The Sarason Sub-Symbol and the Recovery of the Symbol of Densely Defined Toeplitz Operators over the Hardy Space

TL;DR

This paper introduces Sarason Sub-Symbols as a novel tool to recover symbols of densely defined Toeplitz operators on the Hardy space, partially answering a question by Sarason and extending the classification of Toeplitzness.

Contribution

It proposes Sarason Sub-Symbols for densely defined Toeplitz operators, demonstrating their effectiveness in symbol recovery and operator classification.

Findings

Analytic closed densely defined Toeplitz operators are fully determined by Sarason Sub-Symbols.

Sarason Sub-Symbols extend to a broader class of operators of multiplication type.

The work provides a partial answer to Sarason's 2008 question on symbol maps.

Abstract

While the symbol map for the collection of bounded Toeplitz operators is well studied, there has been little work on a symbol map for densely defined Toeplitz operators. In this work a family of candidate symbols, the Sarason Sub-Symbols, is introduced as a means of reproducing the symbol of a densely defined Toeplitz operator. This leads to a partial answer to a question posed by Donald Sarason in 2008. In the bounded case the Toeplitzness of an operator can be classified in terms of its Sarason Sub-Symbols. This justifies the investigation into the application of the Sarason Sub-Symbols on densely defined operators. It is shown that analytic closed densely defined Toeplitz operators are completely determined by their Sarason Sub-Symbols, and it is shown for a broader class of operators that they extend closed densely defined Toeplitz operators (of multiplication type).