Numerical Radius Inequalities Research Articles

Furtherance of numerical radius inequalities of Hilbert space operators

If A, B are bounded linear operators on a complex Hilbert space, then we prove that $$\begin{aligned} w(A)\le & {} \frac{1}{2}\left( \Vert A\Vert +\sqrt{r\left( |A||A^*|\right) }\right) ,\\ w(AB \pm BA)\le & {} 2\sqrt{2}\Vert B\Vert \sqrt{ w^2(A)-\frac{c^2(\mathfrak {R}(A))+c^2(\mathfrak {I}(A))}{2} }, \end{aligned}$$ where $$w(\cdot ),\left\| \cdot \right\| $$ , and $$r(\cdot )$$ are the numerical radius, the operator norm, the Crawford number, and the spectral radius respectively, and $$\mathfrak {R}(A)$$ , $$\mathfrak {I}(A)$$ are the real part, the imaginary part of A respectively. The inequalities obtained here generalize and improve on the existing well known inequalities.

On $${\mathbb {A}}$$-numerical radius equalities and inequalities for certain operator matrices

The main goal of this article is to establish several new {mathbb {A}}-numerical radius equalities for ntimes n circulant, skew circulant, imaginary circulant, imaginary skew circulant, tridiagonal, and anti-tridiagonal operator matrices, where {mathbb {A}} is the ntimes n diagonal operator matrix whose diagonal entries are positive bounded operator A. Some special cases of our results lead to the results of earlier works in the literature, which shows that our results are more general. Further, some pinching type {mathbb {A}}-numerical radius inequalities for ntimes n block operator matrices are given. Some equality conditions are also given. We also provide a concluding section, which may lead to several new problems in this area.

On a new norm on $$\mathcal {B}({\mathcal {H}})$$ and its applications to numerical radius inequalities

We introduce a new norm on the space of all bounded linear operators on a complex Hilbert space, which generalizes the numerical radius norm, the usual operator norm and the modified Davis–Wielandt radius norm. We study basic properties of this norm, including the upper and the lower bounds for it. As an application of the present study, we estimate bounds for the numerical radius of bounded linear operators. We illustrate that our results improve on some of the important existing numerical radius inequalities. Other application of this new norm have also studied.

Norm and numerical radius inequalities for sum of operators

Norm and numerical radius inequalities for sum of operators

On some numerical radius inequalities for normal operators in Hilbert spaces

Let T be a normal bounded operator on a Hilbert space and let ω(T) denote the numerical radius of T. In this paper, we give new inequalities numerical radius of normal operatos on a Hilbert space, one of these inequalties we prove that where T = T 1 + iT 2 the Cartesian decomposition of T and . Moreover, some other related results are also obtained.

Some $${\mathbb {A}}$$-numerical radius inequalities for $$d\times d$$ operator matrices

Let A be a positive (semidefinite) bounded linear operator acting on a complex Hilbert space \(\big ({\mathcal {H}}, \langle \cdot , \cdot \rangle \big )\). The semi-inner product \({\langle x, y\rangle }_A := \langle Ax, y\rangle \), \(x, y\in {\mathcal {H}}\) induces a seminorm \({\Vert \cdot \Vert }_A\) on \({\mathcal {H}}\). Let T be an A-bounded operator on \({\mathcal {H}}\), the A-numerical radius of T is given by $$\begin{aligned} \omega _A(T) = \sup \Big \{\big |{\langle Tx, x\rangle }_A\big |\;; \,\,x\in {\mathcal {H}}, \;{\Vert x\Vert }_A = 1\Big \}. \end{aligned}$$In this paper, we establish several inequalities for \(\omega _{\mathbb {A}}({\mathbb {T}})\), where \({\mathbb {T}}=(T_{ij})\) is a \(d\times d\) operator matrix with \(T_{ij}\) are A-bounded operators and \({\mathbb {A}}\) is the diagonal operator matrix whose each diagonal entry is A. Some of the obtained results generalize some earlier inequalities proved by Bhunia et al. (Math. Inequal. Appl. 24(1), 167–183, 2021).

Refined and generalized numerical radius inequalities for 2 × 2 operator matrices

We give several refined and generalized numerical radius inequalities for 2×2 operator matrices. These inequalities improve and generalize earlier related inequalities. An application of some of our results is also given.

Numerical radius inequalities for operator matrices

In this paper, we firstly establish new numerical radius inequalities which refine a result of Kittaneh in [Studia Math. 168, 73–80 (2005)], then present some numerical radius inequalities involving non-negative increasing convex functions for operator matrices, which generalize the related results of Shebrawi in [Linear Algebra Appl. 523(15), 1–12 (2017)].

Some Refinements of the Numerical Radius Inequalities via Young Inequality

In this paper, we get an improvement of the Hölder-McCarthy operator inequality in the case when r ≥ 1 and refine generalized inequalities involving powers of the numerical radius for sums and products of Hilbert space operators.

