Title : ( An extension of the van Hemmen–Ando norm inequality )
Authors: Hamed Najafi ,Access to full-text not allowed by authors
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 $$\\\\\\\\\\\\\\\\||f(S)X-Xf(T)\\\\\\\\\\\\\\\\|| \\\\\\\\\\\\\\\\leq f^{\\\\\\\\\\\\\\\'}(a) \\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\||SX-XT\\\\\\\\\\\\\\\\||,$$ for each $X\\\\\\\\\\\\\\\\in C_{\\\\\\\\\\\\\\\\||.\\\\\\\\\\\\\\\\||}$. In particular, if ${\\\\\\\\\\\\\\\\rm{sp}}(S), {\\\\\\\\\\\\\\\\rm{sp}}(T) \\\\\\\\\\\\\\\\subseteq \\\\\\\\\\\\\\\\Gamma_a$, then $$\\\\\\\\\\\\\\\\||S^r X-XT^r\\\\\\\\\\\\\\\\|| \\\\\\\\\\\\\\\\leq r a^{r-1} \\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\||SX-XT\\\\\\\\\\\\\\\\||,$$ for each $X\\\\\\\\\\\\\\\\in C_{\\\\\\\\\\\\\\\\||.\\\\\\\\\\\\\\\\||}$ and for each $0\\\\\\\\\\\\\\\\leq r\\\\\\\\\\\\\\\\leq 1$.