Bounded Composition Operators and Multipliers of Some Reproducing Kernel Hilbert Spaces on the Bidisk

TL;DR

This paper investigates the boundedness of composition operators on the Hardy space of the bidisk using reproducing kernel methods, establishing positivity conditions that characterize boundedness and exploring related multiplier spaces.

Contribution

It introduces a positivity criterion for bounded composition operators on the bidisk and studies the associated sub-Hardy Hilbert spaces and their multipliers.

Findings

Boundedness characterized by positive kernel conditions

Identification of sub-Hardy spaces analogous to de Branges-Rovnyak spaces

Classes of bounded composition operators on the bidisk

Abstract

We study the boundedness of composition operators on the bidisk using reproducing kernels. We show that a composition operator is bounded on the Hardy space of the bidisk if some associated function is a positive kernel. This positivity condition naturally leads to the study of the sub-Hardy Hilbert spaces of the bidisk, which are analogs of de Branges-Rovnyak spaces on the unit disk. We discuss multipliers of those spaces and obtain some classes of bounded composition operators on the bidisk.