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Multivariable Operator Theory: A Joint Summer Research Conference on Multivariable Operator Theory July 10-18, 1993 University of Washington Seattle (Contemporary Mathematics) PDF

387 Pages·1995·6.629 MB·English
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Preview Multivariable Operator Theory: A Joint Summer Research Conference on Multivariable Operator Theory July 10-18, 1993 University of Washington Seattle (Contemporary Mathematics)

CONTEMPORARY MATHEMATICS 185 Multivariable Operator Theory A Joint Summer Research Conference on Multivariable Operator Theory July 10 -18, 1993 University of Washington, Seattle Raul E. Curio Ronald G. Douglas Joel D. Pincus Norberto Salinas Editors American Mathematical Society (CONTEMPORARY MATHEMATICS 185 Multivariable Operator Theory A Joint Summer Research Conference on Multivariable Operator Theory July 10-18, 1993 University of Washington, Seattle Rau'l E. Curio Ronald G. Douglas Joel D. Pincus Norberto Salinas Editors American Mathematical Society Providence, Rhode Island Editorial Board Craig Huneke, managing editor Clark Robinson J. T. Stafford Linda Preiss Rothschild Peter M. Winkler The Joint Summer Research Conference on Multivariable Operator Theory was held at the University of Washington, Seattle, Washington, from July 10-18, 1993, with support from the Division of Mathematical Sciences of the National Science Foundation, Grant No. DMS-9221892. 1991 Mathematics Subject Classification. Primary 47-XX, 32-XX, 46-XX; Secondary 30-XX, 43-XX, 19-XX. Library of Congress Cataloging-in-Publication Data Multivariable operator theory : a joint summer research conference on multivariable operator theory, July 10-18, 1993. University of Washington, Seattle / Radl E. Curto ... [et al.], editors. p. cm. - (Contemporary mathematics, ISSN 0271-4132; v. 185) Includes bibliographical references. ISBN 0-8218-0298-4 1. Operator theory-Congresses. 2. Functions of several complex variables-Congresses. 1. Curto, Raul E., 1954- . II. Series: Contemporary mathematics (American Mathematical So- ciety); v. 185. QA329.M85 1995 515'.94-dc2O 95-2345 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionCath.ams.org. The appearance of the code on the first page of an article in this publication (including ab- stracts) indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, providIr. that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyrigl ce` Center, Inc., 222 Rosewood Drive, Danvers, . Massachusetts 01923. This consent dries not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. © Copyright 1995 by the American Mathematical. Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ® The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 0 Printed on recycled paper. This volume was printed from copy prepared by the authors. Some of the articles in this volume were typeset by the authors using ASS-'I X or A M1 X, the American Mathematical Society's 'I X macro systems. 10987654321 009998979695 Contents Preface ix Explicit formulae for Taylor's functional calculus D. W. ALBRECHT A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains JONATHAN ARAZY 7 Homogeneous operators and systems of imprimitivity BHASKAR BAGCHI AND GADADHAR MISRA 67 Duality between A°° and A-O° on domains with nondegenerate corners DAVID E. BARRETT 77 Commutative subspace lattices, complete distributivity and approximation KENNETH R. DAVIDSON 89 Models and resolutions for Hilbert modules R. G. DOUGLAS 109 Positivity, extensions and the truncated complex moment problem LAWRENCE FIALKOW 133 Torsion invariants for finite von Neumann algebras DONGGENG GONG AND JOEL PINCUS 151 Algebraic K-theory invariants for operator theory JEROME KAMINKER 187 Fundamentals of harmonic analysis on domains in complex space STEVEN G. KRANTZ 195 Spectral picture and index invariants of commuting n-tuples of operators R. N. LEVY 219 Schatten class of Hankel and Toeplitz operators on the Bergman space of strongly pseudoconvex domains HUIPING LI AND DANIEL H. LUECKING 237 Operator equations with elementary operators MARTIN MATHIEU 259 Vii Preface This volume consists of a collection of papers presented at the 1993 Summer Research Conference on Multivariable Operator Theory, held at the University of Washington in Seattle, under the auspices of the American Mathematical Society. The articles represent contributions to a variety of areas and topics, which may be viewed as forming an emerging new subject, involving the study of geometric rather than topological invariants associated with the general theme of operator theory in several variables. Beginning with J. L. Taylor's