Christopher Hammond

Which Weighted Composition Operators are Complex Symmetric

Recent work by several authors has revealed the existence of many unexpected classes of normal weighted composition operators. On the other hand, it is known that every normal operator is a complex symmetric operator. We therefore undertake the study of complex symmetric weighted composition operators, identifying several new classes of such operators.

Norms of linear-fractional composition operators

We obtain a representation for the norm of the composition operator C Φ on the Hardy space H 2 whenever Φ is a linear-fractional mapping of the form Φ(z) = b/(cz + d). The representation shows that, for such mappings Φ, the norm of C Φ always exceeds the essential norm of C Φ . Moreover, it shows that a formula obtained by Cowen for the norms of composition operators induced by mappings of the form Φ(z) = sz + t has no natural generalization that would yield the norms of all linear-fractional composition operators. For rational numbers s and t, Cowens formula yields an algebraic number as the norm; we show, e.g., that the norm of C 1/(2-z) is a transcendental number. Our principal results are based on a process that allows us to associate with each non-compact linear-fractional composition operator C Φ , for which ∥C Φ ∥ > ∥C Φ ∥ e , an equation whose maximum (real) solution is ∥C Φ ∥ 2 . Our work answers a number of questions in the literature; for example, we settle an issue raised by Cowen and MacCluer concerning co-hyponormality of a certain family of composition operators.

Adjoints of composition operators with rational symbol

Abstract Building on techniques developed by Cowen and Gallardo-Gutierrez, we find a concrete formula for the adjoint of a composition operator with rational symbol acting on the Hardy space H 2 . We consider some specific examples, comparing our formula with several results that were previously known.

Composition operators with maximal norm on weighted Bergman spaces

We prove that any composition operator with maximal norm on one of the weighted Bergman spaces A 2 α (in particular, on the space A 2 = A 2 0 ) is induced by a disk automorphism or a map that fixes the origin. This result demonstrates a major difference between the weighted Bergman spaces and the Hardy space H 2 , where every inner function induces a composition operator with maximal norm.

The James Function

Summary We investigate the properties of the James function, associated with Bill Jamess so-called “log5 method,” which assigns a probability to the result of a game between two teams based on their respective winning percentages. We also introduce and study a class of functions, which we call Jamesian, that satisfy the same a priori conditions that were originally used to describe the James function.

Zeros of Hypergeometric Functions and the Norm of a Composition Operator

Let φ be an analytic self-map of the unit disk; let Cφ denote the corresponding composition operator acting on the Hardy space H2. Although the precise value of ∥Cφ∥ is quite difficult to calculate, some progress has been made in the case when φ is a linear fractional map. A recent paper by Basor and Retsek demonstrates a connection between the norm of such an operator and the zeros of a particular hypergeometric series. Here we will pursue this line of inquiry further. We shall appeal to several results relating to hypergeometric series — many of which are quite old — to deduce more information about the norm of a composition operator, in particular about the spectrum of Cφ*Cφ. Furthermore, we will use our knowledge of composition operators to establish an apparently new result pertaining to the zeros of hypergeometric series.

Norm Inequalities for Composition Operators on Hardy and Weighted Bergman Spaces

Any analytic self-map of the open unit disk induces a bounded composition operator on the Hardy space H 2 and on the standard weighted Bergman spaces A 2 β. For a particular self-map, it is reasonable to wonder whether there is any meaningful relationship between the norms of the corresponding operators acting on each of these spaces. In this paper, we demonstrate an inequality which, at least to a certain degree, provides an answer to this question.