b are distinct, there is s 2 M1 with kmk D 1; m.a/ O D 0 and m.b/ O D c > 0. Now consider the points ˛n D Ln .˛/ and ˇn D Ln .ˇ/. We must have m.˛n / D m.Ln .˛// ! m.a/ O D 0 and, similarly m.ˇn / ! c. Hence, lim sup ı1 .˛n ; ˇn / > c=2. On the other hand, by the Schwarz-Pick theorem, 8n; 1 > ı0 .˛; ˇ/ ı0 .Ln .˛/; Ln .ˇ// D ı0 .˛n ; ˇn /. Hence, by the previous lemma, ı1 .˛n ; ˇn / ! 0. This contradiction completes the proof.

5.1.1

Proof of the Lemma

Proof. We will give a computational proof based on Taylor estimates. First, however, we offer observations, which, while not leading directly to a proof, perhaps clarify the situation. The metrics ıa are not Riemannian metrics, the distances are not obtained by integrating a Riemannian length element along a curve. However one can compute the Riemannian structure which, in the small, matches ı˛ to second order. For the spaces we consider it is given in terms of the reproducing kernel by ˇ jdsj2 D @z @N log k .z/ˇzD jd zj2 : For discussion of this see [ARSW] and the references there. When we use (1) and compute this for ıa ; 0 ˛ < 1, we obtain a scalar multiple of the density for the hyperbolic metric. That is, for some universal constant c, jds˛ j2 D c.1 ˛/

1 .1 jzj2 /2

jd zj2 :


191

The analogous computation using (2) gives a more awkward formula, but the size of the density is correctly indicated by computing that 1 d2 1 1 log log : dx 2 1x .log.1 x//2 .1 x 2 /2 2 The additional factor of log.1 x 2 / in jds1 j2 shows that, roughly, a very small change in position from z to z C z will produce ı1 .z; z C z/

1 1 1 ı0 .z; z C z/ z: jlog.1 jzj/j jlog.1 jzj/j .1 jzj2 /

In particular, for points near the boundary, ı1 n, the ray e i W 0 1 lies above the circle t I and, similarly, the reflected ray lies below the circle. Hence this polar rectangle contains the circle. We now select z1 D r1 e i1 ; z2 D r2 e i2 2 R. By the triangle inequality ı1 .z1 ; z2 / ı1 .r1 e i1 ; r1 e i2 / C ı1 .r1 e i2 ; r2 e i2 /:

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Now note that ı1 .r1 e i2 ; r2 e i2 / D ı1 .r1 ; r2 / ı1 .n; f /: The equality holds because ı1 .; / is invariant under simultaneous rotation of both variables. The inequality holds because we are replacing r1 with a point nearer the origin and r2 with one further away. The required monotonicity can be verified by calculus. We do a similar analysis for the second term. ı1 .r1 e i1 ; r1 e i2 / D ı1 .r1 e ij1 2 j ; r1 / ı1 .r1 e 2i ; r1 / ı1 .f e 2i ; f /: Again, the equality holds because of rotational invariance. The first inequality holds because if we have two points at the same distance from the origin, and near each other (which was the purpose of first reducing to a situation where variation of arg z on t was small), their ı1 distance in an increasing function of their angular separation. The final inequality holds because, for fixed small and for near 1; ı1 .e 2i ; / is an increasing function of . Again, these monotonicities can be verified by computing the signs of appropriate derivatives. Hence we will be done if we show that, with C fixed, that both

t C t CC ; ; and ı1 .n; f / D ı1 1 tC 1 C tC

t C C 2i t C C ı1 .f e 2i ; f / D ı1 e ; 1 C tC 1 C tC tend to zero as t tends to 1. We now estimate these quantities. The procedures are standard and we will be sketchy. We know ı1 .r1 e

i1

; r2 e

/ D1

i2 2

ˇ ˇ ˇlog 1 rr r2 e i.1 2 / ˇ2 log.1 r12 / log.1 r22 /

(16)

Here we have already canceled some factors involving r1 and r2 . By rotational symmetry we may suppose that 2 D 0. Note that we are interested in the situation where r1 ; r2 ! 1. This lets us replace jlog j with log jj and log.1 ri2 / with log.1 ri / with acceptable errors. Now we define ki by 1 ri D e ki ; i D 1; 2. Also, in the case of interest to us, 1 will be very small; hence we replace e i1 with 1 C i1 . We now estimate the numerator on the right in (16) using ˇ ˇ ˇ ˇ ˇ1 rr r2 e i.1 2 / ˇ ˇ1 1 e k1 1 e k2 .1 C i1 /ˇ


193

ˇ ˇ ˇe k1 C e k2 C i1 ˇ : Suppose now that k2 k1 , and write 1 D j 2k1 with j 0 We continue our estimates with ˇ ˇ ˇ ˇ ˇ1 rr r2 e i.1 2 / ˇ ˇe k1 C ije k1 ˇ ˇ ˇ ˇe k1 .1 C ij /ˇ We will be interested in what happens when j is moderately large, hence we approximate j1 C ij j by j . Combining all this we have the estimate ˇ ˇ ˇlog 1 rr r2 e i.1 2 / ˇ2 log je k1 2

(17)

.k1 log j /2 : When we do a similar analysis of the denominator on the right side of (16) we obtain log.1 r12 / log.1 r22 / k1 k2 :

(18)

We now estimate ı1 .n; f /. The circle t has pseudohyperbolic radius C and both n and f are on that circle. Hence by the strengthened triangle inequality (STI) applied to the metric ı0 and the points n; t , and f we find, ı0 .n; f / 2C =.1 C C 2 / D C 0 < 1. We now use a standard estimate on ı0 I if ı0 .n; f / C 0 < 1 then there is a number K D K.C 0 / such that, if n D 1 2k1 and f D 1 2k2 then jk1 k2 j D O.K/. (K is, roughly a constant multiple of the hyperbolic distance corresponding to pseudohyperbolic distance C I and, in particular, is independent of t; k1 , and k2 :/ Hence ı1 .n; f /2 1

k12 k1 k1 : D1 D1 k1 k2 k2 k1 C O.K/

By (STI) for ı0 and the three points 0; n, and t we have ı0 .0; 1 2k1 / ı0 .0; n/

t C ı0 .0; t / ı0 .n; t / D 1 ı0 .0; t /ı0 .n; t / 1 tC

We know C is fixed and t ! 1. Hence 1 2k1 ! 1, which yields the desired k1 ! 1. To estimate ı1 .f e 2i ; f / we combine (17) and (18) to obtain ı1 .f e 2i ; f /2 1

.k2 log j /2 : k22

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We know k2 > k1 and we just saw that k1 ! 1. Hence the desired estimate ı1 .f e 2i ; f / ! 0 will follow if we establish that j , and hence log j , is O.1/. This will follow from an estimate of . Recall that ne i is the intersection of fjzj D ng with our upper tangent line. Hence n sin D

f n : 2

We are in a situation where we know is small. Hence we continue with

1f n : n 2

Recalling the definitions of the ki , and that k2 k1 this gives

1 2k2 C 2k1 1 2k1 2

O.1/2k1 However, we had 1 D j 2k1 ; and that was after we had used rotational symmetry to set 2 D 0. Hence 2 D j 2k1 which, with the previous estimate, gives the desired boundedness of j .