discovery in 1970 of the right notion of joint spectrum and analytic functional calculus for commuting families of Banach space operators, a number of significant developments have taken place. For instance, a Bochner-Martinelli formula has been generalized to commuting n- tuples in arbitrary C'-algebaas; various forms of the multivariable index theorem have been proved using non-commutative differential geometry and algebraic geometry; and systems of Toeplitz and Hankel operators on Reinhardt domains, bounded symmetric domains, and domains of finite type have been substantially understood from the spectral and algebraic viewpoints, including the discovery of concrete Toeplitz operators with irrational index. These developments have been applied successfully to various types of quanti- zations, and functional spaces on Cartan domains and on pseudoconvex domains with smooth boundary have been thoroughly studied. A generalization of the Berger-Shaw formula to several variables has been proved; and connections with the local multiplicative Lefshetz numbers, analytic torsion, and curvature rela- tions of canonically associated hermitian vector bundles have been established. Moreover, a sophisticated machinery of functional homological algebra suitable for the study of multivariable phenomena has been developed, and a rich theory for invariant pseudodifferential operators on domains with transverse symmetry has been produced. Much of multivariable operator theory involves the interaction between the subspace geometry of defect spaces and algebraic K-theory. For one example, a multivariable index theorem corresponding to commuting pairs of elements with finite defect A, B E End(H), where H is a vector space over an arbitrary field F, has emerged in connection with the Quillen algebraic K-theory. The ix x PREFACE joint torsion r(A,B;f) is a multiplicative Euler characteristic defined solely in terms of the homology of the Koszul complex K. (A, B, H). It has been found to be the obstruction in the homotopy lifting problem from the classifying space BGL(End(H))+ -- BGL(End(H)/F))+, where .F is the finite rank ideal; for it has been shown that r(A, B; N) = O{A +.F, B +.F}, where {A +.F, B + F} is the Steinberg element in BGL(End(H)/.F))+. On a different direction, Calabi-like rigidity phenomena for analytically in- variant subspaces of the Hardy and Bergman spaces have been discovered, and sheaf models for subnormal n-tuples have been formulated, which have led to a substantial understanding of their spectral properties. Results from polynomial convexity have been used to solve intriguing problems on joint quasitriangular- ity, and the polynomially hyponormal conjecture for single operators has been settled using ideas from joint hyponormality. As probably expected during the early stages of a new subject, the recentyears have seen the rise of many new approaches (all quite different) to multivariable operator theory, certainly connected, but with relationships not well understood. The subject has developed in several directions and with the aid of many and varied techniques, and a good number of the advances have been made through cross pollination among different areas of mathematics. The principal goal of the conference was to provide a forum for the discussion of the actual connections among the various approaches, which one hopes will allow researchers to combine their efforts in finding an understanding of the above mentioned relationships and new directions for future research. These proceedings represent the products of those discussions. The Editors December 1994 Contemporary mathematics Volume 166, 1995 EXPLICIT FORMULAE FOR TAYLOR'S FUNCTIONAL CALCULUS D. W. ALBRECHT ABSTRACT. In this paper integral formulae, based on Taylor's functional calculus for several operators, are found. Special cases of these formulae include those of Vasilescu and Janas, and an integral formula for commuting operators with real spectra. §1 Introduction. Let a = (a1,. .. , an) be a commuting tuple of bounded linear operators on a complex Banach space X, let 11 be an open subset containing the Taylor's joint spectrum of a, Sp(a, X), and let f be a holomorphic function defined on St. Then Taylor [T] defined f (a) as follows: f(a)x = 1 j(Ra(z)f(z)x)dZ (27ri)n + for each xEX. However, since Ra(s) is a homomorphism of cohomology, this means, in contrast with the Dunford-Schwartz calculus, that the functional calculus of Taylor is rather inexplicit. In this paper we will show how explicit formulae for Taylor's functional calculus can be obtained, under certain circumstances which include the following: (1) X is a Hilbert space; (2) a