This paper deals with the so-called A-numerical radius associated with a positive (semi-definite) bounded linear operator A acting on a complex Hilbert space H. Several new inequalities involving this concept are established. In particular, we prove several estimates for 2×2 operator matrices whose entries are A-bounded operators. Some of the obtained results cover and extend well-known recent results due to Bani-Domi and Kittaneh. In addition, several improvements of the generalized Kittaneh estimates are obtained. The inequalities given by Feki in his work represent a generalization of the inequalities given by Kittaneh. Some refinements of the inequalities due to Feki are also presented.

/pdf/further-inequalities-for-the-weighted-numerical-radius-of-tjeua992.pdf

Further Inequalities for the Weighted Numerical Radius of Operators

The aim of this paper is to provide new upper bounds of ω(T), which denotes the numerical radius of a bounded operator T on a Hilbert space (H,⟨·,·⟩). We show the Aczél inequality in terms of the operator |T|. Next, we give certain inequalities about the A-numerical radius ωA(T) and the A-operator seminorm ∥T∥A of an operator T. We also present several results related to the A-numerical radius of 2×2 block matrices of semi-Hilbert space operators, by using symmetric 2×2 block matrices.

/pdf/on-some-generalizations-of-cauchy-schwarz-inequalities-and-17gy9ca6.pdf

On Some Generalizations of Cauchy-Schwarz Inequalities and Their Applications

Bounds for A-numerical radius based on an extension of A-Buzano inequality

A refinement of A-Buzano inequality and applications to A-numerical radius inequalities

Bounds for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e24" altimg="si7.svg"><mml:mi mathvariant="double-struck">A</mml:mi></mml:math>-numerical radius based on an extension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e29" al

Let A be a positive (semidefinite) bounded linear operator on a complex Hilbert space $$\big ({\mathcal {H}}, \langle \cdot , \cdot \rangle \big )$$
 . The semi-inner product induced by A is defined by $${\langle x, y\rangle }_A := \langle Ax, y\rangle $$
 for all $$x, y\in {\mathcal {H}}$$
 and defines a seminorm $${\Vert \cdot \Vert }_A$$
 on $${\mathcal {H}}$$
 . This makes $${\mathcal {H}}$$
 into a semi-Hilbert space. For $$p\in [1,+\infty )$$
 , the generalized A-joint numerical radius of a d-tuple of operators $${\mathbf {T}}=(T_1,\ldots ,T_d)$$
 is given by $$\begin{aligned} \omega _{A,p}({\mathbf {T}}) =\sup _{\Vert x\Vert _A=1}\left( \sum _{k=1}^d|\big \langle T_kx, x\big \rangle _A|^p\right) ^{\frac{1}{p}}. \end{aligned}$$
 Our aim in this paper is to establish several bounds involving $$\omega _{A,p}(\cdot )$$
 . In particular, under suitable conditions on the operators tuple $${\mathbf {T}}$$
 , we generalize the well-known inequalities due to Kittaneh (Studia Math 168(1):73–80, 2005).

On some inequalities for the generalized joint numerical radius of semi-Hilbert space operators

. In this article, we present some new inequalities for the numerical radius of products of Hilbert space operators and the generalized Aluthge transform. In particular, we show some upper bounds for ω ( ABC + DEF ) using the celebrated Buzano inequality, then some consequences that generalize some results from the literature are discussed. After that, inequalities that involve the generalized Aluthge transform are shown using some known bounds for the numerical radius of the product of two operators.

Further numerical radius inequalities

The paper deals with the generalized numerical radius of linear operators acting on a complex Hilbert space , which are bounded with respect to the seminorm induced by a positive operator A on . Here A is not assumed to be invertible. Mainly, if we denote by and the generalized and the classical numerical radii respectively, we prove that for every A-bounded operator T we have where is the Moore-Penrose inverse of . In addition, several new inequalities involving for single and several operators are established. In particular, by using new techniques, we cover and improve some recent results due to Najafi [Linear Algebra Appl. 2020;588:489–496].

Some numerical radius inequality for several semi-Hilbert space operators

The main goal of this work is to present new matrix inequalities of the Cauchy-Schwarz type. In particular, we investigate the so-called Lieb functions, whose definition came as an umbrella of Cauchy-Schwarz-like inequalities, then we consider the mixed Cauchy-Schwarz inequality. This latter inequality has been influential in obtaining several other matrix inequalities, including numerical radius and norm results. Among many other results, we show that \[\left\| T \right\|\le \frac{1}{4}\left( \left\| \left| T \right|+\left| {{T}^{*}} \right|+2\mathfrak RT \right\|+\left\| \left| T \right|+\left| {{T}^{*}} \right|-2\mathfrak RT \right\| \right),\] where $\mathfrak RT$ is the real part of $T$.

Cristian Conde

Papers

On some inequalities for the generalized joint numerical radius of semi-Hilbert space operators

Further numerical radius inequalities

Some numerical radius inequality for several semi-Hilbert space operators

On the matrix Cauchy-Schwarz inequality

Norm Inequalities via Convex and Log-Convex Functions