New inequalities for some quadratic forms and related results

Abstract

We prove some new Jensen type inequalities for the Berezin symbol of self-adjoint operators and some class of positive operators on the reproducing kernel Hilbert space $\mathcal{H}(\Omega)$ over some set $\Omega$. Recall that the Berezin symbol $\tilde{A}$ of operator A on $\mathcal{H}(\Omega)$ is defined by the following special type of quadratic form: $\tilde{A}(\lambda):=\langle A \hat{k}_\lambda,\hat{k}_\lambda\rangle, \lambda\in\Omega,$where $ k_\lambda$ is the reproducing kernel of the space $\mathcal{H}(\Omega)$, i.e., $f(\lambda)=\langle f,\hat{k}_\lambda\rangle$ for all $f\in\mathcal{H}(\Omega)$ and $\lambda\in\Omega$;\\$\hat{k}_\lambda :=\frac{k_\lambda}{||K_\lambda||_{\mathcal{H}(\Omega)}}$ is the normalized reproducing kernel.