has real spectrum, i.e. Sp(a, X) C Rn; (3) Il is a bounded convex domain given by a smooth defining function. §2 Notation. Let a = (sl, ... , sn) and dz` = (d21,.. . , den) denote tuples of inde- terminates, and A[c U dz] denote the exterior algebra, over C, generated by these tuples. Let AP[a U d2] denote the subspace of p-forms in A[a U dz]. For 1 < i < n, let Si be operators on A[a U di] defined by Si(d) = si A l;, for every E A[a U di]. We shall call these the creation operators for a on A[a U dz]. Let d1,... , dzn also denote the creation operators for di on A[a U dz], and for each 1 < i < n, let Si` denote the operators on A[a U d2] which satisfy the following anti-commutation relations: S, S3* + S3* S; = 0 ifi=j 1 5, S; + S; S. = ifi54j' 0 1991 Mathematics Subject Classification. Primary 47A60; Secondary 47A56. The detailed version of this paper has been submitted for publication elsewhere Q 1995 American Mathematical Society 0271-4132/95 $1.00 + 1.25 per page 1 2 D. W. ALBRECHT for every 1 < i, j < n. We consider a,_., a,,, and a , ... , as to be operators on C- (Q, X), and set 4 a 0 = d21 + ... + dz 9 zn i9ij and a(z) = (z1 - al)S1 + ... + (z - It follows from Taylor [T] and Vasilescu [VII that (C°° (fz, X ), Co (f2, X ), a, a= ) is a Cauchy-Weil system. Let R°(z) = where s' is the homomorphism induced by the morphism s(C) = l As I A As., for t; E C- (n, X) ® A[dz], i' is the homomorphism induced by the inclusion mapping i in the exact sequence o Co (1z, x) ` > c°°(0, x) r C°°(fz, x)/cc (fz, x) 0, and lr' is the homomorphism induced by the special transformation 7r, where 1r0 is the identity on Co (SZ, X), 7r(si) = 0 for 1 < i < n, and 7r(dz,) = d23 for 1 < j < n. §3 An Explicit Formula. Let A[CC], A[CO°] and A[C°°/Co ] denote respectively Co (12,X)®A[audz],C°°(11,X)®A[aUdz]andC°D(1l, X)/Co (fz,X)®A[aUdz], and consider the following commuting diagram: 0 0 0 I AP-1[C0-] °a±8 ] a±8 l AP [CO- il it AP-1[C°°] AP[C°°] ri rI r1 AP-'[C°°/Co1 I AP[C-ICO ] AP+1 [Co°/C ] I 1 1 0 0 0 We can show that (i')-'[f] = [f - (a+a=)s], EXPLICIT FORMULAE FOR TAYLOR'S FUNCTIONAL CALCULUS 3 where (a + az)r(g) = r(f), (a+BZ)r(f) = 0, and [ ] denotes the cohomology class. Thus, we see that the question of an explicit expression (is)-1, and therefore an explicit formula for f (a) is closely tied to finding an explicit expression for a solution of the above equations. This in turn leads to the following result, which was suggested by results in [V2] and is a special case of a result in (Al. For each z E SZ\ Sp(a, X), let (bl (z),... , b (z)) be a commuting tuple of bounded linear operators on X. Moreover, for z E Il \ Sp(a, X), let (1) 13(z) = bl (z)Sl +... bn (z)S,`,, (2) z --' /3(z) be an infinitely-differentiable mapping, and (3) a(z)+,Q(z) be invertible on A[o,X]. Lemma 1. Suppose f=f0+...+fn, fl, E A[dz, C°C] ®Ap[Q], and (a + 8z)r(f) = 0. Then = n-In-k-I E (-1)"`(a+ (3)-1(8z(a+/3}-1)'nr(fk}m+1) r(g) k=0 m=0 is a solution of (a + &.)r(g) = r(f). We can now obtain explicit formulae for Taylor's functional calculus. Theorem 1. Let D be an open subset of 0, with compact closure, piecewise CI boundary, and containing Sp(a, X). Then for every x E X. Proof. Let F1, F2 be compact subsets of 1, such that Sp(a, X) is contained in the interior of F1, F1 is contained in the interior of F2, and F2 is contained in D. Suppose now that 0 E C°° (11) is such that ¢ = 0 on F1 and 0 = 1 on St \ F2. Let g(z) = 4)(z)(a(z) + N(z))-1(az(a(z) +,8(z))-1)'-1f W81 A ...sn. Then by Lemma 1, we can show the following: 4 D. W. ALBRECHT f(a)x= J Ra(z)f (z)xdz (2 z)71 (-1)n7r*(i*)-18* f (z)xdz (27ri)n Jin jag(z)xdz (21riID d(g(z)x)dz (27ri)n g(z)xdz, (27ri)- J8D for every x E X. Example 1. Let X be a Hilbert space, and )3(z) = (zi - al)`Sl + ... + (zn - an)*Sn, for z E Cn. Then we can show that /3 satisfies the conditions above, and that the explicit formula (1) reduces to Vasilescu's Martinelli type formula in [V2]. Example 2. Let a have real spectrum, i.e. Sp(a, X) C Ht's, and Q(z) = M(z,a)-i(zl -al)Sl +...+M(z,a)-1(zn -an)S,,, where n M(z, a) = E(z= - ai)(zi - ai). i=1 Then we can show that /3 satisfies the conditions above, and that the formula (1) reduces to the following expression: D f (a) _ (2 i)n f (z)M(z, a)-nW(z, a)dz, where n W (z, a) = (n - 1)! E(-1)P-'(2p - ap) A A d2q. P=i 4OP Example 3. Let fl C Cn be a bounded convex domain given by a defining function p, i.e. 1 = {z E cn : p(z) < 0}, where p is infinitely differentiable on an open neighbourhood of 1 and 48zl'"...z onM={zECn:p(z)=0}. 0 We will now show that we can use the formula (1) to obtain the integral formula of Janas [J].

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