Abstract view
Some norm inequalities for operators


Published:19990301
Printed: Mar 1999
Abstract
Let $A_i$, $B_i$ and $X_i$ $(i=1, 2, \dots, n)$ be operators on a
separable Hilbert space. It is shown that if $f$ and $g$ are
nonnegative continuous functions on $[0,\infty)$ which satisfy the
relation $f(t)g(t) =t$ for all $t$ in $[0,\infty)$, then
$$
\Biglvert \,\Bigl\sum^n_{i=1} A^*_i X_i B_i \Bigr^r \,\Bigrvert^2
\leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (X^*_i)^2 A_i \Bigr)^r
\Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (X_i)^2 B_i
\Bigr)^r \Bigrvert
$$
for every $r>0$ and for every unitarily invariant norm. This result
improves some known CauchySchwarz type inequalities. Norm
inequalities related to the arithmeticgeometric mean inequality and
the classical Heinz inequalities are also obtained.
MSC Classifications: 
47A30, 47B10, 47B15, 47B20 show english descriptions
Norms (inequalities, more than one norm, etc.) Operators belonging to operator ideals (nuclear, $p$summing, in the Schattenvon Neumann classes, etc.) [See also 47L20] Hermitian and normal operators (spectral measures, functional calculus, etc.) Subnormal operators, hyponormal operators, etc.
47A30  Norms (inequalities, more than one norm, etc.) 47B10  Operators belonging to operator ideals (nuclear, $p$summing, in the Schattenvon Neumann classes, etc.) [See also 47L20] 47B15  Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B20  Subnormal operators, hyponormal operators, etc.
