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

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