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Pintu Bhunia

Publications and source records attributed to Pintu Bhunia.

At least 19 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 $\mathcal{H}(Ω).$ 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 $n$.

math.FA↗
arXiv2026

Refined numerical radius estimates and Euclidean operator radius

We obtain new lower and upper bounds for the numerical radius of a bounded linear operator $A$ on a complex Hilbert space, which refine the existing ones. In particular, if $w(A)$ and $\|A\|$ denote the numerical radius and operator norm of $A$, respectively, then we show that \begin{eqnarray*} ν(A) + \frac{1}{4} \left\||A|^2+|A^*|^2\right\| \leq w^2(A) \leq \frac12 w\left(\frac{|A|+|A^*|}{2}A \right)+ \frac14 \left\| |A|^2+ \left( \frac{|A|+|A^*|}{2}\right)^2 \right\|, \end{eqnarray*} where $ν(A)\geq 0$ is a real number involving the operator norm of the Cartesian decomposition of $A$. We also develop several new numerical radius inequalities for the products and sums of operators via Euclidean operator radius of $2$-tuples of operators. In addition, we deduce equality characterizations for the inequalities. As an application, we obtain numerical radius inequalities for the commutators of operators, which improves the Fong and Holbrook's inequality $w(AB\pm BA) \leq 2\sqrt{2} w(A) \|B\|$ [Canadian J. Math. 1983].

math.FA↗
arXiv2025

Numerical radius and $\ell_p$ operator norm of Kronecker products and Schur powers: inequalities and equalities

Suppose $A=[a_{ij}]\in \mathcal{M}_n(\mathbb{C})$ is a complex $n \times n$ matrix and $B\in \mathcal{B}(\mathcal{H})$ is a bounded linear operator on a complex Hilbert space $\mathcal{H}$. We show that $w(A\otimes B)\leq w(C),$ where $w(\cdot)$ denotes the numerical radius and $C=[c_{ij}]$ with $c_{ij}= w\left(\begin{bmatrix} 0& a_{ij}\\ a_{ji}&0 \end{bmatrix} \otimes B\right).$ This refines Holbrook's classical bound $w(A\otimes B)\leq w(A) \|B\|$ [J. Reine Angew. Math. 1969], when all entries of $A$ are non-negative. If moreover $a_{ii}\neq 0$ $ \forall i$, we prove that $w(A\otimes B)= w(A) \|B\|$ if and only if $w(B)=\|B\|.$ We then extend these and other results to the more general setting of semi-Hilbertian spaces induced by a positive operator. In the reverse direction, we also specialize these results to Kronecker products and hence to Schur/entrywise products, of matrices: (1)(a) We first provide an alternate proof (using $w(A)$) of a result of Goldberg-Zwas [Linear Algebra Appl. 1974] that if the spectral norm of $A$ equals its spectral radius, then each Jordan block for each maximum-modulus eigenvalue must be $1 \times 1$ ("partial diagonalizability"). (b) Using our approach, we further show given $m \geq 1$ that $w(A^{\circ m})\leq w^m(A)$ - we also characterize when equality holds here. (2) We provide upper and lower bounds for the $\ell_p$ operator norm and the numerical radius of $A\otimes B$ for all $A \in \mathcal{M}_n(\mathbb{C})$, which become equal when restricted to doubly stochastic matrices $A$. Finally, using these bounds we obtain an improved estimation for the roots of an arbitrary complex polynomial.

math.FA↗
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 $σ_μ$-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 $μ\in [0,1]$ and $p\geq 1$, $$\|T\|_{σ_μ\text{-ber}}= \sup_{λ\inΩ}\left\lbrace \left(|\langle T\hat{k}_λ,\hat{k}_λ\rangle |^p~ σ_μ~ \|T\hat{k}_λ\|^p\right)^{\frac{1}{p}}\right\rbrace $$ where $T\in B(\mathcal{H})$ and $σ_μ$ is an interpolation path of the symmetric mean $σ$. We investigate many fundamental properties of the $σ_μ$-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.

math.FA↗
arXiv2025

A new seminorm of $n$-tuple operators and its applications

We introduce a new seminorm of $n$-tuple operators, which generalizes the $A$-Euclidean operator radius of $n$-tuple bounded linear operators on a complex Hilbert space. We introduce and study basic properties of this seminorm. As an application of the present study, we estimate bounds for the $A$-Euclidean operator radius ($A$-joint numerical radius). In addition, we improve on some of the important existing $A$-numerical radius inequalities and related results.

