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 $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$. The semi-inner product ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$, $x, y\in\mathcal{H}$ induces a seminorm ${\|\cdot\|}_A$ on $\mathcal{H}$. Let $T$ be an $A$-bounded operator on $\mathcal{H}$, the $A$-numerical radius of $T$ is given by \begin{align*} \omega_A(T) = \sup\Big\{\big|{\langle Tx\mid x\rangle}_A\big|: \,\,x\in \mathcal{H}, \,{\|x\|}_A = 1\Big\}. \end{align*} 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 $A$.