Spectra for Toeplitz Operators Associated with a Constrained Subalgebra

TL;DR

This paper investigates the spectra of Toeplitz operators on Hardy spaces constrained by a two-point algebra, revealing connections among their spectral properties.

Contribution

It introduces a new class of Hardy spaces associated with a two-point algebra and analyzes the spectral properties of Toeplitz operators acting on these spaces.

Findings

Spectra of Toeplitz operators are interconnected.

Spectral properties depend on the two-point algebra structure.

New insights into Toeplitz operators on constrained Hardy spaces.

Abstract

A two-point algebra is a set of bounded analytic functions on the unit disk that agree at two distinct points $a,b \in \mathbb{D}$. This algebra serves as a multiplier algebra for the family of Hardy Hilbert spaces $H^2_t := \{ f\in H^2 : f(a)=tf(b)\}$, where $t\in \mathbb{C}\cup\{\infty\}$. We show that various spectra of certain Toeplitz operators acting on these spaces are connected.