5.2 Finding Points in Parts Theorem 5. If r 2 X1 is in the closure of an M1 interpolating sequence Z, then P.r/ contains infinitely many points. Our proof works equally well for other spaces and algebras, including the algebras M˛ ; 0 < ˛ 1. However for those algebras it produces a result weaker than those in the previous section. Hence, to minimize notation we present the proof for ˛ D 1. The first step is to present a procedure for perturbing an interpolating sequences to obtain a new interpolating sequence nearby. Our proof of the theorem will then proceed by starting with the sequence Z and constructing other interpolating sequences Z1 ; Z2 ; : : : , each near Z and having closures which each meet P.r/ at points distinct from each other and distinct from r. Suppose now we are given the interpolating sequence fzn g D Z. By Proposition 6 we know 9ı > 0; 8zn ; zm 2 Z; zn ¤ zm ; ı1 .zn ; zm / > ı:


195

Theorem 6. Suppose Z 2 IS1 is as described, and W D fwn g is a nearby sequence such that 8n; ı1 .zn ; wn / < ı=10. Then W 2 IS1 . Proof. By Proposition 6 we need to show W is separated and that the associated P measure, W D kwi .wi /1 ıwi is a D1 Carleson measure. First we prove the separation. For all n; m; n ¤ m we have ı1 .zn ; wn / < ı=10; ı1 .zm ; wm / < ı=10; ı1 .zn ; zm / > ı: Combining these we find ı1 .wn ; wm / > :8ı and hence W is separated. To prove that we have a Carleson measure we use the following lemma which combines a basic fact about Gramm matrices and Carleson measures [AM, Prop 9.5] with the fact that D1 satisfies (DOM) Lemma 3. The measure W 2 CM .D1 / if and only if the Gramm matrix Gr.W / is bounded, if and only if the matrix jGrj .W / is bounded. We know that Z 2 IS1 , hence, by Proposition 6, Z 2 CM .D1 /. Thus, by the lemma, jGrj .Z/ is bounded. We will use this, the fact that the Gramm matrix entries are built from the ı1 distances, and the fact that ı1 satisfies (STI), to show that jGrj .W / is bounded. By the previous lemma, this will complete the proof. Let fjn g D fk1;wn g and fkn g D fk1;zn g be the normalized reproducing kernels associated with the sequences W and Z respectively. We need to show that W D P ki .wi /1 ıwi 2 CM .D1 /. Appealing to the lemma, we can do that by showing that the matrix jGrj .W / is bounded. Our hypotheses insure that jGrj .Z/ is bounded. Hence, it suffices to show that there is a K > 0 so that 8n; m ˇD ˇD Eˇ Eˇ ˇ ˇ ˇ b cˇ ˇ jn ; jm ˇ K ˇ kbn ; kc m ˇ: If n D m then the estimate holds with K D 1, we now consider the other case. Squaring both sides, and recalling that 1 x 2 1 x for 0 < x < 1, we see that it suffices to show, that for some new constant K 0 , .1 ı1 .wn ; wm // K 0 .1 ı1 .zn ; zm //: We first prove the intermediate estimate .1 ı1 .wn ; wm // K 0 .1 ı1 .zn ; wm // : We now use (STI); jı1 .zn ; wm / ı.zn ; wm /j ı1 .wn ; wm /: 1 ı1 .zn ; wm /ı.zn ; wm /

196

R. Rochberg

The expression inside the absolute value signs is negative. Using this and doing a bit of algebra we find, with K 00 a new constant, 1 ı1 .wn ; wm /

1 C ı.wn ; zn / .1 ı1 .zn ; wm // 1 ı.wn ; zn /ı.zn ; wm /

D K 00 .1 ı1 .zn ; wm // : A straightforward variation on this argument gives a comparison of ı1 .zn ; wm / and ı1 .zn ; zm /I and combining the two comparisons completes the proof of the Theorem 6. We now proceed with the proof of Theorem 5. Proof. We are given r 2 X1 is in the closure of an M1 interpolating sequence Z. We consider the nontrivial case of r …D. By Proposition 6 there is ı > 0 so that for all k; j; k ¤ j we have ı1 .zk ; zj / > ı. k k k For n ˚D 1; 2; : : : and k D 1; 2; : : : select zn 2 D with ı1 .zn ; zn / D 2 ı=10. Set k 1 Zk D zn nD1 . To finish the proof we will show: 1. The various sets Zk , the closures of the Zk in X1 , are disjoint. 2. For each k, the set Zk meets P.r/. Fix j; k; j ¤ k. By the previous theorem both Zj and Zk are interpolating sequences. To show Zj [ Zk is an interpolating sequence we need to show that it is separated and that the associated measure satisfies a Carleson measure condition. The separation is a consequence of the separation condition for Z, the choice of the fzsn g for s D j:k, and the triangle inequality. The Carleson measure condition follows from the fact that Zj [Zk D Zj C Zk and that the two measures on the right are Carleson measures. The fact that Zj [Zk is an interpolating sequence insures that there is an m 2 M1 such that mjZs D s for s D j; k. It must then follow that, for s D j; k Zs fz 2 D W m.z/ D sg ft 2 X1 W m.t/ O D sg : The containing sets are disjoint for s D j; k; hence the same must be true for the Zs . To prove (2) we pick j , and we want to show Zj meets P.r/. We have P.r/ D

[

O < 1g : ft W m.t/

m2M1 ;kmkD1;m.r/D0 O

Hence P.r/ is a union of open sets and thus open. Thus the complement, P C D X1 nP.r/, is compact.


197

For each t 2 P C we can find mt 2 M1 with kmt k D 1; m ct .r/ D 0 and mt .t/D1. Thus, for small positive ", the sets Ut D fr W jc mt .r/j > 1 "g form an open cover of ˚ N P C and hence, noting compactness, we can find a finite subcover Utj j D1 . Thus, for each j; mtj .r/ D 0 and, hence, after passing to a subnet if necessary, we have, for each j ,

b

lim mtj .zk / D 0: k

On the other hand, if Zj does not meet P.r/, then every subnet n o of Zj has an j accumulation point in P C and hence there is a cofinal subset z whose closure ˚ meets one of the sets Utj . Hence, for the associated mtj we have ˇ ˇ ˇ ˇ lim sup ˇmtj .zj /ˇ > 1 ":

Finally, our construction of Zj insures that ˇ nˇ o 1 ˇ j ˇ j sup ˇm.zk /ˇ W m 2 M1 ; kmk D 1; m.zk / D 0 ı1 zk ; zk D 2j : 2 Because " is at our disposal, the last three estimates are not compatible. This contradicts the suggestion that Zj does not meet P.r/ and hence completes the proof.

