  In this article, we employ certain properties of the transform
$C_{M,m}(A)=(MI-A^*)(A-mI)$ 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.
