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

Let B(H) be the space of all bounded linear operators on a complex separable Hilbert space H. Let Φ be a unital completely positive map on B(H). Define cov(A,B)=Φ(A * B)-Φ(A) * Φ(B), A,BB(H). The following generalization of the Schwarz inequality is proved:

for any A 1 ,A 2 B(H) the block matrix (cov(A i ,A j ) 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:
46L53Noncommutative probability and statistics
47A63Operator inequalities
60E15Inequalities in probability theory; stochastic orderings
81S25Quantum stochastic calculus
60H05Stochastic integrals