Compactness and Spectra of Hankel Operators

Date

2026-8-7

Author

Baki, Mehmet Orçun

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Spaces of analytic functions and their operators are widely studied. Two of those operators are Toeplitz and Hankel operators. In analytic function spaces that are complemented in a larger space, the Toeplitz operator $T_\phi$ takes the part of the multiplication operator $M_\phi f:=\phi f$ that remains in said space, and the Hankel operator $H_\phi$ takes the rest. Namely, if $P$ is the continuous projection onto said space, we define $T_\phi=PM_\phi$ and $H_\phi=(I-P)M_\phi$. In this thesis, we will introduce Toeplitz and Hankel operators in various spaces and provide a foundation, after which we will narrow our focus to the Hankel operators of the Bergman space $A^2(\Omega)$. We will elaborate on the articles [1] and [2], the first one being about smooth symbols of compact Hankel operators on $A^2(\Omega)$ where $\Omega$ is a pseudoconvex domain in $\C^n$ and the relation between these symbols and analytic discs of the boundary $b\Omega$, and the second one developing an understanding about the spectra and essential spectra of the Hermitian square $\hermsq{H_\phi}$ of Hankel operators $H_\phi$ on the Bergman space $A^2(\D^n)$ of the unit polydisc. We will follow the proofs given in these articles and provide explanations for some claims left to the reader in the original papers.

Subject Keywords

Hankel operator, Analytic function spaces, Essential spectrum, Compact operators, Pseudoconvex domain

URI

https://hdl.handle.net/11511/120570

Collections

Graduate School of Natural and Applied Sciences, Thesis