A new norm on the space of reproducing kernel Hilbert space operators and Berezin number inequalities
Raj Kumar Nayak, Pintu Bhunia
TL;DR
The article defines the $t$-Berezin norm on the bounded operators of a reproducing kernel Hilbert space (RKHS) and demonstrates its placement between the Berezin number and the Berezin norm, with $t\in[0,1]$. It proves that this norm helps characterize invertible operators that are unitary via inequalities on $\|A^*A\|_{t-ber}$ and $\|(A^*A)^{-1}\|_{t-ber}$, and it provides sharp norm relations and counterexamples showing the norm is not an algebra norm. The authors derive various upper bounds for $t$-Berezin norms of operators and 2×2 and general operator matrices, as well as product bounds linked to the spectral radius. In addition, they develop an Orlicz-function framework to obtain refined Berezin-number inequalities, including extensions to $\textbf{ber}^r(A^*B)$ and higher, thereby improving several existing results in RKHS operator theory. Overall, the work offers new tools for precise estimation of Berezin-type quantities and deepens understanding of operator norms in RKHS settings.
Abstract
In this note, we introduce a novel norm, termed the $t-$Berezin norm, on the algebra of all bounded linear operators defined on a reproducing kernel Hilbert space $\mathcal{H}$ as $$\|A\|_{t-ber} = \sup_{ λ, μ\in Ω} \left\{ t|\langle A \hat{k}_λ, \hat{k}_μ\rangle| + (1-t) |\langle A^* \hat{k}_λ, \hat{k}_μ\rangle| \right\}, \quad t\in [0,1],$$ where $A \in \mathcal{B}(\mathcal{H})$ is a bounded linear operator. This norm characterizes those invertible operators which are also unitary. Using this newly defined norm, we establish various upper bounds for the Berezin number, thereby refining the existing results. Additionally, we derive several sharp bounds for the Berezin number of an operator via the Orlicz function.