Inequalities for the Berezin Number and Norms of a Product of Three Operators

We establish several upper bounds for the Berezin number and Berezin norms of products of three bounded operators. Let (HΓ,⟨·,·⟩) be a reproducing kernel Hilbert space (RKHS) on a set Γ with inner product ⟨·,·⟩ and associated norm ∥·∥. Berezin number and the Berezin norms of the operator V∈B(HΓ) are, respectively, defined by ber(V):=supλ∈Γ|⟨Vh^λ,h^λ⟩|, ∥V∥ber:=sup|⟨Vh^λ,h^μ⟩|;λ,μ∈Γ, and ∥V∥ber˜:=supλ∈Γ∥Vh^λ∥. In this paper, we show among others that, if the functions φ and ψ are continuous nonnegative on [0,∞) and satisfying the relation φ(t)ψ(t)=t for all t∈[0,∞), then for any X, Y, V∈BHΓ, we have that berYVX2r≤121pφVX2pr+1qψV*Y*2qrber+berψV*Y*2φVX2r for r≥1, p, q>1 with 1p+1q=1 and pr, qr≥2. Inequalities involving the Moore–Penrose inverse of closed range operators V are provided as well. The sharpness of some obtained inequalities for three, two, and one operator are considered with details in the last section of the paper.