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Numerical radius inequalities of bounded linear operators and $(α,β)$-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 $α$-norm of $T$, which is defined as $\|T\|_α= \sup \left\{ \sqrt{α|\langle Tx,x \rangle|^2+ (1-α)\|Tx\|^2 } : x\in \mathcal{H}, \|x\|=1 \right\}$ for $ 0\leq α\leq 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 $0\leq α\leq 1 \leq β,$ the operator $T$ is called $(α,β)$-normal if $α^2 T^*T\leq TT^*\leq β^2 T^*T$ holds. Note that every invertible operator is an $(α,β)$-normal operator for suitable values of $α$ and $β$. Among other lower bound for the numerical radius of an $(α,β)$-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 $\Re(T)$ and $\Im(T)$ are the real part and imaginary part of $T$, respectively.