Related papers: Advanced Refinements of Numerical Radius Inequalit…

We prove some refinements of a reverse AM-GM operator inequality due to M. Lin [Studia Math. 2013;215:187-194]. In particular, we show the operator inequality \begin{eqnarray*} \Phi^p\left(A\nabla_\nu B+2rMm(A^{-1}\nabla B^{-1}-A^{-1}\sharp…

In this paper, we study the relationship between operator space norm and operator space numerical radius on the matrix space $\mathcal{M}_n(X)$, when $X$ is a numerical radius operator space. Moreover, we establish several inequalities for…

We present several operator and norm inequalities for Hilbert space operators. In particular, we prove that if $A_{1},A_{2},...,A_{n}\in {\mathbb B}({\mathscr H})$, then…

Given a Hilbert module $H$ over a $C^*$-algebra, let $\mathcal{L}(H)$ be the set of all adjointable operators on $H$. For each $T\in\mathcal{L}(H)$, its numerical radius is defined by $w(T)=\sup\big\{\|\langle Tx, x \rangle\|: x\in H,…

We investigate $\rho$-orthogonality and its local symmetry in the space of bounded linear operators. A characterization of Hilbert space operators with symmetric numerical range is established in terms of $\rho$-orthogonality. Further, we…

We prove new inequalities for the spectral radius, essential spectral radius, operator norm, measure of noncompactness and numerical radius of Hadamard weighted geometric means of positive kernel operators on Banach function and sequence…

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…

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…

We observe that the classical notion of numerical radius gives rise to a notion of smoothness in the space of bounded linear operators on certain Banach spaces, whenever the numerical radius is a norm. We demonstrate an important class of…

The paper considers some new properties of the so-called $A$-maximal numerical range of operators, denoted by $W_{\max}^A(\cdot)$, where $A$ is a positive bounded linear operator acting on a complex Hilbert space $\mathcal{H}$. Some…

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…

This paper introduces and investigates the concept of the $q$-numerical range for tuples of bounded linear operators in Hilbert spaces. We establish various inequalities concerning the $q$-numerical radius associated with these operator…

Some new trace inequalities for operators in Hilbert spaces are provided. The superadditivity and monotonicity of some associated functionals are investigated and applications for power series of such operators are given. Some trace…

Utilizing the linking algebra of a Hilbert $C^*$-module $\big(\mathscr{V}, {\|\!\cdot\!\|}\big)$, we introduce $\Omega(x)$ as a definition of numerical radius for an element $x\in\mathscr{V}$ and then show that $\Omega(\cdot)$ is a norm on…

The main purpose of this paper is, in the general setting of the adjointable operators on Hilbert $C^*$-modules, to develop two new tools that can be applied to deal with the positive solutions of certain operator equations, the operator…

In this paper, we introduce a new semi-norm of operators on a semi-Hilbertian space, which generalizes the A-numerical radius and A-operator semi-norm. We study the basic properties of this semi-norm, including upper and lower bounds for…

Let $\mathscr{H}$ be a complex Hilbert space and $A,B\in \mathbb{B}(\mathscr{H})$ such that $0<A,B\leq\frac{1}{2}I$. Setting $A':=I-A$ and $B':=I-B$, we prove $$ A'\nabla_\lambda B'-A'!_\lambda B' \leq A\nabla_\lambda B-A!_\lambda B, $$…

The variance of a bounded linear operator $a$ on a Hilbert space $H$ at a unit vector $h$ is defined by $D_h(a)=\|ah\|^2-|<ah,h>|^2$. We show that two operators $a$ and $b$ have the same variance at all vectors $h\in H$ if and only if there…

It is shown that the numerical range of a linear operator operator in a Hilbert space is a (complete) $(1{+}\sqrt2)$-spectral set. The proof relies, among other things, in the behavior of the Cauchy transform of the conjugates of…

We develop several upper and lower bounds for the $A$-Euclidean operator radius of $2$-tuple operators admitting $A$-adjoint, and show that they refine the earlier related bounds. As an application of the bounds developed here, we obtain…