Spectral properties of weighted Cauchy singular integral transform on S-poly-Barmgann spaces

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 Itô–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 provided results extend to the quaternionic setting those obtained by A. Intissar and A. Intissar in (2006) [32] for the weighted Cauchy transform on the complex plane.