OPERATOR INEQUALITIES IN REPRODUCING KERNEL HILBERT SPACES

In this paper, by using some classical Mulholland type inequality, Berezin symbols and reproducing kernel technique, we prove the power inequalities for the Berezin number ber(A) for some self-adjoint operators A on H(Ω). Namely, some Mulholland type inequality for reproducing kernel Hilbert space operators are established. By applying this inequality, we prove that (ber(A)) ≤ C1ber(A) for any positive operator A on H(Ω).

Let k λ = k λ ∥k λ ∥ be the normalized reproducing kernel of the space H. For any bounded linear operator A on H, the Berezin symbol of A is the function A defined by (see [4]) Recall that the Berezin set and the Berezin number for an operator A ∈ B(H(Ω)) were introduced in [15,16]  . More information about numerical range and numerical radius can be found in [6,7,9,18,19,21]. Using the Hardy-Hilbert type inequalities and some well-known inequalities, some important results about the Berezin number inequalities were obtained in [2,3,10,22,[24][25][26][27][28].
In the present paper, by using inequalities (1), (2) and some ideas of paper [17], we will estimate Berezin number (which is a new numerical value of the bounded linear operators on RKHS) of operators acting in the reproducing kernel Hilbert spaces.

Mulholland Type Inequalities and Berezin Number of Some Operators
In the following result, we prove an analog of inequality (1) for some self-adjoint operators on a RKHS H = H(Ω).
Let f, g be two continuous functions defined on an interval ∆ ⊂ (0, +∞) and f, g ≥ 0. Then the following is true: for all self-adjoint operators A, B, C ∈ B(H(Ω)) with spectra contained in ∆ and for all µ, λ, ξ ∈ Ω.

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Our more general result is the following theorem which gives a sharper estimate than Corollary 1.