**Mathematical Surveys and Monographs**

Volume: 13;
1974;
235 pp;
Softcover

MSC: Primary 47;

**Print ISBN: 978-0-8218-1513-7
Product Code: SURV/13**

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

**Electronic ISBN: 978-1-4704-1241-8
Product Code: SURV/13.E**

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# Topics in Operator Theory

Share this page *Edited by *
*C Pearcy*

The five articles in this volume are expository in nature, and they
all deal with various aspects of the theory of bounded linear operators on
Hilbert space. The volume is very timely, because in the last year or two great
progress has been made on hard problems in this field, and thus operator theory
today is a very exciting part of mathematical research. One particular problem
on which considerable progress has been made recently is the invariant subspace
problem. This is the question whether every bounded linear operator on a
separable, infinite-dimensional, complex Hilbert space \(\mathcal H\) has a
nontrivial invariant subspace. Even though this problem remains unresolved,
there are some operators T on \(\mathcal H\) for which the structure of a
lattice of all invariant subspaces of T is even, and the first article in this
volume, “invariant subspaces” by Donald Sarason, is added to a
discussion of such operators. One of the interesting features of this lucid
presentation is the interplay between operator theory and classical analysis.

The second article is entitled “Weighted shift operators and analytic
function theory” and was written by Allen Shields. He has taken
essentially all of the information presently given about weighted shift
operators (with scalar weights) and incorporated it into this comprehensive
article. A central theme of the composition is the interaction between weighted
shift operators and analytic function theory, and in an added bonus for the
reader, the article contains a list of thirty-two interesting research
problems.

The third article in the volume is a treatise called “A version of
multiplicity theory” by Arlen Brown. The problem treated is how to decide
when two normal operators are unitarily equivalent. (Unitary equivalence is the
analog for operators of the concept of isomorphism for groups, rings, etc.) The
unitary equivalence problem for arbitrary operators is exceedingly difficult,
but the theory of spectral multiplicity, which can be approached in several
different ways, furnishes a reasonable complete set of unitary invariants for
normal operators. The author focuses attention on the concept of a spectral
measure, and his clear presentation of this circle of ideas should lead to a
better understanding of multiplicity theory by beginners and experts alike.

The fourth article in this volume, “Canonical models” by R. G.
Douglas, is concerned with the theory of canonical models for operators on
Hilbert space. The central underlying idea is that if T is any contraction
operator on \(\mathcal H\) (i.e., if the norm of T is at most 1), then there
is a canonical construction that associates with T an operator
\(\mathrm{M}_\mathrm{T}\) that is unitarily equivalent to T, called
its “canonical model”. One can therefore study T by studying
\(\mathrm{M}_\mathrm{T}\) instead, and this theory has made significant
progress in the past ten years. The author, who has contributed substantially
to the geometrization of this theory, exposes in his article various important
components of the theory, and thereby gives the reader much insight into its
successes and failures.

The final article in this volume, “A survey of the Lomonosov technique
in the theory of invariant subspaces” by Carl Pearcy and Allen Shields, is
a survey of some new invariant-subspace theorems that resulted from the
brilliant and elegant method of proof introduced by Victor Lomonosov early in
1973. Further study and refinement of this technique should lead to additional
progress on the invariant subspace problem.

#### Table of Contents

# Table of Contents

## Topics in Operator Theory

- Table of Contents v6 free
- Preface vii8 free
- I. Invariant subspaces 112 free
- 1. Introduction 314
- 2. Some immediate observations 415
- 3. Reducing subspaces of normal operators 617
- 4. Invariant subspaces and operator algebras 1021
- 5. Unitary operators 1425
- 6. The bilateral shift 2031
- 7. Maximal subalgebras 3142
- 8. The Volterra operator 3344
- 9. The Volterra operator plus multiplication by x 3748

- II. Weighted shift operators and analytic function theory 4960
- 1. Introduction 5162
- 2. Elementary properties 5162
- 3. Weighted sequence spaces 5768
- 4. The commutant 6172
- 5. The spectrum 6677
- 6. Analytic structure 7384
- 7. Hyponormal and subnormal shifts 8394
- 8. Algebras generated by shifts 8899
- 9. Strictly cyclic shifts 92103
- 10. Invariant subspaces 102113
- 11. Cyclic vectors 109120
- 12. Notes and comments 116127

- III. A version of multiplicity theory 129146
- Introduction 131148
- 1. The standard spectral measure 133150
- 2. Cyclic subspaces 135152
- 3. Multiplicity : The problem 136153
- 4. The algebra of finite measures 137154
- 5. σ ideals of measures 139156
- 6. Cycles and measures 142159
- 7. σ ideals and subspaces 144161
- 8. Multiplicity : The solution 147164
- 9. Uniform multiplicity one 152169
- 10. The separable case 153170
- 11. The L[sub(2)] space of a σ ideal 156173

- IV. Canonical models 161178
- Introduction 163180
- 1. Canonical models and unitary dilations 164181
- 2. Unitary operators and invariant subspaces 170187
- 3. Absolute continuity and the functional calculus 179196
- 4. Characteristic operator functions 191208
- 5. Contractions in class C[sub(0)] and Jordan models 205222
- 6. Related and more general models 212229

- V. A survey of the Lomonosov technique in the theory of invariant subspaces 219236
- Index 231248 free