6 Comments of Generality Many details of the proofs of the previous results are valid, essentially without change, in more general circumstances; others are not. In this section we offer some brief comments on that topic.

6.1 On Section 4 The results in Sect. 4 used the facts that the Hilbert spaces being considered had a CNPK and the domination property (DOM). This insured access to a convenient characterization of interpolating sequences, Proposition 6. However, other tools were also used in Sect. 4 which are much less generic. In particular Theorem 2 which insured that certain Blaschke products were in the multiplier algebra, and Proposition 7 which insures that, in some cases .BZ / > 0, both involve knowledge about Blaschke products. The proof we offered of

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Theorem 2 used Lemma 1, which is also quite specific to the particular spaces considered. An alternative approach to Theorem 2, using the approach in [V, Theorem 1], although quite different, is again specific to the spaces considered. These comments suggest that the results in that section may be tied to the theory of Blaschke products and might be very specific to Hilbert spaces of functions on the unit disk, having particular additional properties. We do not know if this is so. It would be interesting to obtain these, or similar, results without the use of Blaschke product.

6.2 On Section 5 In Theorem 4, we showed that certain classical constructions will fail for the Dirichlet space. This followed quite easily once we had Lemma 2. Straightforward computations show that the analog of that lemma does not hold ı˛ ; 0 ˛ < 1. In the other direction, it is an immediate corollary of the lemma that an analogous lemma holds for any ı for which ı . ı1 . Little is known beyond that. There were two positive results in the section; a result on perturbation of interpolating sequences, Theorem 6, and Theorem 5 which showed that certain parts had infinitely many points. These results again used a CNPK and (DOM). However these results did not use other information specific to the spaces and hence both results hold in greater generality. Large classes of spaces are known to have a CNPK. The condition (DOM) is less well understood. However there are various cases where (DOM) is known to hold, for instance the Dirichlet type spaces as well as some of their multivariate analogs, the spaces B2 .Bn / described on in [S, pg. 1, Sec. 5.2]. In all of those cases knowing that (DOM) holds is based on computational estimates using explicit formulas for the reproducing kernels. The general question of what abstract properties of H might insure (DOM) has not been studied.

7 A Question Our basic theme has been that classical results about H 1 and its spectrum exhibit interesting interplay between functional analytic and classical analytic ideas; and that those results and the associated interplay suggest questions about the M˛ and X˛ . We developed that theme in the context of analytic structure in the spectra. Here we give another, slightly different, instance of a result that fits that pattern; a classical result about M0 and X0 which again suggests interesting open questions about the M˛ and X˛ . Here is a classical result from [Sc, pgs. 742–744]; we follow the presentation in [Ho1, pgs. 165–166] and refer there for more detail.


199

For each with jj D 1 there is a subset of X0 called the fiber over , denoted by F , which is defined by F D zO1 ./: That is, F is the set on which, zO, the Gelfand transform of the coordinate function z, takes the value . We focus on D 1 and define the sets F1C D

[

F ;

0<arg 0;

8z 2 D;

(C)

kD1

then there exists g 2 H`1 2 solving the Bezout equation, g T .z/f .z/ WD

1 X

gk .z/fk .z/ 1

8z 2 D:

(B)

kD1

Moreover, the function g can be chosen to satisfy (for ı 1=2) kg.z/k

`2

C

1 1 log ı2 ı

8z 2 D;

(1)

with C an absolute constant. Note also that the above condition (C) is necessary for the existence of the function g appearing in (B). Condition (C) is usually called the Carleson Corona Condition (or simply the Corona Condition). Condition (B) is named so after the Bezout equation. The main result of this paper is that if the Corona data f depends smoothly on a parameter s, then one can find a solution g also depending smoothly on the parameter s.

1.2 Preliminaries To give the precise statements we need to introduce some notation.

1.2.1

Function spaces C r .K I H 1 / and H 1 .C r .K //

Let K be a compact Hausdorff space. We say that a function f W D K ! C belongs to C .KI H 1 / if f . ; s/ 2 H 1 for any s 2 K and the function s 7! f . ; s/ is a continuous function on K (with values in the Banach space H 1 ). Since K is compact, the above continuity simply means that the function f is uniformly continuous in the variable s 2 K on D K. Thus the space C .KI H 1 / can be identified with the space of functions f W D K ! C which are analytic in the variable z 2 D and uniformly continuous on D K in the variable s 2 K. In other words, for any " > 0 there exists ı > 0 that for all z 2 D and for all s; s 0 2 K the inequality jf .z; s/ f .z; s 0 /j < " holds whenever js s 0 j < ı.

Corona Solutions Depending Smoothly on Corona Data

203

We can also define the space H 1 .C.K// as the space of functions f W D

K ! C such that the function z 7! f .z; / is a bounded analytic function with values in C.K/. Since for functions with values in a Banach space weak and strong analyticity coincide, see for example [3, Theorem 3.11.4], the space H 1 .C.K// can be identified with the collection of bounded functions f W D K ! C which are analytic in the variable z 2 D and continuous in the variable s 2 K. If K has a smooth structure, for example if K is a compact subset of Rd or a smooth manifold such that K D Clos .Int K/, we can define smooth versions of these function spaces. The spaces C r .KI H 1 / ; r 2 N[1 is the space of functions f W D K ! C such that the vector-valued function s 7! f . ; s/ 2 H 1 has all its partial derivatives (in the strong sense) up to order r existing on Int K and are continuous up to the boundary. We will identify the function with the derivative of order 0, so the function itself is continuous up to the boundary. Similarly, one can say that C r .H 1 / consists of functions f W D K ! C which are analytic in the variable z 2 D and all the partial derivatives in the variable s 2 K up to order r exist and are uniformly continuous in s on D Int K. We will also consider vector-valued versions C r KI H`1 of these spaces, 2 consisting of all functions f W D K ! `2 such that the function s 7! f . ; s/ is a 1 C r function with values in the vector-valued space H`1 with values in `2 . 2 , i.e. H 1 r We can also define the space of functions H .C .K//, which is the space of functions f W D K ! C which are analytic in the variable z 2 D and such that all partial derivatives in s up to the order r exist on D Int K, are continuous and bounded there, and for all z 2 D extend continuously to the boundary of K. Again, in a similar manner, we also consider the vector-valued version H 1 C`r2 .K/ of such spaces, consisting of all functions f W D K ! `2 such that the function z 7! f .z; / is a bounded analytic function with values in the vector-valued space C`r2 .K/, i.e., C r .K/ with values in `2 . Note, that the only difference between the spaces C r .KI H 1 / and H 1 .C r .K// is that in the former we require the uniform continuity in s on D K of the derivatives in s of order r, and in the latter not.

