A-numerical Radius Research Articles

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

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

On approximate A-seminorm and A-numerical radius orthogonality of operators

On approximate A-seminorm and A-numerical radius orthogonality of operators

A collection of seminorms linking the A-numerical radius and the operator A-seminorm

We investigate a novel operator seminorm, QA,mλ,f, for an A-bounded operator Q, where A is a positive operator on a complex Hilbert space (K,⟨·,·⟩). This seminorm is defined using a continuous increasing and bijective function f:R+⟶R+ and an interpolational path mλ of the symmetric mean m. Specifically, QA,mλ,f=supf−1fQy,yAmλfQyA:y∈K,yA=1, where f−1 represents the reciprocal function of f, and ⟨·,·⟩A and ·A denote the semi-inner product and seminorm, respectively, induced by A on K. We explore various bounds and relationships associated with this new concept, establishing connections with existing literature.

Generalized Cauchy–Schwarz Inequalities and A-Numerical Radius Applications

The purpose of this research paper is to introduce new Cauchy–Schwarz inequalities that are valid in semi-Hilbert spaces, which are generalizations of Hilbert spaces. We demonstrate how these new inequalities can be employed to derive novel A-numerical radius inequalities, where A denotes a positive semidefinite operator in a complex Hilbert space. Some of our novel A-numerical radius inequalities expand upon the existing literature on numerical radius inequalities with Hilbert space operators, which are important tools in functional analysis. We use techniques from semi-Hilbert space theory to prove our results and highlight some applications of our findings.

Further Seminorm and Numerical Radius Inequalities for Products and Sums of Operators

Our aim in this article is to establish several new norm and numerical radius inequalities for products and sums of operators acting on a complex Hilbert space . Some of the obtained inequalities improve well known ones. In addition, by using new techniques, we prove certain new inequalities related to and , where and denote the A-numerical radius and the A-operator seminorm of an operator T acting on the semi-Hilbert space respectively. Here for every .

New Results on Boas–Bellman-Type Inequalities in Semi-Hilbert Spaces with Applications

In this article, we investigate new findings on Boas–Bellman-type inequalities in semi-Hilbert spaces. These spaces are generated by semi-inner products induced by positive and positive semidefinite operators. Our objective is to reveal significant properties of such spaces and apply these results to the field of multivariable operator theory. Specifically, we derive new inequalities that relate to the joint A-numerical radius, the joint operator A-seminorm, and the Euclidean A-seminorm of tuples of semi-Hilbert space operators. We assume that A is a nonzero positive operator. Our discoveries provide insights into the structure of semi-Hilbert spaces and have implications for a broad range of mathematical applications and beyond.

On the Joint A-Numerical Radius of Operators and Related Inequalities

In this paper, we study p-tuples of bounded linear operators on a complex Hilbert space with adjoint operators defined with respect to a non-zero positive operator A. Our main objective is to investigate the joint A-numerical radius of the p-tuple.We established several upper bounds for it, some of which extend and improve upon a previous work of the second author. Additionally, we provide several sharp inequalities involving the classical A-numerical radius and the A-seminorm of semi-Hilbert space operators as applications of our results.

An extension of the a-numerical radius on $$C^*$$-algebras

Let a be a positive element in a unital $$C^*$$ -algebra $$\mathfrak {A}$$ . We define a semi-norm on $$\mathfrak {A}$$ , which generalizes the a-operator semi-norm and the a-numerical radius. We investigate the basic properties of this semi-norm and prove inequalities involving it. Further, we derive new upper and lower bounds for the a-numerical radii of elements in $$\mathfrak {A}$$ . Some other related results are also discussed.

Inequalities and Reverse Inequalities for the Joint A-Numerical Radius of Operators

In this paper, we aim to establish several estimates concerning the generalized Euclidean operator radius of d-tuples of A-bounded linear operators acting on a complex Hilbert space H, which leads to the special case of the well-known A-numerical radius for d=1. Here, A is a positive operator on H. Some inequalities related to the Euclidean operator A-seminorm of d-tuples of A-bounded operators are proved. In addition, under appropriate conditions, several reverse bounds for the A-numerical radius in single and multivariable settings are also stated.

On Some Generalizations of Cauchy–Schwarz Inequalities and Their Applications

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.

Bounds for [formula omitted]-numerical radius based on an extension of [formula omitted]-Buzano inequality

Let A be the 2 × 2 diagonal operator matrix determined by a positive bounded linear operator A on a Hilbert space. In this paper, we give several upper bounds for the A-numerical radii of 2 × 2 block matrices. These inequalities, which are based on an extension of A-Buzano inequality, generalize and refine earlier A-numerical radius inequalities. Some applications of our results are also given.

