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.
In this paper, by using of the definition Berezin symbol, we show some Berezin number inequalities. Among other inequalities, it is shown that if A, B, X ∈ B(H ), then ber(AX ± XA) ber 1 2 (A * A + AA * ) ber 1 2 (X * X + XX * ) and ber 2 (A * XB) X 2 ber(A * A)ber(B * B).(ii) ber(αA) = |α|ber(A) for all α ∈ C.(iii) ber(A + B) ber(A) + ber(B).The Berezin symbol is widely applied in the various questions of uniquely determines the operator and analysis. For further information about Berezin symbol we refer the reader to [4,8,9,15] and references therein.2010 Mathematics Subject Classification. Primary: 15A60, Secondary: 47B20.
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