Toeplitz Operators whose Symbols Are Borel Measures

Abstract

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.