1.2.2

Domains with uniformly continuous path metric

For an open connected subset ˝ of a smooth manifold in addition to the metric inherited from the manifold we can define the path metric p .x; y/ as the infimum of the lengths of the paths in ˝ connecting x and y. Clearly p , and p is continuous with respect to . We are interested in domains where the path metric is uniformly continuous, because for such domains H 1 .C r .K// C r1 .KI H 1 /. An example of a domain with uniformly continuous path metric is the so-called weakly uniform domains. We say that a domain ˝ (in Rd or in a smooth manifold) is weakly uniform if there exists M < 1 such that any two points x1 ; x2 2 ˝ can be connected by a smooth path in ˝ of length at most M dist.x1 ; x2 /. For example, a bounded domain

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S. Treil and B.D. Wick

with a smooth boundary is definitely a weakly uniform domain. If Int K is a weakly uniform domain, then an easy application of the Mean Value Theorem shows that H 1 .C r .K// C r1 .KI H 1 /.

1.3 Main Results The first theorem gives us smooth dependence of the solution of the Corona Bezout equation on a parameter. If r 1 we assume that K is a compact subset of Rd or of a smooth manifold, such that K D Clos .Int K/. For r D 0 we only assume that K is a compact Hausdorff space. r Theorem 1. Let r 2 ZC and let f D .fk /1 KI H`1 be such that 2 kD1 2 C 1

1 X

jfk .z; s/j2 ı 2 > 0

8.z; s/ 2 D K:

kD1

r Then there exist g D .gk /1 KI H`1 such that 2 kD1 2 C g T .z; s/f .z; s/ WD

1 X

fk .z; s/gk .z; s/ 1

8.z; s/ 2 D K:

kD1

Moreover, kg.z; s/k`2 WD

1 X

!1=2 jgk .z; s/j2

2C.ı/ < 1;

(2)

kD1

where C.ı/ is the estimate in the Corona Theorem for H 1 , and kgk

C ˛ KIH 1 2 `

C ˛; !f1

1 ; K kf k ˛ C KIH 1 2C.ı/ `2

8˛ 2 ZC ; ˛ rI (3)

here !f is the modulus of continuity of f , and !f1 is its inverse. Note that for r D 0 the estimate (3) is superfluous since it follows immediately from (2). The next theorem can be interpreted as the Corona Theorem for the algebra H 1 .C r .K// ; r 1. We assume here that K is a subset of a smooth manifold, K D Clos .Int K/ such that the path metric on K is uniformly continuous (with respect to the metric inherited from the manifold).


205

r 1 C`2 .K/ be such that Theorem 2. Let r 2 N [ 1 and let f D .fk /1 kD1 2 H 1

1 X

jfk .z; s/j2 ı 2 > 0

8.z; s/ 2 D K:

kD1

r 1 Then there exist g D .gk /1 C`2 .K/ such that kD1 2 H g T .z; s/f .z; s/ WD

1 X

fk .z; s/gk .z; s/ 1

8.z; s/ 2 D K:

kD1

Moreover, kg.z; s/k`2 2C.ı/ < 1

8.z; s/ 2 D K;

(4)

where C.ı/ is the estimate in the Corona Theorem for H 1 , and kgk

H 1 C ˛2 .K/

C ı; ˛; K; kf k

C 12 .K/

kf k

`

`

H 1 C ˛2 .K/

8˛ 2 ZC ; ˛ r:

`

(5)

2 The Proofs Proof (Proof of Theorem 1). Let f and g be the column vectors with entries fk and gk respectively. By the Carleson Corona Theorem for infinitely many functions there exists a constant C.ı/ such that for any f 2 H`1 2 satisfying 1 jf .z/j ı > 0

8z 2 D;

T there exists g 2 H`1 2 ; kgk H 1 C.ı/ such that g .z/f .z/ 1 for all z 2 D. `2

Here is the main idea of how the proof will proceed. We will use this above for each function f . ; s/. However, since s 7! f . ; s/ is continuous we will be able to construct a perturbation of the solutions that remains close to 1 on some open neighborhood. Then, using a partition of unity argument, we can construct the desired function, which is again close to 1, but now for all s and all z. Since s 7! f . ; s/ is continuous, for each point s 2 K let Us be a neighborhood of s such that for all s 0 2 Us f . ; s/ f . ; s 0 /

H1 2 `

1 : 2C.ı/

206


If gs . / 2 H`1 solves the Bezout equation gsT . /f . ; s/ 1 satisfying 2 kgs . /kH 1 C.ı/, then clearly for s 0 2 Us `2

1 g T . /f . ; s 0 /

H1

s

1 : 2

(6)

Since, by assumption, K is compact, we can take a finite cover Uk WD Usk ; k D S 1; 2; : : : ; N such that K N subordinated kD1 Uk , and let k be a partition of unity P to this covering. This means that 0 k 1; supp k Uk and N 1 k .s/ 1. Note, that if K has a smooth structure then one can take k 2 C 1 .K/. Moreover, it is possible to construct a subcover Uk and the C 1 .K/ partition of unity k such that for all ˛ 2 N, N X

kk k C ˛ .K/ C ˛; !f

kD1

1 ;K : 2C.ı/

(7)

For each k let gsk . / be a solution of the Bezout equation gsTk . /f . ; sk / 1 satisfying kgsk . /k H 1 C.ı/. Define `2

g.z; Q s/ WD

N X

k .s/gsk .z/:

kD1

Then (6) clearly implies that for all s 0 2 K 1 gQ T . ; s 0 /f . ; s 0 /

H1

1 : 2

(8)

Indeed, for k such that k .s 0 / > 0 we have s 0 2 Usk , so the estimate (6) holds for Q ; s 0 / is a convex combination of such gsk . /, s D sk . But for any s 0 the function g. so we get (8) as a convex combination of estimates (6). T 0 0 Inequality (8) implies that the (scalar) function gQ . ; s /f . ; s / is invertible in H 1 and that .gQ T . ; s 0 /f . ; s 0 //1 H 1 2. Therefore the function g WD .gQ T f /1 gQ solves the Bezout equation g T f 1 and satisfies (2). Computing the derivatives of g in s and taking into account (7) we easily get (3). Proof (Proof of Theorem 2). As we mentioned above in Sect. 1.2.2, if the path metric p is uniformly continuous then H 1 .C r .K// C r1 .KI H 1 / ;


207

for and the same for the `2 -valued case. In particular, H 1 C`r2 .K/ C KI H`1 2 all r 1. Since s 7! f . ; s/ is continuous, we can construct the partition of unity k as in the proof of Theorem 1 above. Note that if the path metric p is uniformly continuous, the modulus of continuity !f can be estimated via the modulus of continuity of p and !f

1 2C.ı/

!p

1 kf k 1 : C2 2C.ı/ `

where !p is the modulus of continuity of p . The rest follows as in the proof of Theorem 1.

