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 Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.

Keywords:

Unitarily invariant norm, positive operator, arithmetic-geometric mean inequality, Cauchy-Schwarz inequality, Heinz inequality Unitarily invariant norm, positive operator, arithmetic-geometric mean inequality, Cauchy-Schwarz inequality, Heinz inequality

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 Schatten-von 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 Schatten-von Neumann classes, etc.) [See also 47L20]
47B15 - Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B20 - Subnormal operators, hyponormal operators, etc.

top of page | contact us | social media | privacy | site map