math.FA↗
arXiv2025

A new norm on the space of reproducing kernel Hilbert space operators and Berezin number inequalities

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.

math.FA↗
arXiv2025

Strengthening of spectral radius, numerical radius, and Berezin radius inequalities

Suppose $\mathcal{H}_1, \mathcal{H}_2, \ldots, \mathcal{H}_n$ are arbitrary complex Hilbert spaces, and ${\bf A}=[A_{ij}]$ is an $n\times n$ operator matrix with $A_{ij}\in \mathcal{B}(\mathcal{H}_j, \mathcal{H}_i).$ We show that $w({\bf A}) \leq w\left(\begin{bmatrix} a_{ij} \end{bmatrix}_{i,j=1}^n \right),$ where $w(\cdot)$ denotes the numerical radius and the entries $$ a_{ij}=\begin{cases} w(A_{ii}) & \textit{if $i=j$}, \sqrt{ \left( \|A_{ij}\|+\|A_{ji}\| \right)^2- \left(\|A_{ij}\| \|A_{ji}\|-w(A_{ji}A_{ij}) \right)}^{} & \textit{if $i j$.} \end{cases}$$ This bound improves $w({\bf A}) \leq w\left(\begin{bmatrix} a'_{ij} \end{bmatrix}_{i,j=1}^n \right),$ where $a'_{ij}=w(A_{ii})$ if $i=j$ and $a'_{ij}=\|A_{ij}\|$ if $i\neq j$. We deduce an upper bound for the Kronecker products $A\otimes B$, where $A\in \mathcal{M}_n(\mathbb{C})$ and $B\in \mathcal{B}(\mathcal{H}_1)$, which refines Holbrook's classical bound $w(A\otimes B)\leq w(A)\|B\|$, when all entries of $A$ are non-negative. Further, we obtain the Berezin radius inequalities for $n\times n$ operator matrices where the entries are reproducing kernel Hilbert space operators. We provide an example, which illustrates these inequalities for some concrete operators on the Hardy--Hilbert space. Applying the numerical radius bounds, we show that if $A_i \in \mathcal{B}(\mathcal{H}_i, \mathcal{H}_1) $ and $B_i\in \mathcal{B}(\mathcal{H}_1, \mathcal{H}_i)$ for $i=1,2,$ then \begin{eqnarray*} r(A_1B_1+A_2B_2) \leq \frac{ 1 }{2 } \left(w(B_1A_1)+w(B_2A_2) \right) + \frac{ 1 }{2 } \sqrt{ \left(w(B_1A_1)-w(B_2A_2)\right)^2 + 3\|B_1A_2\|\|B_2A_1\| + η}, \end{eqnarray*} where $η=w(B_2A_1 B_1A_2)$, and $r(\cdot)$ denotes the spectral radius. We also achieve a bound for the roots of an algebraic equation.

math.FA↗
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 $0\leq μ\leq 1,$ $$\|a\|_{σ_μ}= \text{sup}\left\lbrace \sqrt{|f(a)|^2 σ_μ f(a^*a)}: f\in \mathcal{A}', \, f(1)=\|f\|=1 \right\rbrace, \quad $$ where $a\in \mathcal{A}$ and $σ_μ$ is an interpolation path of the symmetric mean $σ$. 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.

math.FA↗
arXiv2024

Inequalities for linear functionals and numerical radii on $\mathbf{C}^*$-algebras

