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

387 Pages·1995·6.629 MB·English

by  Raul E. Curio, Ronald G. Douglas, Joel D. Pincus, Norberto Salinas

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
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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].