LONDON MATHEMATICAL SOCIETY STUDENT TEXTS
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LONDON MATHEMATICAL SOCIETY STUDENT TEXTS
Managing editor: Professor E.B. Davies, Department of Mathematics, King's College, Strand, London WC2R 2LS
1
Introduction to combinators and ?.-calculus, J.R. HINDLEY &
J.P. SELDIN 2
Building models by games, WILFRID HODGES
Local fields, J.W.S. CASSELS 4 An introduction to twistor theory, S.A. HUGGETT & K.P. TOD 5 Introduction to general relativity, L. HUGHSTON & K.P. TOD 6 Lectures on stochastic analysis: diffusion theory, DANIEL W. STROOCK 7 The theory of evolution and dynamical systems, J. HOFBAUER & 3
K. SIGMUND 8
9
Summing and nuclear norms in Banach space theory, G.J.O. JAMESON Automorphisms of surfaces after Nielsen and Thurston, A.CAS SON &
S. BLEILER 10 Non-standard analysis and its applications, N.CUTLAND (ed) 11 The geometry of spacetime, G. NABER 12 Undergraduate algebraic geometry, MILES REID 13 An Introduction to Hankel Operators, J.R. PARTINGTON
London Mathematical Society Student Texts. 13
An Introduction to Hankel Operators JONATHAN R. PARTINGTON Fellow and Director of Studies in Mathematics, Fitzwilliam College Cambridge
The right of the University of Cambridge
1
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to prim ..d,,11
all - afbooks was granted by Henry V111 in 1334. The University has printed and published conrtnuously since 1384.
CAMBRIDGE UNIVERSITY PRESS Cambridge
New York New Rochelle Melbourne Sydney
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521366113
© Cambridge University Press 1988
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1988 Re-issued in this digitally printed version 2007
A catalogue record for this publication is available from the British Library ISBN 978-0-521-36611-3 hardback ISBN 978-0-521-36791-2 paperback
To my mother, and in memory of my father
CONTENTS
0
Introduction
1
Compact Operators on a Hilbert Space
2
Hardy Spaces Basic Properties of Hankel Operators
3
4 5
Hankel Operators on the Half Plane Linear Systems and H-
6
Hankel-norm Approximation
7
Special Classes of Hankel Operator Appendix
Exercises Bibliography
Index
0. INTRODUCTION In Riemann, Hilbert or in Banach space Let superscripts and subscripts go their ways. Our asymptotes no longer out of phase, We shall encounter, counting, face to face. Stanislaw Lem (The Cyberiad)
We apologise for the fact that in the title of the Tensors talk in the last newsletter, the words "theoretical physics" came out as "impossible ideas". Arehimedeans' Newsletter, January 1986. Many have been led astray by their speculations, And false conjectures have impaired their judgement. Ecclesiasticus 3, 24.
A Hankel matrix is one of the form
a0 a1 a2 ...
a, a2 a3 ... /I
a2 a3 a4 ...
that is, a matrix {(cij):
i, j = 0,
.
oo}, where cij depends only on i+j, so can be written
cij = ai+j, for some sequence a0, a1, a2, ...
Under suitable conditions such a matrix gives rise in a natural way to a linear map
(an
operator) r on the Hilbert space 12 of square summable sequences, and we have that
(rx)i = E0 ai+jxj, for x = (x0, x1, x2,...) e 12. r is a Hankel operator. Similarly, a Hankel Integral Operator on L2(0, oo) has the representation
Fx(t) = c h(t + s) x(s) ds, so that the kernel, h(t + s), depends on the sum of the two variables involved.
As we shall see in more detail later, 12 is isomorphic to the Hardy space H2 of analytic functions on the unit disc flzl < 1): this is the space of all functions f(z) = EQ anzn with norm
IIfl12 = E Ianl2 < O. We thus have a connection between Hankel operators and complex variable theory, which turns out to be very important. Similarly, L2(0, oo) is easily related to
2
another Hardy space, this time of functions defined on the right half plane C+, using the Laplace transform.
Hankel operators have in recent years been shown to have widespread applications to both Systems Theory and Approximation Theory: we explore these here.
In Chapter 1 we start with some general operator theory. Compact operators on Hilbert spaces
can be written in the form
Fx = 171 ai (x, vi) wi,
with al >- a2 2 ...
0, and (vi) and (wi) orthonormal sequences in the given Hilbert space.
