Refinements of norm and numerical radius inequalities

Abstract

Several refinements of norm and numerical radius inequalities of bounded linear operators on a complex Hilbert space are given. In particular, we show that if $A$ is a bounded linear operator on a complex Hilbert space, then $$ \frac{1}{4}\|A^*A+AA^*\| \leq \frac{1}{8}\bigg( \|A+A^*\|^2+\|A-A^*\|^2 +c^2(A+A^*)+c^2(A-A^*)\bigg) \leq w^2(A)$$ and \begin{eqnarray*} \frac{1}{2}\|A^*A+AA^*\| - \frac{1}{4}\bigg\|(A+A^*)^2 (A-A^*)^2 \bigg\|^{1/2} \leq w^2(A) \leq \frac{1}{2}\|A^*A+AA^*\|, \end{eqnarray*} %$$ \frac{1}{4}\|A^*A+AA^*\| \leq \frac{1}{2}w^2(A) + \frac{1}{8}\bigg\|(A+A^*)^2 (A-A^*)^2 \bigg\|^{1/2}\leq w^2(A),$$ where $\|.\|$, $w(.)$ and $c(.)$ are the operator norm, the numerical radius and the Crawford number, respectively. Further, we prove that if $A,D$ are bounded linear operators on a complex Hilbert space, then \begin{eqnarray*} \|AD^*\| \leq \left\| \int_0^1 \left( (1-t) \left(\frac{ |A|^2+|D|^2}{2}\right) +t\|AD^*\|I \right)^2dt \right\|^{1/2} \leq \frac{1}{2}\left\| |A|^2+|D|^2 \right\|, \end{eqnarray*} where $|A|=(A^*A)^{1/2}$ and $|D|=(D^*D)^{1/2}$. This is a refinement of well known inequality obtained by Bhatia and Kittaneh.