Algebraic and Spectral properties of slant Toeplitz and slant little Hankel Operators on weighted Bergman Space

TL;DR

This paper investigates the algebraic, spectral, and structural properties of slant Toeplitz and slant little Hankel operators on weighted Bergman spaces, providing new criteria for their boundedness, normality, and compactness.

Contribution

It introduces the matrix representations and boundedness criteria for these operators, and characterizes their normality, compactness, and spectral properties in terms of their symbols, advancing understanding of their structure.

Findings

Normality occurs only for constant symbols in slant Toeplitz operators.

Slant little Hankel operators are normal if the symbol is analytic.

Compactness is equivalent to the symbol vanishing.

Abstract

This paper studies the \(k^{th}-\)order slant Toeplitz and slant little Hankel operators on the weighted Bergman space \(\mathcal{A}_\alpha^2(\mathbb{D})\). These operators are constructed using a slant shift operator \(W_k\) composed with classical Toeplitz and Hankel operators, respectively. We derive their matrix representations and establish criteria for boundedness, compactness, and normality. Commutativity conditions are obtained, showing that two such operators commute if and only if their symbols are linearly dependent. Normality is characterized as slant Toeplitz operators are normal only for constant symbols, while slant little Hankel operators are normal if the symbol is analytic. Compactness is shown to occur precisely when the symbol vanishes. Spectral properties, including essential spectra and eigenvalue distributions, are analyzed. Numerical simulations corroborate the theoretical findings and highlight key structural and computational differences between the two operator classes.