Intertwining property for compressions of multiplication operators

TL;DR

This paper investigates operators that intertwine compressions of the unilateral shift within the context of asymmetric model spaces, extending classical results related to Beurling's theorem.

Contribution

It characterizes operators intertwining compressions of the unilateral shift on asymmetric model spaces, introducing new insights into asymmetric truncated Toeplitz operators.

Findings

Operators intertwining shift compressions are characterized.

Asymmetric truncated Toeplitz operators are analyzed.

Results extend classical shift operator theory.

Abstract

Following Beurling's theorem the natural compressions of the multiplication operator in the classical $L^2$ space are compressions to model spaces and to their orthogonal complements. Two possibly different model spaces are considered hence asymmetric truncated Toeplitz and asymmetric dual truncated Toeplitz operators are investigated. The main purpose of the paper is to characterize operators which intertwine compressions of the unilateral shift.