Let be the space of all bounded linear operators on a complex separable Hilbert space . Let be a unital completely positive map on . Define , . The following generalization of the Schwarz inequality is proved:
for any the block matrix 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)].