Toeplitz operators whose symbols are Borel measures (Q2025752)

Summary: In this paper, we are concerned with Toeplitz operators whose symbols are complex Borel measures. When a complex Borel measure \(\mu\) on the unit circle is given, we give a formal definition of a Toeplitz operator \(T_\mu\) with symbol \(\mu\), as an unbounded linear operator on the Hardy space. We then study various properties of \(T_\mu\). Among them, there is a theorem that the domain of \(T_\mu\) is represented by a trichotomy. Also, it was shown that if the domain of \(T_\mu\) contains at least one polynomial, then \(T_\mu\) is densely defined. In addition, we give evidence for the conjecture that \(T_\mu\) with a singular measure \(\mu\) reduces to a trivial linear operator.