3 Some Remarks The method of the proof allows us to get smooth dependence on the parameter for the Corona type problems in very general situations, like in the matrix and operator Corona Problem, as well as in the Corona Problem for multipliers of the reproducing kernel Hilbert spaces. Below, we state some sample results. The proofs are similar to what was presented in Sect. 2 and are left as an exercise for the reader. Let X and Y be Banach spaces, and let A D A .˝I B.X; Y // and B D B .˝I B.Y; X // be some Banach spaces of functions on some set ˝ with values in B.X; Y / (the space of bounded operators from X to Y ) and in B.Y; X / respectively. As in Sect. 1.2 we can introduce the spaces C r .KI A/ ; C r .KI B/, where K is a compact set. The space C r .KI A/ consists of all functions f W ˝ K ! B.X; Y / such that the function s 7! f . ; s/ is an r times continuously differentiable function on K, and similarly for C r .KI B/. As in Sect. 1.2 we assume that K is a Hausdorff compact space if r D 0, and if r 1 we assume that K is a compact subset of a smooth manifold such that K D Clos .Int K/. We also assume that BAB B, i.e. that BA belongs to the multiplier algebra of B. This implies that for f 2 A and g; h 2 B kgf hk B C.A; B/ kgk B kf k A khk B :

(9)

A typical example here would be multiplier spaces of (vector-valued) function spaces. Let for example X and Y be spaces of functions on ˝ with values in X and Y respectively. Define A and B as the multiplier function spaces between X and Y . Namely, A is the space of operator-valued functions ' W ˝ ! B.X; Y / such that the map f 7! 'f is a bounded operator from X to Y ; similarly, the space B will be the collection of functions ' W ˝ ! B.Y; X / such that the map f 7! 'f is a bounded operator from Y to X .

208


For a concrete example, if X D H 2 .X / and Y D H 2 .Y / then A D H .B.X; Y // and B D H 1 .B.Y; X // and we are in the situation of the operator corona. This situation with X D `2 and Y D C gives us the settings of Theorem 1. Another example would be the spaces of multipliers of Bergman or Dirichlet type spaces (in the disc or more general domains in C or Cn ). But in general we do not need to assume that the spaces A and B are multiplier spaces. 1

Theorem 3. Under the above assumptions, let f 2 C r .KI A/ be such that kf . ; s/kA 1 for all s 2 K. Suppose that for all s 2 K there exists gs 2 B such that gs .z/f .z; s/ I

8z 2 ˝;

kgs kB C0 :

Then there exists g 2 C r .KI A/ such that g.z; s/f .z; s/ I

8.z; s/ 2 ˝ K;

and

kg. ; s/kB 2C0 :

Moreover, for all ˛ 2 ZC ; ˛ r we have the estimate kgkC ˛ .KIB/

1 C ˛; !f

1 ; K kf kC ˛ .K;B/ : 2C0 C.A; B/

Here, again, !f is the modulus of continuity of f and C.A; B/ is the constant from (9). Remark. Note that in the above theorem we do not assume any kind of “Corona Theorem” for the spaces A and B; we just postulate the solvability of the Bezout equations for each s 2 K. The proof of Theorem 3 goes along the lines of the proof of Theorem 1. We construct a cover Us ; s 2 K of K such that f . ; s/ f . ; s 0 /

A

1 2C0 C.A; B/

8s 0 2 Us ;

take a finite subcover and construct a partition of unity k D sk , then finally construct the function g, Q etc. Estimate (9) implies that instead of (8) we have the estimate kI g. Q ; s 0 /f . ; s 0 /kM .A/

1 ; 2

where M .A/ is the (left) multiplier algebra of A. But that implies that 1 k g. Q ; s 0 /f . ; s 0 / kM .A/ 2 so all the estimates in Theorem 3 follow.


209

Acknowledgements Research of Sergei Treil supported in part by a National Science Foundation DMS grant #0800876. Research of Brett Wick supported in part by National Science Foundation DMS grant #955342. This note began at the workshop “The Corona Problem: Connections Between Operator Theory, Function Theory and Geometry.” The authors thank the Fields Institute for the pleasant working conditions.

References 1. Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math., 76, 1962, 547–559. 2. John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. 3. Nikolai K.Nikol’ski˘ı, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz, Translated from the French by Andreas Hartmann, Translated from the French by Andreas Hartmann. 4. Nikolai K.Nikol’ski˘ı, Treatise on the shift operator, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hrušˇcev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. 5. Paul A. Fuhrmann, On the corona theorem and its application to spectral problems in Hilbert space, Trans. Amer. Math. Soc., 132, 1968, 55–66. 6. Marvin Roseblum, A corona theorem for countably many functions, Integral Equations Operator Theory, 3, 1980, 125–137. 7. V. A. Tolokonnikov, Estimates in the Carleson corona theorem, ideals of the algebra H 1 , a problem of Sz.-Nagy, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)„ 113, 1981, 178–198, 267 (Russian, with English summary). Investigations on linear operators and the theory of functions, XI. 8. Sergei Treil and Brett D. Wick, Analytic projections, corona problem and geometry of holomorphic vector bundles, J. Amer. Math. Soc., 22, 2009, 55–76. 9. Sergei Treil and Brett D. Wick, The matrix-valued H p corona problem in the disk and polydisk, J. Funct. Anal., 226, 2005, 138–172. 10. A. Uchiyama, Corona Theorems for Countably Many Functions and Estimates for their Solutions, 1980, preprint.

On the Taylor Spectrum of M -Tuples of Analytic Toeplitz Operators on the Polydisk Tavan T. Trent

Abstract We show that if F D .f1 ; : : : ; fm /, where fi is in H 1 .D n / and if Tfi denotes the analytic Toeplitz operator acting on H D H p .D n / for 1 < p < 1, then T ..Tf1 ; : : : ; Tfm /; H / D ran F .D n /

Cm

:

Here T denotes the Taylor spectrum. Keywords Corona theorem • Polydisk • Taylor spectrum Subject Classifications: 30H05, 32A35 When ˝ is equal to the unit ball or even a strictly pseudoconvex domain in Cn , Andersson and Carlsson have computed the Taylor spectrum for analytic Toeplitz operators acting on H p .@˝/. See [AC] for the precise results and further references. Of course, when the dimension is greater than one, bounded analytic solutions have not been found in any of the above cases for the general corona problem; otherwise the computation of the Taylor spectrum would be trivial. For the case of a pair of analytic Toeplitz operators acting on H p .D n /, the computation of the Taylor spectrum for the pair was done in Putinar [P]. Given a finite number of input functions, Li [L] and, independently, Lin [Li] implicitly solved the H p .D n / corona theorem .1 p < 1/ based on the work of Lin. Again, for a finite number of input functions, Boo [B] gave an explicit solution to the H p .D n / corona theorem .1 p < 1/, which was based on integral

T.T. Trent () Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, AL 35487-0350, USA e-mail: [email protected] R.G. Douglas et al. (eds.), The Corona Problem, Fields Institute Communications 72, 211 DOI 10.1007/978-1-4939-1255-1__11, © Springer Science+Business Media New York 2014

