Related papers: Inequalities involving Berezin norm and Berezin nu…
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_{ \lambda, \mu \in \Omega} \left\{…
In this article, we establish the Berezin number and Berezin norm inequalities for bounded linear operators on a reproducing kernel Hilbert space using the Moore-Penrose inverse. The inequalities obtained here refine and generalize the…
For a bounded linear operator $A$ on a reproducing kernel Hilbert space $\mathcal{H}(\Omega)$, with normalized reproducing kernel $\widehat{k}_{\lambda} = \frac{k_{\lambda}}{\lVert k_{\lambda}\lVert}$, the Berezin symbol, Berezin number and…
The Berezin symbol $\widetilde{A}$ of an operator $A$ acting on the reproducing kernel Hilbert space ${\mathscr H}={\mathscr H(}\Omega)$ over some (non-empty) set is defined by $\widetilde{A}(\lambda)=\langle…
Several upper and lower bounds of the Davis-Wielandt-Berezin radius of bounded linear operators defined on a reproducing kernel Hilbert space are given. Further, an inequality involving the Berezin number and the Davis-Wielandt-Berezin…
For a bounded linear operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}(\Omega)$ over some non-empty set $\Omega$, the Berezin range and the Berezin radius of $T$ are defined respectively, by $\text{Ber}(T) := \{\langle…
In this paper, several refinements of the Berezin number inequalities are obtained. We generalize inequalities involving powers of the Berezin number for product of two operators acting on a reproducing kernel Hilbert space $\mathcal…
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…
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…
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…
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…
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…
In this paper, we generalize several Berezin number inequalities involving product of operators. For instance, we show that if $A, B$ are positive operators and $X$ is any operator, then \begin{align*}…
In this article, we present some new general forms of numerical radius inequalities for Hilbert space operators. The significance of these inequalities follow from the way they extend and refine some known results in this field. Among other…
Some elementary inequalities providing upper bounds for the difference of the norm and the numerical radius of a bounded linear operator on Hilbert spaces under appropriate conditions are given.
In this paper, by using of the definition Berezin symbol, we show some Berezin number inequalities. Among other inequalities, it is shown that if $A, B, X\in{\mathbb{B}}(\mathscr H)$, then $$\mathbf{ber}(AX\pm XA)\leqslant…
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…
We introduce a new norm on the space of bounded linear operators on a complex Hilbert space, which generalizes the numerical radius norm, the usual operator norm and the modified Davis-Wielandt radius. We study basic properties of this…
We present some new upper and lower bounds for the numerical radius of bounded linear operators on a complex Hilbert space and show that these are stronger than the existing ones. In particular, we prove that if $A$ is a bounded linear…
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…