Operator inequalities via accretive transforms

Abstract

In this article, we employ certain properties of the transform $\mathscr{C}_{M,m}(A)=(M\mathbf1_{\mathcal{H}}-A^*)(A-m\mathbf1_{\mathcal{H}})$ to obtain new inequalities for the bounded linear operator $A$ on a complex Hilbert space $\mathcal{H}$. In particular, we obtain new relations among $|A|,|A^*|,|\mathfrak{R}A|$ and $|\mathfrak{I}A|$. Further numerical radius inequalities that extend some known inequalities will be presented too.