Paul S. Bourdon

CYCLIC PHENOMENA FOR COMPOSITION OPERATORS

Introduction Preliminaries Linear-fractional composition operators Linear-fractional models The hyperbolic and parabolic models Cyclicity: Parabolic nonautomorphism case Endnotes References.

COMPACT COMPOSITION OPERATORS ON BMOA

We characterize the compact composition operators on BMOA, the space consisting of those holomorphic functions on the open unit disk U that are Poisson integrals of functions on ∂U , that have bounded mean oscillation. We then use our characterization to show that compactness of a composition operator on BMOA implies its compactness on the Hardy spaces (a simple example shows the converse does not hold). We also explore how compactness of the composition operator Cφ : BMOA → BMOA relates to the shape of φ(U) near ∂U , introducing the notion of mean order of contact. Finally, we discuss the relationships among compactness conditions for composition operators on BMOA, VMOA, and the big and little Bloch spaces.

Which linear-fractional composition operators are essentially normal?☆

Abstract We characterize the essentially normal composition operators induced on the Hardy space H2 by linear-fractional maps; they are either compact, normal, or (the nontrivial case) induced by parabolic nonautomorphisms. These parabolic maps induce the first known examples of nontrivially essentially normal composition operators. In addition, we characterize those linear-fractionally induced composition operators on H2 that are essentially self-adjoint, and present a number of results for composition operators induced by maps that are not linear-fractional.

Components of linear-fractional composition operators

Abstract We determine when two linear-fractional composition operators on the Hardy space H 2 belong to the same component in the collection of all composition operators on H 2 . We show that two such composition operators in the same component may fail to have compact difference, which answers a question raised by Joel Shapiro and Carl Sundberg.

Finite-codimensional invariant subspaces of Bergman spaces

ABSTRACT. For a large class of bounded domains in C, we describe thosefinite codimensional subspaces of the Bergman space that are invariant undermultiplication by z. Using different techniques for certain domains in CN, wedescribe those finite codimensional subspaces of the Bergman space that areinvariant under multiplication by all the coordinate functions. Fix a positive integer N, and let V denote Lebesgue volume measure on CN (sothat if N = 1, then V is just area measure). Let Q c C^ be a domain, which, asusual, means that 0 is a nonempty open connected subset of C^. For / an analyticfunction from 0 to C and 1 < p < oo, the norm ||/||n,P is defined by Wfh,r=(jn\f\pdV The Bergman space L^(Q) is defined to be the set of analytic functions from fi toC such that ||/||n,P < oo. Our goal in this paper is to describe the closed finite codimensional subspaces ofLpa (0) that are invariant under multiplication by the coordinate functions zi,...,

Selfcommutators of automorphic composition operators

For each automorphic composition operator C ϕ acting on either the Hardy or Bergman Hilbert space of the unit ball in , we show that is a Toeplitz operator, that is the inverse of a Toeplitz operator, and that the selfcommutator is essentially a Toeplitz operator. We then extract spectral and norm information from these identifications, focusing on selfcommutators. For example, in the setting of the Hardy space of the unit disk, we obtain complete descriptions of the spectrum, essential spectrum, and point spectrum for selfcommutators of automorphic composition operators, which reveal that the spectrum and essential spectrum coincide.

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.

Invertible weighted composition operators

Let X be a set of analytic functions on the open unit disk D, and let phi be an analytic function on D such that phi(D) is contained in D and f |-> f o phi takes X into itself. We present conditions on X ensuring that if f |-> f o phi is invertible on X, then phi is an automorphism of D, and we derive a similar result for mappings of the form f |-> psi.(f o phi), where psi is some analytic function on D. We obtain as corollaries of this purely function-theoretic work, new results concerning invertibility of composition operators and weighted composition operators on Banach spaces of analytic functions such as S^p and the weighted Hardy spaces H^2(beta).

When is zero in the numerical range of a composition operator

AbstractWe work on the Hardy spaceH2 of the open unit disc