numerical radius

New norm and numerical radius inequalities for operators on Hilbert space are given. Among other inequalities, we prove that if $ A, B \in B(H) $, then \[\Vert A \Vert - \frac{3 \Vert A-B^* \Vert }{2} \leq \omega\left(\left[\begin{array}{cc} 0 & A \\ B & 0 \end{array}\right]\right).\] Moreover, $\omega(AB) \leq \frac{3}{2} \Vert Im(A) \Vert \Vert B \Vert + D_{B}\; \omega(A) $. In particular, if $ A $ is self-adjointable, then $\omega(AB) \leq D_{B} \Vert A \Vert$, where $D_{B}=\underset{\lambda \in \mathbb{C}}{\mathop{\inf}}\,\left\| B-\lambda I \right\|$.

In this note, we obtain a reverse version of the Haagerup Theorem. In particular, if A ∈ Mn has a 2 × 2− principal submatrix as  1 α β 1  with β 6= ¯α, then kSAk > 1 where the operator SA : Mn −→ Mn is defined by SA(B) := A ◦ B where ” ◦ ” stands for Schur product

In this paper, several inequalities involving the HilbertSchmidt numerical radius inequalities for 2 × 2 operator matrices operators are established. In particular, we obtain some generalizations and refinements of earlier inequalities. Some upper and lower bounds for the Hilbert-Schmidt numerical radius inequalities for 2 × 2 operator matrices operators is also given.

In this paper, we present several numerical radius and norm inequalities for sum of Hilbert space operators. These inequalities improve some earlier related inequalities. For $A,B\in B\left( H \right)$, we prove that\[\omega \left( {{B}^{*}}A \right)\le \sqrt{\frac{1}{2}{{\left\| A \right\|}^{2}}{{\left\| B \right\|}^{2}}+\frac{1}{2}\omega \left( {{\left| B \right|}^{2}}{{\left| A \right|}^{2}} \right)}\le 4\omega \left( A \right)\omega \left( B \right).\]

We give several inequalities involving numerical radii $\omega \left( \cdot \right)$ and the usual operator norm $\left\| \cdot \right\|$ of Hilbert space operators. These inequalities lead to a considerable improvement in the well-known inequalities                                                                        ½||T||≤ω(T)≤||T||

In this paper, we introduce some numerical radius inequalities for products of two Hilbert space operators. Some of our inequalities improve well-known ones.

By taking into account that the computation of the numerical radius is an optimization problem, we prove, in this paper, several refinements of the numerical radius inequalities for Hilbert space operators. It is shown, among other inequalities, that if A is a bounded linear operator on a complex Hilbert space, thenω(A)≤½√(|| |A|2+|A*|2||+|| |A| |A*|+|A*| |A| ||),where ω(A), ||A||, and |A| are the numerical radius, the usual operator norm, and the absolute value of A, respectively. This inequality provides a refinement of an earlier numerical radius inequality due to Kittaneh, namely,ω(A)≤½(||A||+||A2||)½.Some related inequalities are also discussed.

We present some new numerical radius inequalities of Hilbert space operators. We improve and generalize some inequalities with re- spect to Specht’s ratio. Let A and B be two positive invertible operators on a Hilbert space H and let X be a bounded operator on H. Then ω((A♮B)X) ≤ 1 2S(√h) ∥X∗BX + A∥, (h > 0, h ̸ = 1) where ∥ · ∥, ω(·), S(·), and ♮ denote the usual operator norm, numeri- cal radius, the Specht’s ratio, and the operator geometric mean, respec- tively.

‎The main purpose of this note is to define an analogues of the numerical radius related to the matricial range‎. ‎However‎, ‎we will find relations between the numerical radius and matricial range of an operator‎. ‎The tone of the paper is mostly expository‎.

We obtain some inequalities related to the powers of numerical radius inequalities of Hilbert space operators. Some results that employ the Hermite-Hadamard inequality for vectors in normed linear spaces are also obtained. We improve and generalize some inequalities with respect to Specht's ratio. Among them, we show that, if $A, Bin mathcal{B(mathcal{H})}$ satisfy in some conditions, it follows that begin{equation*} omega^2(A^*B)leq frac{1}{2S(sqrt{h})}Big||A|^{4}+|B|^{4}Big|-displaystyle{inf_{|x|=1}} frac{1}{4S(sqrt{h})}big(biglangle big(A^*A-B^*Bbig) x,xbigranglebig)^2 end{equation*} for some $h>0$, where $|cdot|,,,,omega(cdot)$ and $S(cdot)$ denote the usual operator norm, numerical radius and the Specht's ratio, respectively.

In this paper, the behavior of the pseudopolynomial numerical hull of a square complex matrix with respect to structured perturbations and its radius is investigated.

For A ∈ M n, the Schur multiplier of A is defined as S A(X) = A ◦ X for all X ∈ M n and the spectral norm of S A can be state as ∥S A∥ = supX,0 ∥A ∥X ◦X ∥ ∥. The other norm on S A can be defined as ∥S A∥ω = supX,0 ω(ω S( AX (X ) )) = supX,0 ωω (A (X ◦X ) ), where ω(A) stands for the numerical radius of A. In this paper, we focus on the relation between the norm of Schur multiplier of product of matrices and the product of norm of those matrices. This relation is proved for Schur product and geometric product and some applications are given. Also we show that there is no such relation for operator product of matrices. Furthermore, for positive definite matrices A and B with ∥S A∥ω ⩽ 1 and ∥S B∥ω ⩽ 1, we show that A♯B = n(I − Z)1/2C(I Z)1/2, for some contraction C and Hermitian contraction Z.

For an n-by-n complex matrix A in a block form with the (possibly) nonzero blocks only on the diagonal above the main one, we consider two other matrices whose nonzero entries are along the diagonal above the main one and consist of the norms or minimum moduli of the diagonal blocks of A. In this paper, we obtain two inequalities relating the numeical radii of these matrices and also determine when either of them becomes an equality.

In this paper we will generalize a singular value inequality that was proved before. In particular we obtain an inequality for numerical radius as follows: begin{equation*} 2 sqrt{t (1-t)} omega(t A^{nu}B^{1-nu}+(1-t)A^{1-nu}B^{nu}) leq omega(t A + (1- t) B), end{equation*} where, $ A $ and $ B $ are positive semidefinite matrices, $ 0 leq t leq 1 $ and $ 0 leq nu leq frac{3}{2}.$