Let $\mathcal{A}$ be a unital $\mathbf{C}^*$-algebra with unit $e$. We develop several inequalities for a positive linear functional $f$ on $\mathcal{A}$ and obtain several bounds for the numerical radius $v(a)$ of an element $a\in \mathcal{A}$. Among other inequalities, we show that if $ a_k, b_k, x_k\in \mathcal{A}$, $r\in \mathbb{N}$ and $f(e)=1$, then \begin{eqnarray*} \left| f \left( \sum_{k=1}^n a_k^*x_kb_k\right)\right|^{r} &\leq& \frac{n^{r-1}}{\sqrt{2}} \left| f\left( \sum_{k=1}^n \big( (b_k^*|x_k| b_k)^{r}+ i (a_k^*|x_k^*|a_k)^{r} \big) \right) \right| \quad (i=\sqrt{-1}), \end{eqnarray*} \begin{eqnarray*} \left| f\left( \sum_{k=1}^n a_k\right)\right|^{2r} &\leq& \frac{n^{2r-1}}{2} f \left( \sum_{k=1}^n Re(|a_k|^r|a_k^*|^r) + \frac12 \sum_{k=1}^n (|a_k|^{2r}+ |a_k^*|^{2r} ) \right). \end{eqnarray*} We find several equivalent conditions for $v(a)=\frac{\|a\|}{2}$ and $v^2(a)={\frac{1}{4}\|a^*a+aa^*\|}$. We prove that $v^2(a)={\frac{1}{4}\|a^*a+aa^*\|}$ (resp., $v(a)=\frac{\|a\|}{2}$) if and only if $\mathbb{S}_{\frac12{ \| a^*a+aa^*\|}^{1/2}} \subseteq V(a) \subseteq \mathbb{D}_{\frac12 {\| a^*a+aa^*\|}^{1/2}}$ (resp., $\mathbb{S}_{\frac12 \| a\|} \subseteq V(a) \subseteq \mathbb{D}_{\frac12 \| a\|}$), where $V(a)$ is the numerical range of $a$ and $\mathbb{D}_k$ (resp., $\mathbb{S}_k$) denotes the circular disk (resp., semi-circular disk) with center at the origin and radius $k$. We also study inequalities for the $(α,β)$-normal elements in $\mathcal{A}.$

math.FA↗
arXiv2024

Numerical Radius Inequalities via Orlicz function

Employing the Orlicz functions we extend the Buzano's inequality which is a refinement of the Cauchy-Schwarz inequality. Also using the Orlicz functions we obtain several numerical radius inequalities for a bounded linear operator as well as the products of operators. We deduce different new upper bounds for the numerical radius. It is shown that \begin{eqnarray*} {w(T)} \leq \sqrt[n]{ \log \left[ \frac{1}{2^{n-1}} e^{w(T^n)} + \left( 1-\frac{1}{2^{n-1}}\right) e^{\|T\|^n}\right]} &\leq& \|T\| \quad \forall n=2,3,4, \ldots \end{eqnarray*} where $w(T)$ and $\|T\|$ denote the numerical radius and the operator norm of a bounded linear operator $T$, respectively.

math.FA↗
arXiv2024

$A$-Davis-Wielandt Radius Bounds of Semi-Hilbertian Space Operators

Consider $\mathcal{H}$ is a complex Hilbert space and $A$ is a positive operator on $\mathcal{H}.$ The mapping $\langle\cdot,\cdot\rangle_A: \mathcal{H}\times \mathcal{H} \to \mathbb {C}$, defined as $\left\langle y,z\right\rangle_{A}=\left\langle Ay,z\right\rangle $ for all $y,z$ $\in $ ${\mathcal{H}}$, induces a seminorm $ \left\Vert \cdot\right\Vert_{A}$. The $A$-Davis-Wielandt radius of an operator $S$ on $\mathcal{H}$ is defined as $dω_{A}\left( S\right) =\sup \left\{ \sqrt{\left\vert \left\langle Sz,z\right\rangle_{A}\right\vert ^{2}+\left\Vert Sz\right\Vert_{A}^{4}} :\left\Vert z\right\Vert_{A}=1\right\} \text{.} $ We investigate some new bounds for $dω_{A}\left( S\right)$ which refine the existing bounds. We also give some bounds for the $2\times 2$ off-diagonal block matrices.

math.FA↗
arXiv2024

Sharper bounds for the numerical radius of $n \times n$ operator matrices II

Let $A=[A_{ij}]$ be an $n\times n$ operator matrix where each $A_{ij}$ is a bounded linear operator on a complex Hilbert space $\mathcal{H}$. With other numerical radius bounds via contraction operators, we show that $w(A) \leq w(\tilde{A}),$ where $\tilde{A}=[a_{ij}]$ is an $n\times n$ complex matrix with \begin{eqnarray*} a_{ij}=\begin{cases} w(A_{ii}) \quad \text{if } i=j\\ \underset{0\leq t \leq 1}{\min} \left\| |A_{ij}|^{2t} + |A_{ji}^*|^{2t} \right\|^{1/2} \left\| |A_{ij}^*|^{2(1-t)}+ |A_{ji}|^{2(1-t)} \right\|^{1/2} \quad \text{if } i< j 0 \quad \text{if } i> j. \end{cases} \end{eqnarray*} This bound refines the well known bound $w(A) \leq w(\hat{A}),$ where $\hat{A}=[\hat{a}_{ij}]$ is an $n\times n$ matrix with $\hat{a}_{ij}= w(A_{ii}) $ \text{if } $i=j$ and $\hat{a}_{ij}= \|A_{ij}\| $ \text{if } $i\neq j$ [Linear Algebra Appl. 468 (2015), 18--26]. We deduce that if $A$, $B$ are bounded linear operators on $\mathcal{H},$ then \begin{eqnarray*} w\left(\begin{bmatrix} 0&A\\ B&0 \end{bmatrix}\right) \leq \frac12 \left\| |A|^{2t} + |B^*|^{2t} \right\|^{1/2} \left\| |A^*|^{2(1-t)}+ |B|^{2(1-t)} \right\|^{1/2} \quad \text{for all } t\in [0,1]. \end{eqnarray*} Further by applying the numerical radius bounds of operator matrices, we deduce some numerical radius bounds for a single operator, the product of two operators, the commutator of operators. We show that if $A$ is a bounded linear operator on $\mathcal{H},$ then $w(A) \leq \frac12 \|A\|^t \left\| |A|^{1-t}+|A^*|^{1-t} \right\| \quad \text{for all } t\in [0,1],$ which refines as well as generalizes the existing ones.

