Berezin number and Berezin norm inequalities via Moore-Penrose inverse

TL;DR

The paper develops new upper bounds for the Berezin number and Berezin norm of bounded operators on reproducing kernel Hilbert spaces by utilizing the Moore–Penrose inverse. Through a suite of lemmas on Jensen–Young-type inequalities, inner-product bounds, and 2×2 block-positivity, the authors establish bounds for $\mathrm{ber}^{2r}(A)$ in terms of operator combinations such as $|A|^{4r}+(AA^{\dag})^{r}$ and $|A^{*}|^{4r}+(A^{\dag}A)^{r}$, with parameters $\lambda\in[0,1]$ and $r\ge 1$. They further provide corollaries for sharp special cases (e.g., $\mathrm{ber}(A) \le \tfrac{1}{2}\mathrm{ber}(|A|^{2}+I)$) and extend the framework to sums, products, and mixed operator forms, illustrating improvements over existing bounds. The work enhances the toolkit for operator inequalities in RKHS settings, with implications for spaces possessing the Ber property and related transforms.

Abstract

In this article, we establish the Berezin number and Berezin norm inequalities for bounded linear operators on a reproducing kernel Hilbert space using the Moore-Penrose inverse. The inequalities obtained here refine and generalize the earlier inequalities.

Theorems & Definitions (54)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Theorem 3.1
• proof
• Remark 3.2