Intertwining relations and extended eigenvalues for analytic Toeplitz operators

TL;DR

This paper investigates the relationships between analytic Toeplitz operators, focusing on intertwining relations, symbol containment, and extended eigenvalues, providing new insights into their structural properties on the Hardy space.

Contribution

It establishes connections between intertwining relations, symbol containment, and the nature of intertwining operators for analytic Toeplitz operators, and explores extended eigenvalues.

Findings

Intertwining relations correspond to symbol containment.

Characterization of the nature of intertwining operators.

Analysis of extended eigenvalues for Toeplitz operators.

Abstract

We study the intertwining relations between analytic Toeplitz operators induced on the Hardy space H^2 by analytic functions bounded on the open unit disc. Our work centers on the connection between intertwining between the Toeplitz operators the image containment between their symbols, as well as on the nature of the intertwining operator. We use our results to study the "extended eigenvalues" of analytic Toeplitz operators, i.e., the special case where the operator is intertwined with a scalar multiple of itself.