Bounded linear operators Interpolation of Banach spaces Integral operators on $L^p$ spaces Bergman spaces Bloch and Besov spaces The Berezin transform Toeplitz operators on the Bergman space Hankel operators on the Bergman space Hardy spaces and BMO Hankel operators on the Hardy space Composition operators Bibliography Index.

http://www.gbv.de/dms/goettingen/52319126X.pdf

Operator theory in function spaces

Real and Complex Analysis

THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

Introduction Semigroups on Banach spaces and evolution semigroups Evolution families and Howland semigroups Characterizations of dichotomy for evolution families Two applications of evolution semigroups Linear skew-product flows and Mather evolution semigroups Characterizations of dichotomy for linear skew-product flows Evolution operators and exact Lyapunov exponents Bibliography List of notations Index.

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Evolution Semigroups in Dynamical Systems and Differential Equations

Operators with dense, invariant, cyclic vector manifolds

The study of composition operators forges links between fundamental properties of linear operators and results from the classical theory of analytic functions This book provides a self-contained introduction to both the subject and its function-theoretic underpinnings The development is geometrically motivated, and accessible to anyone who has studied basic graduate-level real and complex analysis The work explores how operator-theoretic issues such as boundedness, compactness and cyclicity evolve Each of these classical topics is developed fully, and particular attention is paid to their common geometric heritage as descendants of the Schwarz Lemma

Composition Operators: and Classical Function Theory

The essential norm of a composition operator

A vector x in a linear topological space X is called universal for a linear operator T on X if the orbit {Tnx: n > 0} is dense in X. Our main result gives conditions on T and X which guarantee that T will have universal vectors. It applies to the operators of differentiation and translation on the space of entire functions, where it makes contact with Polya's theory of final sets; and also to backward shifts and related operators on various Hilbert and Banach spaces.

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Universal vectors for operators on spaces of holomorphic functions

Let U denote the open unit disc of the complex plane, and φ a holomorphic function taking U into itself. In this paper we study the linear composition operator Cφ defined by Cφf = f o φ for f holomorphic on U. Our goal is to determine, in terms of geometric properties of φ, when Cφ is a compact operator on the Hardy and Bergman spaces of φ. For Bergman spaces we solve the problem completely in terms of the angular derivative of φ, and for a slightly restricted class of φ (which includes the univalent ones) we obtain the same solution for the Hardy spaces Hp (0 < p < ∞). We are able to use these results to provide interesting new examples and to give unified explanations of some previously discovered phenomena.

Joel H. Shapiro

Papers

Composition Operators: and Classical Function Theory

Operators with dense, invariant, cyclic vector manifolds

The essential norm of a composition operator

Universal vectors for operators on spaces of holomorphic functions

Angular Derivatives and Compact Composition Operators on the Hardy and Bergman Spaces