An operator $T$ acting on a Hilbert space is called $(\alpha ,\beta)$-normal ($0\leq \alpha \leq 1\leq \beta $) if \begin{ equation*} \alpha ^{ 2}T^{ \ast }T\leq TT^{ \ast}\leq \beta ^{ 2}T^{ \ast}T. \end{ equation*} In this paper we establish various inequalities between the operator norm and its numerical radius of $(\alpha ,\beta)$-normal operators in Hilbert spaces. For this purpose, we employ some classical inequalities for vectors in inner product spaces.
Categories: math.FA math.OA