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Raj Kumar Nayak

Publications and source records attributed to Raj Kumar Nayak.

10 recordsLinked to original sources
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.

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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.

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arXiv2024

Enhancement of the Cauchy-Schwarz Inequality and Its Implications for Numerical Radius Inequalities

In this article, we establish an improvement of the Cauchy-Schwarz inequality. Let $x, y \in \mathcal{H},$ and let $f: (0,1) \rightarrow \mathbb{R}^+$ be a well-defined function, where $\mathbb{R}^+$ denote the set of all positive real numbers. Then \[|\langle x, y \rangle|^2 \leq \frac{f(t)}{1+f(t)} \|x\|^2 \|y\|^2 + \frac{1}{1+ f(t)} |\langle x, y \rangle | \|x\|\|y\|. \] We have applied this result to derive new and improved upper bounds for the numerical radius.

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arXiv2023

Weighted numerical radius inequalities for operator and operator matrices

The concepts of weighted numerical radius has been defined in recent times. In this article, we obtain several upper bound for weighted numerical radius of operators and $2 \times 2$ operator matrices which generalize and improves some well known famous inequality for classical numerical radius. We also obtain an upper bound for the weighted numerical radius of the Aluthge transformation, $\tilde{T}$ of an operator $T \in \mathcal{B}(\mathcal{H}),$ where $\tilde{T} = |T|^{1/2} U |T|^{1/2}$ and $T = U |T|$ be the canonical polar decomposition of $T.$

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arXiv2020

Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators

We give new inequalities for $A$-operator seminorm and $A$-numerical radius of semi-Hilbertian space operators and show that the inequalities obtained here generalize and improve on the existing ones. Considering a complex Hilbert space $\mathcal{H}$ and a non-zero positive bounded linear operator $A$ on $\mathcal{H},$ we show with among other seminorm inequalities, if $S,T,X\in \mathcal{B}_A(\mathcal{H})$, i.e., if $A$-adjoint of $S,T,X$ exist then $$2\|S^{\sharp_A}XT\|_A \leq \|SS^{\sharp_A}X+XTT^{\sharp_A}\|_A.$$ Further, we prove that if $T\in \mathcal{B}_A(\mathcal{H})$ then \begin{eqnarray*} \frac{1}{4}\|T^{\sharp_{A}}T+TT^{\sharp_{A}}\|_A \leq \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg), ~~\textit{and} \end{eqnarray*} \begin{eqnarray*} \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg) +\frac{1}{8}c_A^2\big(T+T^{\sharp_{A}}\big)+\frac{1}{8}c_A^2\big(T-T^{\sharp_{A}}\big) \leq w^2_A(T). \end{eqnarray*} Here $w_A(.), c_A(.)$ and $\|.\|_A $ denote $A$-numerical radius, $A$-Crawford number and $A$-operator seminorm, respectively.

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arXiv2020

Improvement of $A$-numerical radius inequalities of semi-Hilbertian space operators

Let $\mathcal{H}$ be a complex Hilbert space and let $A$ be a positive operator on $\mathcal{H}$. We obtain new bounds for the $A$-numerical radius of operators in semi-Hilbertian space $\mathcal{B}_A(\mathcal{H})$ that generalize and improve on the existing ones. Further, we estimate bounds for the $B$-operator seminorm and $B$-numerical radius of $2\times 2$ operator matrices, where $B=\mbox{diag}(A,A)$. The bounds obtained here improve on the existing ones.

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arXiv2020

Refinements of A-numerical radius inequalities and their applications

We present sharp lower bounds for the A-numerical radius of semi-Hilbertian space operators. We also present an upper bound. Further we compute new upper bounds for the $B$-numerical radius of $2 \times 2$ operator matrices where $B = \textit{diag}(A,A)$, $A$ being a positive operator. As an application of the A-numerical radius inequalities, we obtain a bound for the zeros of a polynomial which is quite a bit improvement of some famous existing bounds for the zeros of polynomials.

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arXiv2020

On inequalities for A-numerical radius of operators

Let $A$ be a positive operator on a complex Hilbert space $\mathcal{H}.$ We present inequalities concerning upper and lower bounds for $A$-numerical radius of operators, which improve on and generalize the existing ones, studied recently in [A. Zamani, A-Numerical radius inequalities for semi-Hilbertian space operators, Linear Algebra Appl. 578 (2019) 159-183]. We also obtain some inequalities for $B$-numerical radius of $2\times 2$ operator matrices where $B$ is the $2\times 2$ diagonal operator matrix whose diagonal entries are $A$. Further we obtain upper bounds for $A$-numerical radius for product of operators which improve on the existing bounds.

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arXiv2020

Estimations of zeros of a polynomial using numerical radius inequalities

We present new bounds for the numerical radius of bounded linear operators and $2\times 2$ operator matrices. We apply upper bounds for the numerical radius to the Frobenius companion matrix of a complex monic polynomial to obtain new estimations for zeros of that polynomial. We also show with numerical examples that our new estimations improve on the existing estimations.

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