Some classical inequalities and their applications

In this paper, we define analogies of classical H?lder-McCarthy and Young type inequalities in terms of the Berezin symbols of operators on a reproducing kernel Hilbert space H = H (?). These inequalities are applied in proving of some new inequalities for the Berezin number of operators. We also define quasi-paranormal and absolute-k-quasi paranormal operators and study their properties by using the Berezin symbols.

Some extended Berezin number inequalities

We present generalized extensions of Berezin number inequalities involving the Euclidean Berezin number and f-connection of operators.

Berezin Number Inequalities of an Invertible Operator and Some Slater Type Inequalities in Reproducing Kernel Hilbert Spaces

A reproducing kernel Hilbert space (shorty, RKHS) H=H(Ω) on some set Ω is a Hilbert space of complex valued functions on Ω such that for every λ∈Ω the linear functional (evaluation functional) f→f(λ) is bounded on H. If H is RKHS on a set Ω, then, by the classical Riesz representation theorem for every λ∈Ω there is a unique element kH,λ∈H such that f(λ)=〈f,kH,λ〉; for all f∈H. The family {kH,λ:λ∈Ω} is called the reproducing kernel of the space H. The Berezin set and the Berezin number of the operator A was respectively given by Karaev in [26] as following Ber(A)={A(λ):λ∈Ω} and ber(A):=|A(λ)|. In this chapter, the authors give the Berezin number inequalities for an invertible operator and some other related results are studied. Also, they obtain some inequalities of the slater type for convex functions of selfadjoint operators in reproducing kernel Hilbert spaces and examine related results.