Refinements of norm and numerical radius inequalities (2010.12750v1)

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.