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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.

The aim of this article was to provide improved estimates for the (α,β)-norm of a bounded linear operator. In particular, our results enabled the determination of new upper bounds involving both the Berezin number and the Berezin norm of bounded linear operators that act on reproducing kernel Hilbert spaces. Through our analysis, we hoped to enhance the understanding of the properties and behavior of such operators and contribute to the development of new mathematical tools for their characterization and application.

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).

Let $\mathbf{T}=(T_1,\ldots,T_d)$ be a $d$-tuple of operators on a complex Hilbert space $\mathcal{H}$. The spherical Aluthge transform of $\mathbf{T}$ is the $d$-tuple given by $\widehat{\mathbf{T}}:=(\sqrt{P}V_1\sqrt{P},\ldots,\sqrt{P}V_d\sqrt{P})$ where $P:=\sqrt{T_1^*T_1+\ldots+T_d^*T_d}$ and $(V_1,\ldots,V_d)$ is a joint partial isometry such that $T_k=V_k P$ for all $1 \le k \le d$. In this paper, we prove several inequalities involving the joint numerical radius and the joint operator norm of $\widehat{\mathbf{T}}$. Moreover, a characterization of the joint spectral radius of an operator tuple $\mathbf{T}$ via $n$-th iterated of spherical Aluthge transform is established.

This paper is concerned with the study of certain properties of operator tuples on a complex Hilbert space H when a semi-inner product induced by a positive operator A on H is considered. In particular, we show that r A ( T ) ≤ ω A ( T ) for every commuting operator tuple T = ( T 1 , … , T d ) such that each T k admits an A-adjoint operator, where r A ( T ) and ω A ( T ) denote respectively the A-joint spectral radius and the A-joint numerical radius of T. This study allows to establish that r A ( T ) = ω A ( T ) = ‖ T ‖ A for every A-normal commuting tuple of operators T, where ‖ T ‖ A is denoted to be the A-joint operator seminorm of T. In addition, the A-joint spectral radius of an ( A , m ) -isometric tuple of commuting operators is studied.

Let A be a positive bounded linear operator acting on a complex Hilbert space $${\mathcal {H}}$$ . Our aim in this paper is to prove some A-numerical radius inequalities of bounded linear operators acting on $${\mathcal {H}}$$ when an additional semi-inner product structure induced by A is considered. In particular, an alternative proof of a recent result proved in Moslehian et al. (Linear Algebra Appl 591:299–321 2020) is given.

In this paper, we aim to introduce and characterize the numerical radius orthogonality of operators on a complex Hilbert space $${\mathcal {H}}$$ which are bounded with respect to the seminorm induced by a positive operator A on $${\mathcal {H}}$$ . Moreover, a characterization of the A-numerical radius parallelism for A-rank one operators is obtained. As applications of the results obtained, we derive some $${\mathbb {A}}$$ -numerical radius inequalities of operator matrices, where $${\mathbb {A}}$$ is the operator diagonal matrix whose each diagonal entry is a positive operator A on a complex Hilbert space $${\mathcal {H}}.$$

In this paper we generalize the concept of Davis–Wielandt shell of operators on a Hilbert space when a semi-inner product induced by a positive operator A is considered. Moreover, we investigate the parallelism of A-bounded operators with respect to the seminorm and the numerical radius induced by A. Mainly, we characterize A-normaloid operators in terms of their A-Davis–Wielandt radii. In addition, a connection between A-seminorm-parallelism to the identity operator and an equality condition for the A-Davis–Wielandt radius is proved. This generalizes the well-known results in Chan and Chan (Oper Matrices 11(3):885–890, 2017), Zamani et al. (Linear Multilinear Algebra 67(11):2147–2158, 2019). Some other related results are also discussed.

Fil: Bottazzi, Tamara Paula. Universidad Nacional de Río Negro. Centro Interdisciplinario de Telecomunicaciones, Electrónica, Computación y Ciencia Aplicada. Río Negro, Argentina. Fil: Conde, Cristian Marcelo. Instituto de Ciencias, Universidad Nacional de General Sarmiento Fil: Feki, Kais. Laboratory Physics-Mathematics and Applications (LR/13/ES-22), Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia In this paper, we establish several characterizations of the A-parallelism of bounded linear operators with respect to the semi-norm induced by a positive operator A acting on a complex Hilbert space. Among other things, we investigate the relationship between A- seminorm-parallelism and A-Birkhoff-James orthogonality of A-bounded operators. In particular, we characterize A-bounded operators which satisfy the A-Daugavet equation. In addition, we relate the A-Birkhoff- James orthogonality of operators and distance formulas and we give an explicit formula of the center mass for A-bounded operators. Some other related results are also discussed En este artículo, establecemos varias caracterizaciones del A-paralelismo de operadores lineales acotados respecto de la seminorma inducida por un operador positivo A que actúa sobre un espacio de Hilbert complejo. Entre otras cosas, investigamos la relación entren A-paralelismo en seminorma y A- ortogonalidad Birkhoff-James de operadores A-acotados. En particular, caracterizamos operadores A-acotados que satisfacen la ecuación A-Daugavet. Además, relacionamos A- ortogonalidad Birkhoff-James de operadores y fórmulas de distancia y dame una fómrula explícita del centro de masa de operadores A-acotados.