math.FA↗
arXiv2024

Schatten $p$-norm and numerical radius inequalities with applications

We develop a new refinement of the Kato's inequality and using this refinement we obtain several upper bounds for the numerical radius of a bounded linear operator as well as the product of operators, which improve the well known existing bounds. Further, we obtain a necessary and sufficient condition for the positivity of $2\times 2$ certain block matrices and using this condition we deduce an upper bound for the numerical radius involving a contraction operator. Furthermore, we study the Schatten $p$-norm inequalities for the sum of two $n\times n$ complex matrices via singular values and from the inequalities we obtain the $p$-numerical radius and the classical numerical radius bounds. We show that for every $p>0$, the $p$-numerical radius $w_p(\cdot): \mathcal{M}_n(\mathbb C)\to \mathbb R$ satisfies $ w_p(T) \leq \frac12 \sqrt{\left\| |T|^{2(1-t)}+|T^*|^{2(1-t)} \right\|^{} \, \big \||T|^{2t}+|T^*|^{2t} \big\|_{p/2}^{} } $ for all $t\in [0,1]$. Considering $p\to \infty$, we get a nice refinement of the well known classical numerical radius bound $w(T) \leq \sqrt{\frac12 \left\| T^*T+TT^* \right \|}.$ As an application of the Schatten $p$-norm inequalities we develop a bound for the energy of graph. We show that $ \mathcal{E}(G) \geq \frac{2m}{ \sqrt{ \max_{1\leq i \leq n} \left\{ \sum_{j, v_i \sim v_j}d_j\right\}} },$ where $\mathcal{E}(G)$ is the energy of a simple graph $G$ with $m$ edges and $n$ vertices $v_1,v_2,\ldots,v_n$ such that degree of $v_i$ is $d_i$ for each $i=1,2,\ldots,n.$

math.FA↗
arXiv2024

Norm Inequalities for Hilbert space operators with Applications

Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank operator $A,$ it is shown that \begin{eqnarray*} \|A\|_{p} &\leq &\left(\textit{rank} \, A\right)^{1/{2p}} \|A\|_{2p} \,\, \leq \,\, \left(\textit{rank} \, A\right)^{{(2p-1)}/{2p^2}} \|A\|_{2p^2}, \quad \textit{for all $p\geq 1 $} \end{eqnarray*} where $\|\cdot\|_p$ is the Schatten $p$-norm. If $\{ λ_n(A) \}$ is a listing of all non-zero eigenvalues (with multiplicity) of a compact operator $A$, then we show that \begin{eqnarray*} \sum_{n} \left|λ_n(A)\right|^{p} &\leq& \frac12 \| A\|_{ p}^{ p} + \frac12 \| A^2\|_{p/2}^{p/2}, \quad \textit{for all $p\geq 2$} \end{eqnarray*} which improves the classical Weyl's inequality $\sum_{n} \left|λ_n(A)\right|^{p} \leq \| A\|_{ p}^{ p}$ [Proc. Nat. Acad. Sci. USA 1949]. For an $n\times n$ matrix $A$, we show that the function $p\to n^{-{1}/{p}}\|A\|_p$ is monotone increasing on $p\geq 1,$ complementing the well known decreasing nature of $p\to \|A\|_p.$ \indent As an application of these inequalities, we provide an upper bound for the sum of the absolute values of the zeros of a complex polynomial. As another application we provide a refined upper bound for the energy of a graph $G$, namely, $\mathcal{E}(G) \leq \sqrt{2m\left(\textit{rank Adj(G)} \right)},$ where $m$ is the number of edges, improving on a bound by McClelland in $1971$.

