SOME REFINEMENTS OF BEREZIN NUMBER INEQUALITIES VIA CONVEX FUNCTIONS.

The Berezin transform ˜A and the Berezin number of an operator A on the reproducing kernel Hilbert space over some set Ω with normalized reproducing kernel k̂λ are defined, respectively, by ˜A(λ) = ⟨ Ak̂λ, k̂λ ⟩, λ ∈ Ω and ber ( A ) := supλ∈Ω ∣ ˜A(λ) ∣. A straightforward comparison between these characteristics yields the inequalities ber (A) ≤ 1/2 ( ∥ A ∥ber + ∥ A 2 ∥ber1/2) . In this paper, we study further inequalities relating them. Namely, we obtained some refinement of Berezin number inequalities involving convex functions. In particular, for A ∈ B (H) and r ≥ 1 we show that ber2r (A) ≤ 1/4 ( ∥ A* A + A A* ∥berr + ∥ A* A − A A* ∥berr ) + 1/2 berr ( A² ) . [ABSTRACT FROM AUTHOR]