Convexity of Berezin Range and Berezin Radius Inequalities via a class of Seminorm
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910046234869760 |
|---|---|
| author | Das, P. Hiran Augustine, Athul Bhunia, Pintu Shankar, P. |
| author_facet | Das, P. Hiran Augustine, Athul Bhunia, Pintu Shankar, P. |
| contents | 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08184 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convexity of Berezin Range and Berezin Radius Inequalities via a class of Seminorm Das, P. Hiran Augustine, Athul Bhunia, Pintu Shankar, P. Functional Analysis 47A12, 47A30, 26E60, 46L05 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 $n$. |
| title | Convexity of Berezin Range and Berezin Radius Inequalities via a class of Seminorm |
| topic | Functional Analysis 47A12, 47A30, 26E60, 46L05 |
| url | https://arxiv.org/abs/2603.08184 |