Spectral estimates for multiparametric operator products via the $\mathbb{A}$-Berezin norm in RKHS

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 $\mathbb{A}$. A fundamental challenge in this setting is the geometric discrepancy between the normalized reproducing kernels and the unit $\mathbb{A}$-sphere, which renders classical numerical radius techniques inapplicable. By overcoming this structural obstacle, we establish sharp inequalities for the $\mathbb{A}$-Berezin number and $\mathbb{A}$-Berezin norm. Our main contribution involves the derivation of multiparametric estimates for triple operator products of the form ${\mathrm{P}}^{\alpha }X{\mathrm{R}}^{\alpha }$ 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.