212

T.T. Trent

formulas. For all of these results, the size of solutions depended on the number of input functions. Our estimates are independent of the number of input functions. For general corona data on the polydisk, it was shown in Trent [T2] that corona solutions exist, which, although not bounded or even in BMOA, still belong to T 1 p n 2n1 /. a space better than 1 pD1 H .D /; namely the Orlicz-type space, exp.L p Moreover, the vector-valued H -corona theorem on the polydisk was established in the above context. In this paper we modify those techniques and compute the Taylor spectrum of m-tuples of analytic Toeplitz operators acting on H p .D n / for 1 < p < 1. We will use the H p .D n / corona theorem in the vector-valued case. This establishes the last node in the computation of the Taylor spectrum for an m-tuple of analytic Toeplitz operators. The main technique involves constructing integral operators based on considering explicit mappings, arising from the Koszul complex. Our approach is to find candidates for the appropriate solutions to this problem in the smooth case and then provide estimates for them. To find these candidates, we use the algebra of the Koszul complex and are left with constructing appropriate integral operators. The integral operators to be constructed are not unique and are easily given by appropriate matrices. This is the main reason for our matricial approach. The basic idea for the corona estimates involves iterating the one-variable Littlewood-Paley results, motivated by T. Wolff’s proof of Carleson’s corona theorem on the unit disk. (See Garnett [G].) However, to simplify the estimates we will appeal to the remarkable H 1 .D n / weak factorization theorem of LaceyTerwilleger [LT]. A careful reading of our proof shows that these estimates are independent of the number of input functions ffj gm j D1 , but depend on the dimension of the polydisk, n. We will use the following notation: open unit disk in the complex plane, C

D

unit circle, T D @ D

T D

polydisk, D n D D D; n times

n

distinguished boundary of D n ; T n D T T;

Tn

n times d

normalized Lebesgue measure on Œ;

dk

denotes d 1 .t1 / : : : d k .tk /

dA

area Lebesgue measure on D

dL

measure on D defined by dL.z/ D ln denotes dL1 .z1 / : : : dLk .zk /

d Lk p

1 d A.z/ jzj2

n

H .D /

Hardy space of analytic functions on D n ; 1 p 1

Taylor Spectrum of M -Tuples

213

[We will also identify this space with ff 2 Lp .T n / j for fkj gnj D1 Z with at least one kj < 0; Z f .e it1 ; : : : ; e itn /e ik1 t1 : : : e ikn tn d .t1 / : : : d .tn / D 0g Tn

L .T / p

n

ff W T n ! l 2 j f is strongly measurable and Z def p kf .e it1 ; : : : ; e itn /kl 2 d .t1 / : : : d .tn / < 1g j f j pp D Tn

for 1 p < 1 L 1 .T n /

ff W T n ! l 2 j f is strongly measurable and def

jf j1 D ess H p .D n /

sup

u1 ;:::;un 2T

kf .u1 ; : : : ; un /kl 2 < 1g

ff W D n ! l 2 j f is analytic; l 2 -valued on D n and Z def p kf .r e it1 ; : : : ; r e itn /kl 2 d .t1 / : : : d .tn // j f j pp D sup. r"1

Tn

< 1g for 1 p < 1 H 1 .D n /

ff W D n ! l 2 j f is analytic; l 2 -valued on D n and def

jf j1 D O j .u/

sup kf .z1 ; : : : ; zn /kl 2 < 1g

z1 ;:::;zn 2D

the jth Cauchy transform of a (possibly l 2 -valued) C .1/ function on D

n

1 O j .u1 ; : : : ; ' ; : : : ; un / D z jth

TF

Z D

jth

.u1 ; : : : ; z ; : : : ; un / dA.w/ wz

Toeplitz operator with symbol F acting on H p .D n / for any 1 p < 1

F

the operator of pointwise multiplication by the matrix it1 itn p n Œfj k .e it1 ; : : : ; e itn / 1 j;kD1 D F .e ; : : : ; e / on L .T /

Q.z1 ; : : : ; zn /

the operator on l 2 gotten by applying the matrix Q.z1 ; : : : ; zn / D Œfj k .z1 ; : : : ; zn / 1 j;kD1 to the standard basis of l 2 : Here fj k 2 H 1 .Dn /:

T .T; H /

the Taylor spectrum of the m-tuple T D .T1 ; : : : ; Tm / for Ti Tj D Tj Ti in B.H /; where H is a Banach space.

214

T.T. Trent

We begin with a brief outline of the Taylor spectrum. Following Curto [Cu], let n Œe denote the exterior algebra on n generators e1 ; : : : ; en with e0 D 1 and ei ^ ej C ej ^ ei D 0. Let Ei W n Œe ! n Œe be defined by Ei x D ei ^ x. So Ei Ej C Ej Ei D 0. For H a Banach space and T D .T1 ; : : : ; Tm / a commuting m-tuple of bounded operators on H , we define DT W H ˝ n Œe ! H ˝ n Œe by DT .h ˝ x/ D

n X

Tj h ˝ Ej x:

j D1

Since DT2 D 0, we have ran DT ker DT . Then the Taylor spectrum of T on H is defined by T .T; H / D f 2 Cm W ran DT ¤ ker DT g: Taylor [Ta] showed that T .T; H / is nonempty, compact and contained in T0 .T /, the joint algebraic spectrum of T with respect to the commutant of T . In addition, T .T; H / satisfies, for pi polynomials in m variables: p D .p1 ; : : : ; pk / W Cm ! Ck ..p1 .T /; : : : ; pk .T //; H / D p. ..T1 ; : : : ; Tm /; H // and ..T1 ; : : : ; Ts /; H / D ProjCs . ..T1 ; : : : ; Tm /; H // 2n

n

for all permutations. We identify H ˝ n Œe with ˚ H D H 2 in the natural n

iD1

way. Thus we may consider DT as acting on H 2 . For example, let m D 3 and T D .T1 ; T2 ; T3 / with Ti Tj D Tj Ti . Then, with 3 3 3 the natural identification of H ˝ n Œe with H 2 ; DT W H 2 ! H 2 has the following matrix D with respect to the natural basis: 2

0 6T1 6 6T 6 2 6 6T DT D 6 3 60 6 60 6 40 0

0 0 0 0 T2 T3 0 0

0 0 0 0 T1 0 T3 0

0 0 0 0 0 T1 T2 0

0 0 0 0 0 0 0 T3

0 0 0 0 0 0 0 T2

0 0 0 0 0 0 0 T1

3 0 07 7 2 0 07 7 7 T 07 6 Q 7D6 2 07 4 0 7 07 0 7 05 0

0 0 TQ1 0

0 0 0 TQ0

3 0 07 7: 05 0


215

Here TQj D DT j j Œe , where j Œe is the subspace of n Œe , where all elements have degree j in the basis e. We see that ran TQj C1 ker TQj , i.e. TQj TQj C1 D 0. Then 0 … T ..T1 ; T2 ; T3 /; H / means that the following complex is exact:

0!H

T1 TQ2 D T2 T3

!

! 3

˚H

T2 T1 0 TQ1 D T3 0 T1 0 T3 T2

!

!

1

3

˚H

TQ0 D.T3 ;T2 ;T1 /

!