The ai are called singular values (approximation numbers, generalised eigenvalues) and have many important properties. For example we can consider what it means to say that E ai < -, or that E ai < 00 (nuclear operators and Hilbert-Schmidt operators).
Hardy spaces are introduced in Chapter 2. For the applications to Hankel operators we are
only concerned with H2, H and (occasionally) Hl, and we give a more elementary discussion than is customary (for example we are able to avoid the use of maximal functions entirely). We also treat Hardy spaces on C+ by considering their equivalence with Hardy spaces on the disc.
Having established the background we are able to introduce Hankel operators in Chapter 3.
Nehari's Theorem and the Carathdodory-Fejdr and Nevanlinna-Pick problems are treated. In addition we establish Hartman's theorem on compact Hankel operators.
Hankel integral operators on L2(0, oo) and their equivalent forms on H2(C+) are discussed in Chapter 4. Most results here are obtained using equivalences with Hankel operators on the disc.
An elementary treatment of linear systems and H,, is presented in Chapter 5. Some infinitedimensional systems (where the associated Hankel operator is of infinite rank) are discussed. Here
we give the physical motivation for Model Reduction - approximation by simpler functions in suitable norms.
3
In Chapter 6 we present Beurling's Theorem and the Adamjan-Arov-Krein results on Hankel-
norm approximation. Here we follow Power's simplified treatment, giving additional proofs, examples and explanations of this rather deep problem.
The final chapter connects the general operator theory of the first chapter with the Hardy space theory. Various results on Hilbert-Schmidt and nuclear Hankel operators are presented,
culminating in the recent results of Peller, Coifman and Rochberg, Bonsall and Walsh, and including various inequalities which give L1 and H,, error bounds for model reduction.
We conclude with an appendix covering various background results in functional analysis which may be unfamiliar to some readers. These include standard results from Operator Theory and Measure Theory, and we give them in their simplest form.
These notes are based on those for a Part III Mathematics course on Hankel operators given
to an audience of Mathematicians and Engineers at Cambridge University in the Michaelmas
Term, 1987. I am grateful to Dr B. Bollobds, Dr T.K. Came, Dr K. Glover, Dr T.W. KOmer,
Mr D.C. McFarlane and Dr R. Ober for useful discussions and comments; also
to the
Departments of Engineering and of Pure Mathematics and Mathematical Statistics of Cambridge
University, to Fitzwilliam College, Cambridge, and to the Science and Engineering Research Council for their assistance.
4
1. COMPACT OPERATORS ON A HILBERT SPACE
In this first chapter, we begin by considering linear operators in general - we specialise to Hankel operators in Chapter 3. Although it is possible to discuss operators defined on a general nonmed space, we shall not do so, but just consider linear operators defined on a complete inner-
product space, a Hilbert space. The properties in which we are interested are of greatest importance when the operator is compact, that is, close to being a finite-rank operator (a formal definition will be given later).
For compact operators which are also Hermitian there is the Spectral Theorem, which shows how the action of the operator is fully determined by its eigenvalues and eigenvectors. From this
we move to the Schmidt expansion of a general compact operator, and come naturally to the definition of the approximation numbers (singular values) of a compact operator (to be denoted (as))-
A brief discussion of the polar decomposition follows: this enables us to refer to the modulus of an operator, itself an operator with several useful properties.
We spend the remainder of the chapter in considering operators of the class Cp (1
_ 0 and (T*Tx, y) = (x, T*Ty), we have T*T >_ 0. Let X1, X2, ... be the nonzero eigenvalues of T*T, ordered in decreasing size, v1, v2, ... the
corresponding eigenvectors (orthonormal), and ai = X112. Now write wi = Tvi/ai. We thus
have (wi, wj) = (Tvi, Tvj)laiaj = (T*Tvi, vj)laiaj = ai(vi, vj)laj = Sid,
i.e.
the (wi) are
orthonormal.
Note that T*Tx = 0 if and only if Tx = 0, so that Tx = Z-1 (x, vi)Tvi = EI ai(x, vi)wi. Also
Tvi = aiwi and T*wi = aivi; 7T*wi = ai wi = Xiw and T*x =
7 ai(x, wi)vi.
The numbers (ai) are called singular values (sometimes approximation numbers, s-numbers or generalised eigenvalues.)
Corollary 1.3 If A is an m-by-m matrix, we can find unitary matrices U and V and a positive semi-definite diagonal matrix D such that A = UDV.
