An extension of the van Hemmen–Ando norm inequality

Abstract

Let $C_{\||.\||}$ be an ideal of compact operators with symmetric norm $\||.\||$ . In this paper, we extend the van Hemmen–Ando norm inequality for arbitrary bounded operators as follows: if f is an operator monotone function on $[0,\infty)$ and S and T are bounded operators in $\mathbb{B}(\mathscr{H}\;\,)$ such that ${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a=\{z\in \mathbb{C} \ | \ {\rm{re}}(z)\geq a\}$ , then

\begin{equation*}\||f(S)X-Xf(T)\|| \leq\;f'(a) \ \||SX-XT\||,\end{equation*}

for each $X\in C_{\||.\||}$ . In particular, if ${\rm{sp}}(S), {\rm{sp}}(T) \subseteq \Gamma_a$ , then

\begin{equation*}\||S^r X-XT^r\|| \leq r a^{r-1} \ \||SX-XT\||,\end{equation*}

for each $X\in C_{\||.\||}$ and for each $0\leq r\leq 1$ .