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

Abstract

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).