math.FA↗
arXiv2023

Euclidean operator radius and numerical radius inequalities

Let $T$ be a bounded linear operator on a complex Hilbert space $\mathscr{H}.$ We obtain various lower and upper bounds for the numerical radius of $T$ by developing the Euclidean operator radius bounds of a pair of operators, which are stronger than the existing ones. In particular, we develop an inequality that improves on the inequality $$ w(T) \geq \frac12 {\|T\|}+\frac14 {\left|\|Re(T)\|-\frac12 \|T\| \right|} + \frac14 { \left| \|Im(T)\|-\frac12 \|T\| \right|}.$$ Various equality conditions of the existing numerical radius inequalities are also provided. Further, we study the numerical radius inequalities of $2\times 2$ off-diagonal operator matrices. Applying the numerical radius bounds of operator matrices, we develop the upper bounds of $w(T)$ by using $t$-Aluthge transform. In particular, we improve the well known inequality $$ w(T) \leq \frac12 {\|T\|}+ \frac12{ w(\widetilde{T})}, $$ where $\widetilde{T}=|T|^{1/2}U|T|^{1/2}$ is the Aluthge transform of $T$ and $T=U|T|$ is the polar decomposition of $T$.

math.FA↗
arXiv2023

Estimations of Euclidean operator radius

We develop several Euclidean operator radius bounds for the product of two $d$-tuple operators using positivity criteria of a $2\times 2$ block matrix whose entries are $d$-tuple operators. From these bounds, by using the polar decomposition of operators, we obtain Euclidean operator radius bounds for $d$-tuple operators. Among many other interesting bounds, it is shown that \begin{eqnarray*} w_e(\mathbf{A}) &\leq&\frac1{\sqrt2} \mathbf{A}\|^{1/2}\sqrt{\left\|\sum_{k=1}^{d} (|A_k|+|A_k^*|)\right\|}, \end{eqnarray*} where $w_e(\mathbf{A})$ and $\|\mathbf{A}\|$ are the Euclidean operator radius and the Euclidean operator norm, respectively, of a $d$-tuple operator $\mathbf{A}=(A_1,A_2, \ldots,A_d).$ Further, we develop an upper bound for the Euclidean operator radius of $n\times n$ operator matrix whose entries are $d$-tuple operators. In particular, it is proved that if $\begin{bmatrix} \mathbf{A_{ij}} \end{bmatrix}_{n\times n}$ is an $n\times n$ operator matrix then $$ w_e\left( \begin{bmatrix} \mathbf{A_{ij}} \end{bmatrix}_{n\times n}\right)\leq w \left(\begin{bmatrix} a_{ij} \end{bmatrix}_{n\times n}\right),$$ where each $\mathbf{A_{ij}}$ is a $d$-tuple operator, $1\leq i,j\leq n$, $a_{ij}=w_e(\mathbf{A_{ij}})\, \textit{ if i=j}$, $a_{ij}= \sqrt{w_e\left(|\mathbf{A_{ji}|}+|\mathbf{A_{ij}^*}|\right)w_e\left(|\mathbf{A_{ij}|}+|\mathbf{A_{ji}^*}|\right)}\,\textit{ if $i j$}.$ Other related applications are also discussed.

math.FA↗
arXiv2023

Berezin number and Berezin norm inequalities for operator matrices

We establish new upper bounds for Berezin number and Berezin norm of operator matrices, which are refinements of the existing bounds. Among other bounds, we prove that if $A=[A_{ij}]$ is an $n\times n$ operator matrix with $A_{ij}\in\mathbb{B}(\mathcal{H})$ for $i,j=1,2\dots n$, then $\|A\|_{ber} \leq \left\|\left[\|A_{ij}\|_{ber}\right]\right\|$ and $\textbf{ber}(A) \leq w([a_{ij}]),$ where $a_{ii}=\textbf{ber}(A_{ii}),$ $a_{ij}=\big\||A_{ij}|+|A^*_{ji}|\big\|^{\frac{1}{2}}_{ber} \big\||A_{ji}|+|A^*_{ij}|\big\|^{\frac{1}{2}}_{ber}$ if $i j$. Further, we give some examples for the Berezin number and Berezin norm estimation of operator matrices on the Hardy-Hilbert space.

math.FA↗