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).

In this paper, we introduce the concept of normality of a d-tuple of bounded linear operators acting on a complex Hilbert space H when an additional semi-inner product induced by a positive operato...

Let $A$ be a non-zero positive bounded linear operator on a complex Hilbert space $(\mathcal{H},\langle\cdot,\cdot\rangle)$. Let $\omega_A(T)$ denote the $A$-numerical radius of an operator $T$ acting on the semi-Hilbert space $(\mathcal{H},\langle\cdot,\cdot\rangle_A)$, where $\langle x, y\rangle_{A} :=\langle Ax, y\rangle$ for all $x,y\in \mathcal{H}$. Let $N_A(\cdot)$ be a seminorm on the algebra of all $A$-bounded operators acting on $\mathcal{H}$ and let $T$ be an operator which admits $A$-adjoint. Then, we define the generalized $A$-numerical radius as $$\omega_{N_A}(T)=\displaystyle{\sup_{\theta \in \mathbb{R}}}\; N_A\left(\frac{e^{i\theta}T+e^{-i\theta}T^{\sharp_A}}{2}\right),$$ where $T^{\sharp_A}$ denotes a distinguished $A$-adjoint of $T$. We develop several generalized $A$-numerical radius inequalities from which follows the existing numerical radius and $A$-numerical radius inequalities. We also obtain bounds for generalized $A$-numerical radius of sum and product of operators. Finally, we study $\omega_{N_A}(\cdot)$ in the setting of two particular seminorms $N_A(\cdot)$.

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.

Our aim in this paper is to prove the equality between two kinds of joint operator norms in the case of doubly commuting tuples of hyponormal operators on Hilbert spaces. Moreover, a sharp bound for the distance between two doubly commuting tuples of hyponormal operators in terms of the distance between their Harte spectra is proved.

For a given bounded positive (semidefinite) linear operator $A$ on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$, we consider the semi-Hilbertian space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle_A \big)$ where ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$ for every $x, y\in\mathcal{H}$. The $A$-numerical radius of an $A$-bounded operator $T$ on $\mathcal{H}$ is given by \begin{align*} \omega_A(T) = \sup\Big\{\big|{\langle Tx\mid x\rangle}_A\big|\,; \,\,x\in \mathcal{H}, \,{\langle x\mid x\rangle}_A= 1\Big\}. \end{align*} Our aim in this paper is to derive several $\mathbb{A}$-numerical radius inequalities for $2\times 2$ operator matrices whose entries are $A$-bounded operators, where $\mathbb{A}=\text{diag}(A,A)$., Comment: It is submitted to a research journal since 1 May 2020

Let $r_A(T)$ denote the $A$-spectral radius of an operator $T$ which is bounded with respect to the seminorm induced by a positive operator $A$ on a complex Hilbert space $\mathcal{H}$. In this paper, we aim to establish some $A$-spectral radius inequalities for products, sums and commutators of $A$-bounded operators. Moreover, under suitable conditions on $T$ and $A$ we show that \begin{equation*} r_A\left( \sum_{k=0}^{+\infty}c_{k}T^{k}\right) \leq \sum_{k=0}^{+\infty}|c_{k}|\left[r_A(T)\right]^{k}, \end{equation*} where $c_k$ are complex numbers for all $k\in \mathbb{N}$.

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, it is proved for operators $T$ and $S,$ having $A$-adjoint, that $$ \omega_A(TS) \leq \frac{1}{2}\omega_A(ST)+\frac{1}{4}\Big(\|T\|_A\|S\|_A+\|TS\|_A\Big),$$ where $\omega_A(T)$ and $\|T\|_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$.

Our aim in this paper is to give a new formula of the joint spectral radius of commuting d-tuples of operators on a complex Hilbert space. Also we show that r ( T ) ≤ ω ( T ) for a commuting d-tuple of operators T, where r ( T ) and ω ( T ) denote respectively the joint spectral radius and the joint numerical radius of T. This generalizes the well known relation between the spectral and the numerical radii of Hilbert space operators which is proved in [10] .

In this paper we aim to investigate the concept of numerical range and maximal numerical range relative to a positive operator of a d-tuple of bounded linear operators on a Hilbert space. Some properties and applications of these sets are studied. Mainly, it is proved that they are convex for d = 1 , this generalizes the well known Toeplitz–Hausdorff Theorem [24] , [16] and Stampfli's result [23] . Moreover, under additional hypotheses, we show that these sets are convex for d ≥ 2 .

In the original article published, during the final typesetting stage the equation in the proof of the Theorem 3.5 was published incorrectly.