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An extension of the van Hemmen–Ando norm inequality

Part of: General theory of linear operators Linear spaces and algebras of operators Special classes of linear operators

Published online by Cambridge University Press:  03 August 2022

Hamed Najafi
Hamed Najafi*
Affiliation:
Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran E-mail: hamednajafi20@gmail.com
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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 $0\leq r\leq 1$ .

MSC classification

Primary: 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
Secondary: 47B47: Commutators, derivations, elementary operators, etc. 47L20: Operator ideals
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 65 , Issue 1 , January 2023 , pp. 121 - 127
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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