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More operator versions of the Schwarz inequality. (English) Zbl 0984.46040

Let $B\left(H\right)$ be the space of all bounded linear operators on a complex separable Hilbert space $H$. Let ${\Phi }$ be a unital completely positive map on $B\left(H\right)$. Define $cov\left(A,B\right)={\Phi }\left({A}^{*}B\right)-{\Phi }{\left(A\right)}^{*}{\Phi }\left(B\right)$, $A,B\in B\left(H\right)$. The following generalization of the Schwarz inequality is proved:

for any ${A}_{1},{A}_{2}\in B\left(H\right)$ the block matrix $\left(cov{\left({A}_{i},{A}_{j}\right)}_{i,j=1}^{2}$ is positive.

An operator version of the well-known Wielandt inequality is proved. The proof uses an operator version of Kantarovich inequality, which was proved by the authors in [Am. Math. Monthly 107, 353-356 (2000; Zbl 1009.15009)].

##### MSC:
 46L53 Noncommutative probability and statistics 47A63 Operator inequalities 60E15 Inequalities in probability theory; stochastic orderings 81S25 Quantum stochastic calculus 60H05 Stochastic integrals