Weighted composition operators in functional Banach spaces: an axiomatic approach

Abstract

We work with very general Banach spaces of analytic functions in the disk or other domains which satisfy a minimum number of natural axioms. Among the preliminary results, we discuss some implications of the basic axioms and identify all functional Banach spaces in which every bounded analytic function is a pointwise multiplier. Next, we characterize (in various ways) the weighted composition operators among the bounded operators on such spaces, thus generalizing some well-known results on multiplication or composition operators. We also characterize the invertible weighted composition operators on the disk and on general Banach spaces of analytic functions on bounded domains under different sets of axioms whose connections we discuss by providing appropriate examples. This generalizes and complements various recent results by Gunatillake, Bourdon, and Hyvärinen-Lindström-Nieminen-Saukko.

Invertible and isometric weighted composition operators

“…The situation is more complicated in the Dirichlet space where other conditions have to be added to the boundedness assumption on u; we refer the reader to [33]. The spaces of the disk for which M (X ) = H ∞ have been characterized in [2] in terms of a domination property.…”

“…We review a partial list of spaces of analytic functions in the disk that satisfy all of the above axioms. The list is a bit larger than the list of the spaces satisfying the axioms from [2].…”

“…The point of view of the present paper is to study the properties of surjective WCOs in a unified way on general Banach spaces of analytic functions that satisfy only a handful of axioms while still obtaining that same conclusions as in the known special situations, thus covering many cases in one stroke. Several recent papers have been guided by a similar philosophy, for example, [4], [10], [24], [2]. Two general theorems regarding invertibility were proved in [2], assuming different sets of five axioms that the space should satisfy.…”

“…Several recent papers have been guided by a similar philosophy, for example, [4], [10], [24], [2]. Two general theorems regarding invertibility were proved in [2], assuming different sets of five axioms that the space should satisfy. Theorem 1 of this note proves a more general result under slightly modified axioms that seem easier to verify than those in [2].…”

“…Two general theorems regarding invertibility were proved in [2], assuming different sets of five axioms that the space should satisfy. Theorem 1 of this note proves a more general result under slightly modified axioms that seem easier to verify than those in [2]. Isometries of various spaces defined in terms of derivatives were characterized in [23].…”