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).
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