Some norm inequalities for accretive Hilbert space operators

Abstract

New norm inequalities for accretive operators on Hilbert space are given. Among other inequalities, we prove that if \(A, B \in \mathbb{B(H)}\) and \(B\) is self-adjoint and also \(C_{m,M}(iAB)\) is accretive, then \begin{eqnarray*} \frac{4 \sqrt{Mm}}{M+m} \Vert AB\Vert \leq \omega(AB-BA^*),\end{eqnarray*} where \(M\) and \(m\) are positive real numbers with \(M > m\) and \(C_{m,M}(A) = (A^* - mI)(MI - A)\). Also, we show that if \(C_{m,M}(A)\) is accretive and \((M-m) \leq k \Vert A \Vert\), then \begin{eqnarray*} \omega(AB) \leq ( 2 + k)\omega(A)\omega(B).\end{eqnarray*}