1

H ! 0:

Pm 1 n 2 2 n Let ffj gm j D1 jfj .z/j > 0 for all z 2 D . Fix j D1 H .D / with 1 < p < 1 and let Tj D Tfj denote analytic Toeplitz operators acting on H D H p .D n / by Tfj .h/ D fj h. Then ..1/m1 Tm ; : : : ; T1 / D ..1/m1 Tfm ; : : : ; Tf1 / is onto. This is the H p .D n / corona ! theorem mentioned in the introduction. Tf1

Moreover, we always have that

:: :

is injective.

Tfm

We are now ready to state our theorem. Fix any 1 < p < 1. Theorem. Let H D H p .T n /. Then T ..Tf1 ; : : : ; Tfm /; H / D ran F .D n /

Cm

.

Notice that ran F .D n /

Cm

D f.f1 .z/; : : : ; fm .z// W z 2 D n gC D f 2 Cm W infn z2D

m X

m

jfj .z/ j j2 D 0g

j D1

./

D f 2 Cm W .Tf1 1 ; : : : ; Tfm m / is not ontog

T ..Tf1 ; : : : ; Tfm /; H / ; where ./ follows from the H p corona theorem on the polydisk. See Boo [B], Li [L], Lin [Li], Trent [T2] and Varopoulos [V]. So we need only show that if .Tf1 ; : : : ; Tfm / is onto, then 0 … T ..Tf1 ; : : : ; Tfm /; H /: As a model of what needs to be shown, consider our previous example with m D 3. We always have that TQ2 is 1-1 and we assume that TQ0 is ont o. We must show that ran TQ2 D ker TQ1 and ran TQ1 D ker TQ0 . So if 0 1 a1 TQ0 a D .T3 ; T2 ; T1 / @a2 A D 0; ai 2 H p .T n /; a3

216

T.T. Trent

then we want 0 1 0 10 1 a1 T2 T1 0 u1 a D @a2 A D @T3 0 T1 A @u2 A D TQ1 u a3 0 T3 T2 u3 where ui 2 H p .T n / and similarly TQ1 r D 0 ) r D TQ2 m; with the entries of m in H p .T n /. We will first state the vector-valued corona theorem. Theorem (H p .D n /-corona Theorem). Let F 2 H 1 .D n / satisfy 0 < 2 F .z/ F .z/ 1 for all z2D n . Then TF acting from H p .D n / to H p .D n / is onto for each 1 1 and 1 j n, Q1 K1 : : : Qj Kj D

ŒQ1 P2 : : : Pn 1 ; K1C .n

C K1 0 1/ Q2 0 K2C .n 1/ #

" 0 KjC1 : : : Qj 0 KjC1 .n 1/

D ŒQ1 P2 : : : Pn 1 .Q2 K1C .n 1/ : : : Qj KjC1 .n 1/; Q1 K1C .n 1/ : : : Qj KjC .n 1/ : By induction on the number of analytic variables, Q1 K1C .n 1/ : : : Qj KjC .n 1/ D ŒB 2˘.j / 1…

and

Q2 K1C .n

1/ : : : Qj KjC1 .n

1/ D ŒC 2˘.j 1/ : 1…

Here B and C denote operators formed from finite products of Ql ’s, Pk ’s, and Pk? ’s for 1 l j and 2 k n. But 1 C D B1; 1 D B ;

for some B

where 2 ˘.j / and 1 2 . To see this, first notice than neither expression Q1 K1C .n 1/ : : : Qj KjC .n 1/ or Q2 K1C .n 1/ : : : Qj KjC1 .n 1/ involves the projection P1 and 1 Pk D Pk 1 for 2 k n. Also, we have X 1 Ql Y D X P1? Ql 1 Y:

(2)

Continuing this procedure with 1 commuting across the Pk ’s and using (2) to move 1 to the right of the Ql ’s, (a) follows. As for (b), first note that Qj FF

D j

Qj Qj 1 F : : : D D Dj Qj1 : : : D2 Q1 D1 F : 1 FF FF .FF /j C1


227

This follows since Ql Ql1 Ql 1 1 X Dl . Dl Y DX /Q Y C Dl Ql1 Y FF FF FF FF l1 FF DX

Ql Dl Ql1 Y; using Ql Ql1 D 0: .FF /2

Now Dj Qj1 : : : D1 F D .Dj Qj1 / : : : .D1 F / is formally (see [T1]) the same as 1 1 0 @1 Qj1 @1 F B : C B :: C @ : A ^ ^ @ :: A 0

@n Qj1 X

D

2˘.j / D.i1 ;:::;ij /

D jŠ

X

2 4

@n F X

3

.1/sgn ˛ @˛.ij / Qj1 : : : @˛.i1 / F 5 e

˛2P ./

@ij Qj1 : : : @i1 F e :

2˘.j /

This last equality comes from the fact that @˛.ij / Qj1 : : : @˛.i1 / F D .@˛.ij / F / ^ ^ .@˛.i1 / F / D .1/sgn ˛ @ij F ^ ^ @i1 F D .1/sgn ˛ @ij Qj1 : : : @i1 F : Thus, Qj FF

Dj

Qj1 FF

Qj F : : : D1 D jŠ @i Q : : : @i1 F FF .FF /j C1 j j 1

2˘.n/ D.i1 ;:::;in /

as an jn 1 vector. Consider elements from B , which from (a) involves finite products using only j elements from fQl glD1 ; fPk gnkD1 , and fPk? gnkD1 . Replace any Pk? ’s by I Pk and write B as a finite sum of terms containing no Pk? ’s. Consider a term in B of the form X Pk S Pk Y . Then Pk Y is analytic in the kth variable. Since S involves Ql ’s which are analytic and Pl ’s, we have S Pk Y is analytic in the kth variable. Thus X Pk S Pk Y D X S Pk Y . Using this procedure, we may assume that in each term of B at most one occurrence of Pk for 1 k n appears. Thus (c) follows. We note that in Lemma 10 (a) can be replaced by

228

T.T. Trent

Qj C2 K1 : : : Qj ClC1 Kl D ŒB 2˘.l/ ; where B is a finite product of operators belonging to fQs ; Pk ; Pk? W j C 2 s l C j C 1; 1 k ng. Also, (b) can be replaced by " ElCj C1 Dl : : : Ej C2 D1 Ej C1 a D lŠ

QlCj C1

@i Q .FF /lC1 l lCj

# : : : @i1 QjC1 a

: 2˘.l/ D.i1 ;:::;il /

We are now ready to complete the proof for the computation of the Taylor spectrum of TF . Cm

Proof. Suppose that 0 … ran F .D n / . Then we may assume that for some > 0, we have 2 F .z/F .z/ 1 for all z 2 D n . Form E0 .z/ D

