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

中文翻译：

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符号为Borel度量的Toeplitz运算符

在本文中，我们关注符号为复Borel测度的Toeplitz算子。当给出单位圆上的复Borel测度时，我们给出符号为Toeplitz算子的正式定义，作为Hardy空间上的无界线性算子。然后，我们研究的各种性质。其中有一个定理，其域由三分法表示。此外，还表明，如果的域包含至少一个多项式，则会被严格定义。此外，我们为猜想提供了证据，即用奇异测度可简化为平凡的线性算子。