New berezin symbol inequalities for operators on the reproducing kernel hilbert space

Official URL: https://dx.doi.org/10.7153/oam-2021-15-64

We use Kittaneh and Manasrah inequality and Kian’s functional calculus method to prove some new inequalities for Berezin symbols and Berezin numbers of operators. In particular, we prove that ( ber f (A)2) (f(A) p ≥ ber +f )(A)q p q for all self-adjoint operators A on the reproducing kernel Hilbert space H (Ω) with spectrum in J ⊂ (−∞,+∞) and all continuous nonnegative functions f defined on J. We also prove new upper and lower bounds for Berezin numbers of reproducing kernel Hilbert space operators. Among our results, we prove that if A: H (Ω) → H (Ω) is a bounded pseudo-hypornormal operator on the reproducing kernel Hilbert space H (Ω), then for all non-negative non-decreasing pseudo-operator convex function f on [0,∞), we have ⎛ ⎞ ⎛ ⎞ f (ber(A)) ≥12 ∥f ⎝|A| ⎠ + f ⎝|A∗ | ⎠ 1 +ξ|A|2 8 1 +ξ|A|2, ∥ 8 Ber where ‖.‖Ber denotes the Berezin norm of operator.