Proof A corresponds to a finite rank operator T: Cm
Cm. With respect to the orhonormal
bases (vi) and (wi) (extended if necessary by adding vectors from the kernels of T and T*), T has the diagonal matrix D. Changing back to the standard orthonormal basis transfonns D into UDV, where U and V are unitary matrices.
We can interpret a1(T) as 11711. More generally we have the following result.
Theorem 1.4 For n >_ 1,
(n(T) = inf (IIT - SII: rank(S) < n). The infimum is actually attained.
Proof We may assume without loss of generality that n is at least 2, since for n = 1, S = 0 will do. Clearly, taking
Sx =
E7-1
ai(x vi)wi,
we have rank(S) < n and
(T - S)(x) = En ai(x, vi)wi, and so IIT - SII = on
Suppose now that R is any operator of rank k, say, and consider L, the linear span of the
vk+1 Since dim(L) > rank(R), we see that the restriction R: L -4 Im R is not
vectors v1,
injective and there exists a vector x of norm I with x E L and Rx = 0. But IITxll ? ak+lllxll, since the coordinates of x are each magnified at least that much, and so
II(T - R)xII ? ak+lllxll, which implies that IIT - RII z ak+l
an, and the result follows.
This explains why the ai are sometimes called approximation numbers of T. When T is not compact, but merely bounded, we can still define
ai(T) = inf (IIT - SII: rank(S) < i), and clearly ai(T) -* 0 if and only if T is compact.
Corollary
1.5
am+n-1(S + T) 5 am(S) + an(T)
and
am+n-1(ST) _ 1. In particular, am(ST) _ 0, although we shall
9
not prove this.
Definition We say that a compact operator T is in the class Cp (1 _ 0, as it does not change the value of the expression to multiply the vectors by scalars of modulus 1. Z-1
(Txj, yj) = El (Txj, Uxj) where U is the norm I map (partial isometry) taking xj to yj for
each j (and zero on the orthogonal complement of the (xj)). Thus
EI (Txj, yj) = tr (U*T)
i.e. IV(z)E N bnznllL2 5 K]IE N bnzr`II
L2
for all trigonometric polynomials. But then, given any g e L2 the analogous inequality will hold, since we can find a sequence (gk) of trigonometric polynomials converging to g in L2 and almost everywhere. Then, by Fatou's Lemma (see the Appendix),
18
Ilf (z)g(z)IIL2 5 lim inf Ilf (z)gk(z)IIL2 5 KIIgfl
,
and hence, by (i), IVIIL_ s.
Proof (i) Jo" P(reie, w)f(reie)de/27t = (1/27ti)Jlzl=r (r2 - ww)f(z)dzl(z(z - w)(r2/z
-
w)),
setting z = ree, dz = izde. The result now follows by the Residue theorem, since the only pole is at w, with residue f(w).
(ii) Take f(z) = 1, and use (i). (iii) 0
n
P(z, sel0)4l21c = (1/27ti)J1wI=s (zY - s2)dwl(w(z - w)(z - s2/w)
(1/2ici)J1w1=s (zY - s2)dwl((z - w)(iw - s2). The result follows again from the residue theorem: this time the only pole is at w = s2/r. Corollary 2.11 Ifs < r < 1 and f is analytic in
(Izl
< 1), then
Dx fAId412c S a" f(re'0)Ide/2n, and thus, if f e H1, then
Ilfllyl = limr41o(reie)Ide/2n. tn f Proof ?-7c (f(set4Id4l27c
=0I
J ' f(rei0) P(reie, sets) de/2711 dtp127c
_
I(fz)I for all z in the
.
Theorem 2.13 (Riesz Factorization Theorem) f(z) E H j if and only if there exist g(z), h(z) E H2 such that f = gh, and RAW, = IIgIIH2IIhIIH2 In other words we can regard H j functions as products of H2 functions, rather in the same way as we regarded C j operators as the products of C2 operators. Proof It follows immediately from the Cauchy-Schwarz inequality that we always have IIghIIH
j5
IIgIIH2IIhIIH2
Conversely, given any f(z) E H1, we may write f(z) = fj(z)B(z), where B is the Blaschke product as in Theorem 2.12, f, has no zeroes in the disc, and II/IH1 = analytic and nonzero, it has an analytic square root, g say, and IiAIH1
=
Il f j II H
j Since f, is
IIgIIH2' Thus
.Rz) = g(z)g(z)B(z), and IVIIH
j=
IIgIIH2IIgBIIH2,
since we already have ' 0, and taking gn = znhn E H1 + C(T) gives the required result. Thus a compact Hankel operator is actually the limit of a sequence of finite rank Hankel operators.
