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The semi-inner product ${\\langle x, y\\rangle}_A := \\langle Ax, y\\rangle$, $x, y\\in\\mathcal{H}$ induces a semi-norm ${\\|\\cdot\\|}_A$ on $\\mathcal{H}$. Let ${\\|T\\|}_A$ and $w_A(T)$ denote the $A$-operator semi-norm and the $A$-numerical radius of an operator $T$ in semi-Hilbertian space $\\big(\\mathcal{H}, {\\|\\cdot\\|}_A\\big)$, respectively. 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