Hardy-Hilbert's inequality and power inequalities for Berezin numbers of operators

Abstract

Abstract. We give operator analogues of some classical inequalities, including Hardy and HardyHilbert 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 thatfor 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 (Ω) over some set Ω, for some operator A on H (Ω) .Mathematics subject classification (2010): 47A63.

Refinements of Kantorovich type, Schwarz and Berezin number inequalities

“…In this article, motivated mainly by the paper [34], we will prove some new inequalities for the Berezin number of powers of operators by using Kantorovich and Kantorovich type inequalities and also a refinement of Schwarz inequality due to Dragomir [10]. For the related results, the reader can see in [4,5,13,15,16,17,18,21,25,26,30,32,33,36,40,41,42,44]…”

Some results related to the Berezin number inequalities

“…More information about numerical range and numerical radius can be found in [1,4,14,19,21,24,25]. Recently, some results about the Berezin number were obtained in [3,18,20,29,30].…”

Cebysev’s type inequalities and power inequalities for the Berezin number of operators

“…But, an example with a nonzero nilpotent operator shows that there exists operators for which inequality (P) does not hold. It was given some predictions for the constant C by using Hardy-Hilbert type inequalities [15,16,30,31].…”