Generalized A-Numerical Radius of Operators and Related Inequalities

Abstract

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.