The Sarason sub-symbol and the recovery of the symbol of densely defined Toeplitz operators over the Hardy space

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).