By using Hardy-Hilbert's inequality, some power inequalities for the Berezin number of a selfadjoint operators in Reproducing Kernel Hilbert Spaces (RKHSs) with applications for convex functions are given.

For a given function ϕ ∈ H ∞ with ϕ (z) < 1 (z ∈ D), we associate some special operators subspace and study some properties of these operators including behavior of their Berezin symbols. It turns that such boundary behavior is closely related to the Blaschke condition of sequences in the unit disk D of the complex plane. In terms of Berezin symbols the trace of some nuclear truncated Toeplitz operator is also calculated. Following the definition of [12], we say that a RKHS H is standard, if k H,λ → 0 weakly as λ → ξ for any point ξ ∈ ∂Ω. For example, the Hardy Hilbert space is a standard RKHS. Recall that if B (H) denotes the space of all bounded and linear operators on H, then the Berezin symbol A of any operator A ∈ B (H) is the function defined on Ω by A(λ) := A k H,λ , k H,λ , λ ∈ Ω.

We give operator analogues of some classical inequalities, includingČebyšev type inequality for synchronous and convex functions of selfadjoint operators in Reproducing Kernel Hilbert Spaces (RKHSs). We obtain some Berezin number inequalities for the product of operators. Also, we prove the Berezin number inequality for the commutator of two operators. Hardy et al. [19] in their book in 1934 mentioned the inequality (1) in the more general setting of synchronous sequences, i.e., if a, b are synchronous (asynchronous), this means that a i − a j b i − b j ≥ (≤) 0, for each i, j ∈ {1,. .. , n}, then the (1) is valid.

Let A be a Banach algebra with a unit e, and let a ∈ A be an invertible element. We define the following algebra: B loc a := x ∈ A : a n xa −n ≤ c x n α(x) for some α (x) ≥ 0 and c x > 0. In this article we study some properties of this algebra; in particular, we prove that B loc e+p = x ∈ A : px e − p = 0 , where p is an idempotent in A. We also investigate the following Deddens subspace. Let a, b ∈ A be two elements. Fix any number α, 0 ≤ α < 1, and consider the following subspace of A : D α a,b := {x ∈ A : a n xb n = O (n α) as n → ∞}. Here we study some properties of the subspaces D α a,b and D α b,a .