We wish to show that H1 + C(T) is closed in L (T), or equivalently, using the isometry U,
that H + C(T) is closed in L,,. It is convenient now to prove one or two further facts about the Poisson kernel P(z, w) =
(Izl2 -
IwI2) / (Iz - w12), which relate to Proposition 2.10.
For f e L (T), r < I and Iw) = 1, write Prf(w) =Jo P(eie, rw) ,fei0) dO/27c. Note that Proposition 2.10 implies that IIPrfll 5 ML.,. Lemma 3.17 (i) If, f(w) = wn, then Prf(w) = rl nl wn;
(ii) If f E H,,, then Prf E A0; (iii) If f E C(T), then Prf -4 f uniformly as r -a 1. Proof (i) For n Z 0, this follows from Proposition 2.10 (i); for n < 0 the result can be obtained by taking complex conjugates.
(ii) This follows since Prf = fr where fr(z) = f(rz); (iii)
Given e > 0, choose g such that g(ei0) = E N anein0 and IV -
prg(eie) _
N
anrlnlein0.
_
1, with x(t) given for 0 0, and h e LI, let h = hl + h2, where hl is continuous with compact support and 1Ih211 < e/2. Then
IEp I(ren, en)l
- (1/2)IIhIIII
5 Ep I(r'2en. en)l + IEp I(r'len, en)I -
5 IIh2111 + 1Z' I(rlen, en)I
- (1/2)IlhIIIII + (1/2)IIh2111
(1/2)IlhlljI
< e if a is sufficiently small.
But, by Theorem 1.12, IITIIN ? IF I(rea, en)l, and hence InIN >_ (1/2)Ilhlll, as required. The above result has a corollary which gives some useful bounds for model reduction. More such results may be found in the papers of Glover and Partington, and of Glover et al.
Corollary 7.5 Let H = Lh a
determine a nuclear Hankel operator. If I'1 is an
optimal rank-n approximant to r (as in Chapter 6) and H1 = LhI a RH (C+) a rational symbol for F1, then Ilh
- hllil 5
5 4nan+1(r) + 2(an+1(r) + an+2(r) +
)
Proof The singular values of F - rl satisfy
(a) ai(r - r1) 5 an+1(r) for all i, since Ilr
-
r'lll 5 an+1(r); and
(b) aj+n(r - rl) _ 1/n, f(0) = n, and f is linear on [0, 1/n] and []/n, oo). Then f fn = 1/2 for all n, and fn(x) -a 0 as n - oo, for all
x;
so the limit function has integral zero. It is therefore useful to establish conditions under which lira
(f fn) = f (lim fn). The
theorem which follows gives one such condition.
A.11. Suppose that (Q is a sequence of real functions, that fn(x) -* f(x) almost everywhere and that the sequence fn(x) is monotonically increasing for almost all x. Then f fn
f f.
The above result is used in Chapter 2 during the proof of Theorem 2.12. Other convergence theorems are available, of which the following result (the Dominated Convergence Theorem) is perhaps the most generally useful.
A.12. Suppose that fn(x) -4 f(x) almost everywhere and that there exists an integrable function g such that Vn(x)I n am). The limit on the right hand side is easily seen to be a supremum as well. Fatou's Lemma may be stated in the following forth.
86
A.D. Let (fn) be any sequence of non-negative functions. Then
I lim inf fm _ an+i for each i, and deduce that IIT - TnIIN
an+l + an+2 +
i.
Show that
----
6. Show that for an arbitrary compact operator T there is in general more than one rank-n approximant Tn such that IIT -
Tall = a,+1
7. Show that Coo(R), the space of all continuous functions with compact support, is dense in Lp(R) for 1 1. Show that IIg(z)
- (a0 + a1z + ...
+
anzn)Ilo, = O(K-n)
for some K > 1. Deduce that g determines a nuclear Hankel operator whose singular values satisfy an = O(K-n). 45. Given A > 0 show how to find a Hilbert-Schmidt Hankel operator t with kernel h(t) such that
lh(t)I > AIII'IIHS on
Ih(t)I