Abstract
We give operator analogues of some classical inequalities, including Hardy and Hardy-Hilbert type inequalities for numbers. We apply these operator forms of such inequalities for proving some power inequalities for the so-called Berezin number of self-adjoint and positive operators acting on Reproducing Kernel Hilbert Spaces (RKHSs). More precisely, we prove that
(ber(f(A)))(2) <= Cber ((f(A))(2))
for some constants C > 1. We also use reproducing kernels technique to estimate dist (A, U), where U is the set of all unitary operators on a RKHS H = H (Omega) over some set Omega, for some operator A on H (Omega).