Spectral estimates for multiparametric operator products via the A-Berezin norm in RKHS

Abstract

This paper addresses the spectral analysis of operators acting on reproducing kernel Hilbert spaces equipped with a semi-inner product induced by a positive operator A. A fundamental challenge in this setting is the geometric discrepancy between the normalized reproducing kernels and the unit A-sphere, which renders classical numerical radius techniques inapplicable. By overcoming this structural obstacle, we establish sharp inequalities for the A-Berezin number and A-Berezin norm. Our main contribution involves the derivation of multiparametric estimates for triple operator products of the form P alpha XR alpha involving Schatten-type exponents. These results generalize and refine existing bounds in the literature. Furthermore, we provide a qualitative analysis of the obtained bounds through weighted Toeplitz operators on Hardy spaces and verify the theoretical findings with concrete matrix examples involving the geometric behavior of weight functions.