Improved Berezin radius inequalities for functional Hilbert space operators

The Berezin number of an operator S is defined by ber(S) = supλ∈Θ |Ŝ(λ)| = supλ∈Θ |⟨Sĸ̂λ, ĸ̂λ⟩|, where Ŝ(λ) = ⟨Sĸ̂λ, ĸ̂λ⟩ (λ ∈ Θ) is the Berezin transform of an operator S on the functional Hilbert space ℋ = ℋ(Θ) and ĸ̂λ = ĸλ / ‖ĸλ‖ is the normalized reproducing kernel of ℋ. In this paper, we show an improvement of the classical Cauchy–Schwarz inequality involving the Berezin number. Moreover, some Berezin numbers inequalities are proven for invertible operators.