arXiv subjects

Athul Augustine

Publications and source records attributed to Athul Augustine.

7 recordsLinked to original sources

arXiv2026

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

Let $B(\mathcal{H})$ denote the $C^*$-algebra of all bounded linear operators acting on a reproducing kernel Hilbert space $\mathcal{H}(\Omega).$ In this paper, we introduce a new family of seminorms on $B(\mathcal{H})$, called the $\sigma_t$-Berezin norm, defined as $$ \|A\|_{{ber}_{\sigma_t}} = \sup_{\lambda,\mu\in \Omega} \left\{ \left( \left|\left\langle A\hat{k}_\lambda,\hat{k}_\mu\right\rangle\right|^p \, \sigma_t \, \left|\left\langle A^*\hat{k}_\lambda,\hat{k}_\mu\right\rangle\right|^p \right)^{\frac{1}{p}} \right\}, $$ where $A\in B(\mathcal{H}), ~p \geq 1, ~t \in [0,1]$ and ~$\sigma_t$ denotes an interpolation path of a symmetric mean $\sigma$. We show that this family of seminorms characterizes invertible operators that are unitary. Several fundamental properties of the $\sigma_t$-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 $\phi(z)=Az$, where $A$ is a scalar matrix of order $n$.

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arXiv2025

On the convexity of Berezin range and Berezin radius inequalities via a class of semi-norms

This paper introduces a new family of semi-norms, say $\sigma_\mu$-Berezin norm on the space of all bounded linear operators $B(\mathcal{H})$ defined on a reproducing kernel Hilbert space $\mathcal{H}$, namely, for each $\mu \in [0,1]$ and $p\geq 1$, $$\|T\|_{\sigma_{\mu}\text{-ber}}= \sup_{\lambda\in\Omega}\left\lbrace \left(|\langle T\hat{k}_\lambda,\hat{k}_\lambda\rangle |^p~ \sigma_{\mu}~ \|T\hat{k}_\lambda\|^p\right)^{\frac{1}{p}}\right\rbrace $$ where $T\in B(\mathcal{H})$ and $\sigma_{\mu}$ is an interpolation path of the symmetric mean $\sigma$. We investigate many fundamental properties of the $\sigma_\mu$-Berezin norm and develop several inequalities associated with it. Utilizing these inequalities, we derive improved bounds for the Berezin radius of bounded linear operators, enhancing previously known estimates. Furthermore, we study the convexity of the Berezin range of a class of composition operators and weighted shift operators on both the Hardy space and the Bergman space.

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arXiv2025

A Family of Semi-norms in $C^*$-algebras

We introduce a new family of non-negative real-valued functions on a $C^*$-algebra $\mathcal{A}$, i.e., for $0\leq \mu \leq 1,$ $$\|a\|_{\sigma_{\mu}}= \text{sup}\left\lbrace \sqrt{|f(a)|^2 \sigma_{\mu} f(a^*a)}: f\in \mathcal{A}', \, f(1)=\|f\|=1 \right\rbrace, \quad $$ where $a\in \mathcal{A}$ and $\sigma_{\mu}$ is an interpolation path of the symmetric mean $\sigma$. These functions are semi-norms as they satisfy the norm axioms, except for the triangle inequality. Special cases satisfying triangle inequality, and a complete equality characterization is also discussed. Various bounds and relationships will be established for this new family, with a connection to the existing literature in the algebra of all bounded linear operators on a Hilbert space.

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arXiv2024

Convexity of the Berezin range of finite rank operators

For a bounded linear operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}(\Omega)$ over a nonempty set $\Omega$, the Berezin range of $T$ is defined by \[ \mathrm{Ber}(T)=\left\{\langle T\hat{k}_{\lambda},\hat{k}_{\lambda}\rangle_{\mathcal{H}} : \lambda \in \Omega \right\} \] and the Berezin radius is given by \[ \mathrm{ber}(T)=\sup\left\{ |\gamma| : \gamma \in \mathrm{Ber}(T) \right\}, \] where $\hat{k}_{\lambda}$ denotes the normalized reproducing kernel at $\lambda \in \Omega$. In this paper, we study the convexity of the Berezin range of finite rank operators on the Hardy space and the Bergman space over the unit disc $\mathbb{D}$. We present applications of some scalar inequalities to get some operator inequalities. A characterization of closure of the numerical range of reproducing kernel Hilbert space operator in terms of convex hull of its Berezin range is also discussed.

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arXiv2024

On the convexity of the Berezin range of composition operators and related questions

The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}$ is the set $B(T)$ := $\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle_{\mathcal{H}} : x \in X\}$, where $\hat{k}_{x}$ is the normalized reproducing kernel for $\mathcal{H}$ at $x \in X$. In general, the Berezin range of an operator is not convex. Primarily, we focus on characterizing the convexity of the Berezin range for a class of composition operators acting on the Fock space on $\mathbb{C}$ and the Dirichlet space of the unit disc $\mathbb{D}$. We prove an analogue of the elliptic range theorem for the unitarily equivalent Berezin range of an operator on a two-dimensional reproducing kernel Hilbert space and characterize the convexity of the unitarily equivalent Berezin range for a bounded operator $T$ on a reproducing kernel Hilbert space $\mathcal{H}$.

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arXiv2023

Composition operators, convexity of their Berezin range and related questions

The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}$ is the set $\text{Ber}(T)$ := $\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle_{\mathcal{H}} : x \in X\}$, where $\hat{k}_{x}$ is the normalized reproducing kernel for $\mathcal{H}$ at $x \in X$. In general, the Berezin range of an operator is not convex. In this paper, we discuss the convexity of range of the Berezin transforms. We characterize the convexity of the Berezin range for a class of composition operators acting on the Hardy space and the Bergman space of the unit disk. Also for so-called superquadratic functions, we prove the Berezin set mapping theorem for positive self-adjoint operators $A$ on the reproducing kernel Hilbert space $\mathcal{H}(Ω)$, namely we prove that $f(\mathrm{Ber}(Φ(A)))=\mathrm{Ber}(Φ(f(A)))$, where $Φ:\mathcal{B}%\left( \mathcal{H}\left( Ω\right) \right) \mathcal{\rightarrow}\mathcal{B}\left( \mathcal{K(}Q\mathcal{)}\right) $ is a normalized positive linear map.

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arXiv2022

The structure of twisted power partial isometries

Let $n>1$ and let $\{U_{ij}\}_{1\leq i<j\leq n}$ be $n\choose 2$ commuting unitaries on a Hilbert space $\mathcal{H}$. Suppose $U_{ji}:=U^*_{ij}$, $1\leq i<j\leq n$. An n-tuple of power partial isometries $(V_1,...,V_n)$ on Hilbert space $\mathcal{H}$ is called $\mathcal{U}_n$-twisted power partial isometry with respect to $\{U_{ij}\}_{i<j}$ (or simply $\mathcal{U}_n$-twisted power partial isometry if $\{U_{ij}\}_{i<j}$ is clear from the context) if $V_i^*V_j=U_{ij}V_jV^*_i, ~~ V_iV_j=U_{ji}V_jV_i ~~\text{and}~~ V_kU_{ij}=U_{ij}V_k~~(i,j,k=1,2,...,n,~\text{and}~i\neq j).$ We prove that each $\mathcal{U}_n$-twisted power partial isometry admits a Halmos and Wallen \cite{HW70} type orthogonal decomposition.

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