Further inequalities for the 𝔸-numerical radius of certain 2 × 2 operator matrices

Abstract Let 𝔸 = ( A O O A ) \mathbb{A}={\bigl{(}\begin{smallmatrix}A&O\\ O&A\\ \end{smallmatrix}\bigr{)}} be a 2 × 2 2\times 2 diagonal operator matrix whose each diagonal entry is a bounded positive (semi-definite) linear operator A acting on a complex Hilbert space ℋ \mathcal{H} . In this paper, we derive several 𝔸 \mathbb{A} -numerical radius inequalities for 2 × 2 2\times 2 operator matrices whose entries are bounded with respect to the seminorm induced by the positive operator A on ℋ \mathcal{H} . Some applications of our inequalities are also given.

Further Inequalities for the Weighted Numerical Radius of Operators

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.

Generalized A-Numerical Radius of Operators and Related Inequalities

Let \(\mathcal {H}\) be a complex Hilbert space with inner product \(\langle \cdot , \cdot \rangle \) and let A be a non-zero bounded positive linear operator on \(\mathcal {H}.\) Let \(\mathbb {B}_A(\mathcal {H})\) denote the algebra of all bounded linear operators on \(\mathcal {H}\) which admit A-adjoint, and let \(N_A(\cdot )\) be a seminorm on \(\mathbb {B}_A(\mathcal {H})\). The generalized A-numerical radius of \(T\in \mathbb {B}_A(\mathcal {H})\) is defined as $$\begin{aligned} \omega _{N_A}(T)=\displaystyle {\sup _{\theta \in \mathbb {R}}}\; N_A\left( \frac{e^{i\theta }T+e^{-i\theta }T^{\sharp _A}}{2}\right) , \end{aligned}$$where \(T^{\sharp _A}\) stands for a distinguished A-adjoint of T. In this article, we focus on the development of several generalized A-numerical radius inequalities. We also develop bounds for the generalized A-numerical radius of sum and product of operators.

Numerical radius inequalities for products and sums of semi-Hilbertian space operators

New inequalities for the A-numerical radius of the products and sums of operators acting on a semi-Hilbert space, i.e. a space generated by a positive semidefinite operator A, are established. In particular, for every operators T and S which admit A-adjoints, it is proved that ?A(TS) ? 1/2?A(ST) + 1/4 (||T||A||S||A + ||TS||A), where ?A(T) and ||T||A denote the A-numerical radius and the A-operator seminorm of an operator T respectively.

Further results on A-numerical radius inequalities

Let A be a bounded linear positive operator on a complex Hilbert space {mathcal {H}}. Furthermore, let {mathcal {B}}_Amathcal {(H)} denote the set of all bounded linear operators on {mathcal {H}} whose A-adjoint exists, and {mathbb {A}} signify a diagonal operator matrix with diagonal entries are A. Very recently, several {mathbb {A}}-numerical radius inequalities of 2times 2 operator matrices were established. In this paper, we prove a few new {mathbb {A}}-numerical radius inequalities for 2times 2 and ntimes n operator matrices. We also provide a new proof of an existing result by relaxing a sufficient condition “A is strictly positive”. Our proofs show the importance of the theory of the Moore–Penrose inverse of a bounded linear operator in this field of study.

Some generalizations of A-numerical radius inequalities for semi-Hilbert space operators

Some generalizations of A-numerical radius inequalities for semi-Hilbert space operators

A-numerical radius and A-norm inequalities for semi-Hilbertian space operators

ABSTRACT Let (H, 〈 · , · 〉) be a complex Hilbert space and A be a positive bounded linear operator on H. The semi-inner product 〈x, y〉 A : = 〈Ax, y〉, x, y ∈ H, induces a semi-norm on H. Let ω A (T) and denote the A-numerical radius and the A-operator semi-norm of an operator T in semi-Hilbertian space (H, 〈 · , · 〉 A ), respectively. In this paper, some new bounds for the A-numerical radius of operators in semi-Hilbertian space are obtained, which improve the existing ones. In particular, a refinement of the triangle inequality for A-operator semi-norm is also shown.

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).

Improvement of A-Numerical Radius Inequalities of Semi-Hilbertian Space Operators

Let \(\mathcal {H}\) be a complex Hilbert space and let A be a positive operator on \(\mathcal {H}\). We obtain new bounds for the A-numerical radius of operators in semi-Hilbertian space \(\mathcal {B}_A(\mathcal {H})\) that generalize and improve on the existing ones. Further, we estimate an upper bound for the \(\mathbb {A}\)-operator seminorm of \(2\times 2\) operator matrices, where \(\mathbb {A}=\text{ diag }(A,A)\). The bound obtained here generalizes the earlier related bound.