Ql .z/ F .z/ and E .z/ D for l D 1; : : : : l F .z/F .z/ F .z/F .z/

Then by Lemma 3, the Q’s and E’s satisfy the algebraic equations that enable us to invoke Lemma 9. Suppose that a 2 H p .D n / and TQj a D 0. So Qj .z/a.z/ D 0 for all z2D n . We lose no generality (by considering Fr .z/ D F .rz/ and ar .z/ D a.rz/) in assuming that F and a are smooth across @D n . Then we must show that our estimates are independent of r and apply a compactness argument to complete the proof. Let u.r/ D Ej C1 a C

n X .1/k Qj C2 K1 : : : QkCj C1 Kk Ej CkC1 Dk : : : Ej C2 D1 Ej C1 a; kD1

where the Ej ’s and Qj ’s are evaluated at z 2 D n . We use the subscript “.r/” to remind us that all the terms are defined using Fr in place of F and ar in place of a. Fix 1 < p < 1. Now by Lemma 10, u.r/ D

n X kD1

.1/

k

X 2˘.k/ D.i1 ;:::;ik /

" kŠ B

QjCkC1

@i Q .FF /kC1 k j Ck

# : : : @i1 QjC1 a

;

where D i1 ;:::;ik . By Lemma 11, B is a finite sum (of at most nŠ terms) of finite products of contractions Ql and of at most n projections among fPk gnkD1 . Now as an operator on L p .T n /, for 1 < p < 1; Pk has norm kPk k ˛.p/. [See Garnett [G] for this fact on Lp .T /.] Thus kB kB.L p .T n // C0 ˛.p/n ;


229

where C0 is independent of r and p (but depends on n). Estimating, we get 1

0 n X

X

kD1

2˘.k/ D.i1 ;:::;ık /

B j u.r/j p B @

QjCkC1 kŠkŒ @i Q .FF /kC1 k j Ck

C n : : : @i1 QjC1 a kL p .T n / C A C0 ˛.p/ :

We estimate the worst term; the others follow more easily. Let " lD

#

QjCnC1

@n QjCn .FF /nC1

: : : @1 QjC1 a

:

We will show that j l 1 ;:::;nj p

C1 ˛.p/n

4nC2

to complete the proof. By duality, since L p .T n / L q .T n / for 1 p < 1 (see Edwards [E], p. 607) and l 1 ;:::;n D P1? : : : Pn? l 1 ;:::;n , we have j l 1 ;:::;nj p D

sup

jhl 1 ;:::;n ; kij

k2L q .T n / j kjjq 1

D

sup k2L q .T n /

jhl 1 ;:::;n ; P1? : : : Pn? kij

j kjjq 1

sup H0 2z1 :::zn H q .T n / j H0j q ˛.p/n

jhl 1 ;:::;n ; H 0 ij:

Applying Lemma 1 n times, we must estimate Z hl

1 ;:::;n

; H 0i D

@1 : : : @n hl; H 0 i dL1 : : : dLn Z

Dn

D

@1 : : : @n h Dn

D

1 Z X lD1

QnCj C1

.FF /nC1

@1 : : : @n h Dn

@n QnCj : : : @1 QjC1 a; H 0 i d Ln

QnCj C1

.FF /nC1

@n QnCj : : : @1 QjC1 el ; el i al hl d Ln

(3) for H0 2 z1 : : : zn H 1 .T n / and j H0j 1 ˛.p/n .

230

T.T. Trent

n Since QnCj C1 @n QnCj : : : @1 Qj C1 is co-analytic in D , all @1 ; : : : ; @n derivatives

in (3) apply to either .FF 1/nC1 or to al hl . Let f1; : : : ; ng D I [ J . If J D fj1 ; : : : ; jp g, let @J denote @j1 : : : @jp and similarly for I . Then we must estimate sums of terms of the form ˇZ ˇ ˇ ˇ 1 ˇ ˇ ˇ n h@I . .FF /nC1 /Qj CnC1 @n Qj Cn : : : @1 Qj C1 el ; el i@J .al bl / d Ln ˇ D Z 1 k@I . /k kQjCnC1 kop k@n QjCn kop : : : k@1 QjC1 kj@J .al hl /j d Ln /nC1 op n .FF D Z C3 4nC2 jal hl j d n :

Tn Recall that k@k Qj .z/kop kFk .z/kl 2 , for z 2 D n ; k D 0; : : : ; n and j D 1; : : : . So the last inequality follows from Lemma 7. Thus, applying Holder’s R inequality twice 1 Z C3 X jhl 1 ;:::;n ; H 0 ij 4nC2 jal hl j d n n

lD1 T Z C3 4nC2 kakl 2 kH0 kl 2 d n

Tn

C3 4nC2

C3 ˛.p/n j ajjp :

4nC2

j ajjp j H0j q

In summary, we have shown that if a 2 H p .D n / and TQj a D 0 and 0 < r < 1 is fixed, then there exists u.r/ 2 H p .D n / with T.Qj C1 /r .u.r/ / D ar and

j u.r/j p

C3 ˛.p/2n j ar j p

4nC2

C3 ˛.p/2n j ajjp :

4nC2

Now, since 1 < p < 1, a routine compactness argument shows that there exists a u 2 H p .D n / with TQj C1 u D a and with the same norm estimates. This completes the proof of the Theorem.


231

References [AC] M. Andersson and H. Carlsson, Estimates of solutions of the H p and BMOA corona problem, Math. Ann. 316 (2000), 83–102. [B] J. Boo, The H p corona theorem in analytic polyhedra, Ark. Mat. 35 (1997), 225–251. [Ch] S. A. Chang, Carleson measures on the bi-disc, Annals of Math. 109 (1979), 613–620. [Cu] R. Curto, Applications of several complex variables to multiparameter spectral theory, Surveys of some recent results in operator theory, Vol.II, Pitman Res. Notes Math. Soc. 192 (1988), 25–90. [E] R. E. Edwards, Functional Analysis, Theory and Applications, Dover Pub., New York, 1995. [G] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. [K] P. Koosis, Introduction to H p Spaces, Cambridge University Press, New York, 1980. [LT] M. Lacey Nn and E. Terwilleger, Hankel operators in several complex variables and product BMO. 1 CC /, Houston J. Math. 135 (2009), no.1, 159–183. [L] S.-Y. Li, Corona problems of several complex variables, Madison Symposium of Complex Analysis: Contemporary Mathematics, vol. 137, Amer. Math. Soc., 1991. [Li] K. C. Lin, The H p -corona theorem for the polydisc, Trans. Amer. Math. Soc. 341 (1994), 371–375. [N] N. K. Nikolski, Treatise on the Shift Operator, Springer-Verlag, New York, 1985. [P] M. Putinar, On joint spectra of pairs of analytic Toeplitz operators , Studia Math. 115 (1995), 129–134. [S] E. M. Stein, Harmonic Analysis, Princeton University Press, Princeton, New Jersey, 1993. [Ta] J. Taylor, A joint spectrum for several commuting operators, J. Func. Anal. 6 (1970), 172–191. [T1] T. T. Trent, A new estimate for the vector-valued corona theorem, J. Func. Anal. 189 (2002), 267–282. 1 [T2] T. T. Trent, Solutions for the H 1 .D n / corona problem belonging to exp.L 2n1 /, Recent Advances in Matrix and Operator Theory 179 (2008), 309–228. [V] N. Th. Varopoulos, Probabilistic approach to some problems in complex analysis, Bull. Sci. Math. 105 (1981), 181–224.

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