Holomorphic semi-almost periodic functions

TL;DR

This paper investigates Banach algebras of bounded holomorphic functions on the unit disk with boundary values that are semi-almost periodic, connecting to Toeplitz operator theory.

Contribution

It characterizes a class of holomorphic functions with semi-almost periodic boundary values, extending Sarason's algebra and exploring their algebraic properties.

Findings

Identifies the structure of these Banach algebras.

Links semi-almost periodic functions to Toeplitz operator problems.

Provides new insights into boundary behavior of holomorphic functions.

Abstract

We study the Banach algebras of bounded holomorphic functions on the unit disk whose boundary values, having, in a sense, the weakest possible discontinuities, belong to the algebra of semi-almost periodic functions on the unit circle. The latter algebra contains as a special case an algebra introduced by Sarason in connection with some problems in the theory of Toeplitz operators.