The classical theory of Toeplitz operators on Hardy space over the unit disk is a well-developed area in Operator Theory. If we substitute the domain disk $\Delta$ with a bounded multiply connected domain $D$, where $\partial D$ consists of finitely many smooth closed curves, what kinds of difficulties arise? This question motivates us to explore the theory for Toeplitz operators on Hardy space over a multiply connected domain $D$. In 1974, M.B. Abrahamse’s Ph.D. thesis made significant contributions in this topic, extending well-known results for the disk like characterizations of commutator ideals for the Banach Algebra generated by Toeplitz operators with continuous $\mathbb{C}(\partial D)$ or $H^\infty + C(\partial D)$ symbols, and the characterization of Fredholm operators with $H^\infty+C$ symbols to those for the multiply connected domain $D$. Also, he came up with the striking reduction theorem, which roughly says that modulo compact operators, the Toeplitz operator defined on the Hardy space over a multiply connected domain $D$, can be written as the direct sum of Toeplitz operators defined on the Hardy space over the unit disk.

In this talk, we will provide the definition of the Hardy Space $H^p$ over multiple connected domains $D$, where $1 \leq p \leq \infty$, and build some prerequisites to present the aforementioned characterization theorems obtained by Abrahamse for the case of multiple connected domains $D$. We will present the proofs of some of these theorems originally done by Abrahamse. Following that, we will examine the proof of the reduction theorem and explore some of its applications.

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Last updated: 15 Jul 2024