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Convexity of Berezin Range and Berezin Radius Inequalities via a class of Seminorm

P. Hiran Das, Athul Augustine, Pintu Bhunia, P. Shankar

Abstract

Let $B(\mathcal{H})$ denote the $C^*$-algebra of all bounded linear operators acting on a reproducing kernel Hilbert space $\mathcal{H}(Ω).$ In this paper, we introduce a new family of seminorms on $B(\mathcal{H})$, called the $σ_t$-Berezin norm, defined as $$ \|A\|_{{ber}_{σ_t}} = \sup_{λ,μ\in Ω} \left\{ \left( \left|\left\langle A\hat{k}_λ,\hat{k}_μ\right\rangle\right|^p \, σ_t \, \left|\left\langle A^*\hat{k}_λ,\hat{k}_μ\right\rangle\right|^p \right)^{\frac{1}{p}} \right\}, $$ where $A\in B(\mathcal{H}), ~p \geq 1, ~t \in [0,1]$ and ~$σ_t$ denotes an interpolation path of a symmetric mean $σ$. We show that this family of seminorms characterizes invertible operators that are unitary. Several fundamental properties of the $σ_t$-Berezin norm are established, along with a collection of new inequalities that yield refined upper bounds for the Berezin radius of bounded linear operators, thereby improving existing results in the literature. Furthermore, we investigate the convexity of the Berezin range of operators acting on weighted Hardy space and Fock space over $\mathbb{C}^n$. We characterised the convexity of the Berezin range of composition operator with elliptic automorphism and finite rank operators with different weights on the weighted Hardy space. We also characterized convexity of the Berezin range of composition operator on Fock space over $\mathbb{C}^n$ with symbol $φ(z)=Az$, where $A$ is a scalar matrix of order $n$.

Convexity of Berezin Range and Berezin Radius Inequalities via a class of Seminorm

Abstract

Let denote the -algebra of all bounded linear operators acting on a reproducing kernel Hilbert space In this paper, we introduce a new family of seminorms on , called the -Berezin norm, defined as where and ~ denotes an interpolation path of a symmetric mean . We show that this family of seminorms characterizes invertible operators that are unitary. Several fundamental properties of the -Berezin norm are established, along with a collection of new inequalities that yield refined upper bounds for the Berezin radius of bounded linear operators, thereby improving existing results in the literature. Furthermore, we investigate the convexity of the Berezin range of operators acting on weighted Hardy space and Fock space over . We characterised the convexity of the Berezin range of composition operator with elliptic automorphism and finite rank operators with different weights on the weighted Hardy space. We also characterized convexity of the Berezin range of composition operator on Fock space over with symbol , where is a scalar matrix of order .
Paper Structure (6 sections, 34 theorems, 118 equations, 3 figures)

This paper contains 6 sections, 34 theorems, 118 equations, 3 figures.

Table of Contents

  1. Introduction
  2. preliminaries
  3. The $\sigma_t$- Berezin norm of Operators
  4. convexity of the berezin range
  5. On weighted Hardy space
  6. On Fock Space

Key Result

Lemma 2.1

generalisedinequality Let $A, B \in B(\mathcal{H})$ have polar decompositions and let $f,g :[0,\infty) \to [0,\infty)$ be nonnegative continuous. Then for all $x, y \in \mathcal{H}$ the following inequality holds: In particular, if $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ such that $f(t)g(t)=t,$ then

Figures (3)

  • Figure 1: $\textit{Ber}(C_{\phi})$ on $H^2(\beta)$ for $\eta=-0.75$ , $\beta=0.25$ (left, apparently convex) and $\eta=0.6i$ , $\beta=0.5$ (right, apparently not convex)
  • Figure 2: The Berezin range $\textit{Ber}(C_{\phi_{\alpha}})$ on $H^2(\beta)$ for $\alpha=0.5$ and $\beta=0.7$ ( apparently not convex).
  • Figure 3: The Berezin range $\textit{Ber}(C_{\phi})$ on $\mathcal{F}^2_{\alpha}(\mathbb{C}^n)$ for $\alpha=1$ and $\lambda=0.5i$ ( apparently not convex).

Theorems & Definitions (67)

  • Definition 1.1
  • Definition 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Proposition 3.1
  • ...and 57 more