Numerical radius inequalities of bounded linear operators and $(\alpha ,\beta )$-normal operators

We obtain various upper bounds for the numerical radius $w(T)$ of a bounded linear operator $T$ defined on a complex Hilbert space $\mathcal{H}$, by developing the upper bounds for the $\alpha$-norm of $T$, which is defined as $\mathrm{\parallel }T{\mathrm{\parallel }}_{\alpha }=\sup \{\sqrt{\alpha \mathrm{\mid }⟨Tx,x⟩{\mathrm{\mid }}^2+(1-\alpha )\mathrm{\parallel }Tx{\mathrm{\parallel }}^2}:x\in \mathcal{H},\mathrm{\parallel }x\mathrm{\parallel }=1\}$ for $0\le \alpha \le 1$. Further, we prove that \begin{eqnarray*} w(T) &\leq & \sqrt{\left( \min_{α\in [0,1]}\left| α|T|+(1-α)|T^| \right| \right) |T|} ,,,, \leq ,, ,, |T|. \end{eqnarray} For $0\le \alpha \le 1\le \beta ,$ the operator $T$ is called $(\alpha ,\beta )$-normal if ${\alpha }^2{T}^{\ast }T\le T{T}^{\ast }\le {\beta }^2{T}^{\ast }T$ holds. Note that every invertible operator is an $(\alpha ,\beta )$-normal operator for suitable values of $\alpha$ and $\beta$. Among other lower bound for the numerical radius of an $(\alpha ,\beta )$-normal operator $T$, we show that \begin{eqnarray*} w(T) &\geq & \sqrt{\max \left{ 1+α^2, 1+\frac{1}{β^2}\right} \frac{|T|^2}{4}+ \frac {\left| |\Re(T)|^2-|\Im(T)|^2 \right|}2} &\geq & \max \left{ \sqrt{1+α^2}, \sqrt{1+\frac{1}{β^2}} \right} \frac{|T|}{2} & > & \frac{|T|}2, \end{eqnarray*} where $\mathrm{\Re }(T)$ and $\mathrm{\Im }(T)$ are the real part and imaginary part of $T$, respectively.