The Smirnov Class for Sub-Bergman Hilbert Spaces

TL;DR

This paper explores the properties of Smirnov classes within sub-Bergman spaces, revealing their boundary behavior and the impact of the defining functions on their structure.

Contribution

It provides new insights into the Smirnov property of sub-Bergman spaces and characterizes the boundary values and defect operator range in these classes.

Findings

Smirnov classes in sub-Bergman spaces have non-tangential boundary values almost everywhere.

The range of the defect operator over Smirnov-sub-Bergman classes is characterized.

Relations between the Smirnov property and the defining functions are established.

Abstract

In this work we consider the Smirnov classes for sub-Bergman spaces. First we point out some observations about the Smirnov property of sub-Bergman space $H^\alpha(\varphi)$ and its relation to the defining $H^{\infty}_{1}$ function $\varphi$. The first main result of the work deals with the range of the defect operator over Smirnov-sub-Bergman class whereas in the last part we show that contrary to classical Bergman spaces, Smirnov-sub-Bergman classes have non-tangential boundary values almost everywhere on the unit circle.