Abstract
AbstractLet
$C_{\||.\||}$
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
$0\leq r\leq 1$
.
Publisher
Cambridge University Press (CUP)