Weighted quaternionic Cauchy singular integral

TL;DR

This paper studies the spectral properties of the weighted quaternionic Cauchy transform, including boundedness, compactness, and eigenfunctions, extending complex plane results to the quaternionic setting.

Contribution

It provides explicit expressions for the transform's action on quaternionic Itô-Hermite polynomials and characterizes its spectral properties in quaternionic Hilbert spaces.

Findings

Determined the boundedness and compactness conditions.

Identified orthogonal eigenfunctions and singular values.

Extended complex Cauchy transform results to quaternionic analysis.

Abstract

We investigate some spectral properties of the weighted quaternionic Cauchy transform when acting on the right quaternionic Hilbert space of Gaussian integrable functions. We study its boundedness, compactness, and memberships to the $k$-Schatten class, and we identify its range. This is done by means of its restriction to the n-th S-polyregular Bargmann space of the second kind, for which we provide an explicit closed expression for its action on the quaternionic It\^o--Hermite polynomials constituting an orthogonal basis. We also exhibit an orthogonal basis of eigenfunctions of its n-Bergman projection leading to the explicit determination of its singular values. The obtained results generalize those given for weighted Cauchy transform on the complex plane to the quaternionic setting.