Dual-band general Toeplitz operators

TL;DR

This paper explores dual-band general Toeplitz operators, establishing their relationships with block truncated Toeplitz operators and matrix-symbol Toeplitz operators, and analyzes their spectral properties and boundary behaviors.

Contribution

It introduces new connections between dual-band Toeplitz operators and matrix-symbol Toeplitz operators, and investigates their spectral and boundary properties.

Findings

Dual-band Toeplitz operators relate to block truncated Toeplitz operators.

Spectral properties of the dual-band shift are complex and linked to boundary behavior.

New connections between operator theory and boundary analysis are established.

Abstract

We relate dual-band general Toeplitz operators to block truncated Toeplitz operators and, via equivalence after extension, with Toeplitz operators with $4 \times 4$ matrix symbols. We discuss their norm, their kernel, Fredhomlness, invertibility and spectral properties in various situations, focusing on the spectral properties of the dual-band shift, which turns out to be considerably complex, leading to new and nontrivial connections with the boundary behaviour of the associated inner function.