Some Refinements of Numerical Radius Inequalities

We propose some refinements for the second inequality in $$ \frac{1}{2}\left\Vert A\right\Vert \le w(A)\le \left\Vert A\right\Vert, $$ where A ∈ B(H). In particular, if A is hyponormal, then, by refining the Young inequality with the Kantorovich constant K K(⋅, ⋅), we show that $$ w(A)\le \frac{1}{2{\operatorname{inf}}_{\left\Vert x\right\Vert =1}\zeta (x)}\left|\left\Vert A\right.\right|++\left|\left.{A}^{\ast}\right\Vert \right|\le \frac{1}{2}\left|\left\Vert A\right.\right|+\left|\left.{A}^{\ast}\right\Vert \right|, $$ where $$ \upzeta (x)=K{\left(\frac{\left\langle \left|A\right|x,x\right\rangle }{\left\langle \left|{A}^{\ast}\right|x,x\right\rangle },2\right)}^r,r=\min \left\{\uplambda, 1-\uplambda \right\}, $$ and 0 ≤ λ ≤ 1. We also give a reverse for the classical numerical radius power inequality w(An) ≤ wn(A) for any operator A ∈ B(H) case where n = 2.

Numerical Radius Inequalities Concerning with Algebra Norms

We present an expression for the generalized numerical radius associated with a norm on the algebra of bounded linear operators on a Hilbert space and then apply it to obtain upper and lower bounds for the generalized numerical radius. We also establish some generalized numerical radius inequalities involving the product of two operators. Applications of our inequalities are also provided.

On some numerical radius inequalities for Hilbert space operators

This article is devoted to studying some new numerical radius inequalities for Hilbert space operators. Our analysis enables us to improve an earlier bound for numerical radius due to Kittaneh. It is shown, among other, that if $A\in \mathcal{B}(\mathcal{H})$, then \[ \frac{1}{8}\left( {{\left\| A+{{A}^{*}} \right\|}^{2}}+{{\left\| A-{{A}^{*}} \right\|}^{2}} \right)\le \omega ^{2}\left( A \right) \le \left\| \frac{{{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}}}{2} \right\|-m\left( {{\left( \frac{\left| A \right|-\left| {{A}^{*}} \right|}{2} \right)}^{2}} \right ). \] <br><br> Отримані нові нерівності для числового радіуса операторів у гільбертовім просторі. Зокрема, покращено попередній результат Кіттане. Показано, що для $A\in B(H)$, \[ \frac{1}{8}\left( {{\left\| A+{{A}^{*}} \right\|}^{2}}+{{\left\| A-{{A}^{*}} \right\|}^{2}} \right)\le \omega ^{2}\left( A \right) \le \left\| \frac{{{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}}}{2} \right\|-m\left( {{\left( \frac{\left| A \right|-\left| {{A}^{*}} \right|}{2} \right)}^{2}} \right ). \]

Some generalizations of numerical radius inequalities for Hilbert space operators

Some generalizations of numerical radius inequalities for Hilbert space operators

Some general quadratic numerical radius inequalities for the off-diagonal parts of $2 ⨉ 2$ block operator matrices

Some general quadratic numerical radius inequalities for the off-diagonal parts of $2 ⨉ 2$ block operator matrices

A Chain of numerical radius inequalities in complex Hilbert space

A Chain of numerical radius inequalities in complex Hilbert space

Some refinements of numerical radius inequalities

UDC 517.5 In this paper, we give some refinements for the second inequality in where In particular, if is hyponormal by refining the Young inequality with the Kantorovich constant we show that where and . We also give a reverse for the classical numerical radius power inequality for any operator in the case when

Generalized numerical radius inequalities of operators in Hilbert spaces

This paper is concerned with linear operators on a complex Hilbert space $$\mathcal {H}$$ , which are bounded with respect to the seminorm induced by a positive operator A on $$\mathcal {H}$$ . In particular, several inequalities involving the A-numerical radius of operators are established.

New estimates for the numerical radius of Hilbert space operators

New numerical radius inequalities for Hilbert space operators are given. These inequalities improve a well-known inequality of F. Kittaneh. We also show that the inequalities obtained here are sharp. A refinement of the triangle inequality is also shown.

On 𝔸-numerical radius inequalities for 2 × 2 operator matrices

Let be a complex Hilbert space, and A be a positive bounded linear operator on Let denote the set of all bounded linear operators on whose A-adjoint exists. Let denote a operator matrix of the form . Very recently, for a strictly positive operator A, Bhunia et al. [On inequalities for A-numerical radius of operators. Electron J Linear Algebra. 2020;36:143–157] proved an important lemma (Lemma 2.4) to establish several -numerical radius inequalities for operator matrices in . In this article, we first prove an analogous result and then provide a new proof of the same lemma by dropping the assumption ‘A is strictly positive’. We then establish several new upper and lower bounds for the -numerical radius of an operator matrix whose entries are operators in Further, we prove some refinements of earlier A-numerical radius inequalities for operators in .

Numerical Radius Inequalities in 2-Inner Product Spaces

In this paper, we have obtained the analogue results on numerical radius inequalities from the classical inner product spaces to 2-inner product spaces. We have established several related reverse inequalities and some well known results in 2-inner product spaces