Essential Spectra of Quasi-parabolic Composition Operators on Hardy Spaces of the Poly-disc

TL;DR

This paper extends the analysis of quasi-parabolic composition operators to the bi-disc, characterizing their essential spectra through a tensor product of C*-algebras and revealing new spectral properties.

Contribution

It generalizes previous results to the bi-disc case, expressing operators as Toeplitz and Fourier multipliers, and describes the essential spectra via tensor products of C*-algebras.

Findings

Identification of the C*-algebra as a tensor product in bi-disc case

Description of essential spectra for quasi-parabolic operators

Discovery of a nontrivial set within the essential spectra

Abstract

This work is a generalization of the results in [Gul] to bi-disc case. As in [Gul], quasi-parabolic composition operators on the Hilbert-Hardy space of the bi-disc are written as a linear combination of Toeplitz operators and Fourier multipliers. The C*-algebra generated by Toeplitz operators and Fourier multipliers on the Hilbert-Hardy space of the bi-disc is written as the tensor product of the similar C*-algebra in one variable with itself. As a result we find a nontrivial set lying inside the essential spectra of quasi-parabolic composition operators.