Some ${\mathbb {A}}$-numerical radius inequalities for $d\times d$ operator matrices

Abstract

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).