Algebraic and Graphical Analysis of H-Toeplitz operators on Fock space

TL;DR

This paper thoroughly analyzes H-Toeplitz operators on Fock space, deriving matrix representations, exploring algebraic and spectral properties, and introducing graphical models to visualize their structure, bridging operator theory and graph theory.

Contribution

It introduces explicit matrix forms, characterizes spectral properties, and develops a novel graph-theoretic visualization for H-Toeplitz operators, advancing understanding of their structure and properties.

Findings

H-Toeplitz operators with harmonic symbols commute under specific conditions.

Non-zero H-Toeplitz operators are not Hilbert-Schmidt.

Graphical patterns reveal structural insights into operator adjacency relations.

Abstract

This paper presents a comprehensive study of H-Toeplitz operators on the Fock space, a class of operators that synthesizes structural elements of both Toeplitz and Hankel operators. We derive explicit matrix representations for these operators with respect to the standard orthonormal basis of monomials, providing a foundational tool for their analysis. Central to our investigation are the algebraic and spectral properties of these operators. We establish precise conditions for commutativity, particularly for operators with harmonic symbols, and prove that non-zero H-Toeplitz operators cannot be Hilbert-Schmidt. Furthermore, we develop a Mellin transform-based framework to characterize the hyponormality and normality of operators with quasi-homogeneous symbols, deriving verifiable analytical criteria. Finally, we introduce the novel concept of directed H-Toeplitz graphs to visualize the adjacency relations encoded by the matrix structure of these operators. This graphical representation reveals distinct and interpretable patterns in indegree and outdegree sequences, offering a new combinatorial lens through which to understand their structure. Our results forge a significant connection between the analytic theory of operators on function spaces and the discrete structures of graph theory, enriching both fields.