Generalized $A$-numerical radius of operators and related inequalities

Abstract

Let $A$ be a non-zero positive bounded linear operator on a complex Hilbert space $(\mathcal{H},⟨\cdot ,\cdot ⟩)$. Let ${\omega }_A(T)$ denote the $A$-numerical radius of an operator $T$ acting on the semi-Hilbert space $(\mathcal{H},⟨\cdot ,\cdot {⟩}_A)$, where $⟨x,y{⟩}_A:=⟨Ax,y⟩$ 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 $${\omega }_{N_A}(T)=\underset{\theta \in \mathbb{R}}{\sup }{N}_A\left(\frac{e^{i\theta }T+e^{-i\theta }{T}^{{♯}_A}}{2}\right),$$ where $T^{{♯}_A}$ 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 $N_A(\cdot )$.

References in corpus (2)

• An extension of Birkhoff--James orthogonality relations in semi-Hilbertian space operators
• New semi-norm of semi-Hilbertian space operators and its application

Cited by in corpus (1)

• $A$-numerical radius : New inequalities and characterization of equalities