Inequalities for the Berezin Number and Norms of a Product of Three Operators
4 affiliations
- 1Applied Mathematics Research Group, Institute for Sustainable Industries & Liveable Cities, Victoria University, P.O. Box 14428, Melbourne, VIC 8001, Australia
- 2Mathematical Sciences, School of Science, Royal Melbourne Institute of Technology University, Melbourne, VIC 3001, Australia
- 3Department of Mathematical Sciences, College of Sciences, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia
- 4Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia
Abstract
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.
Related
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BibTeX
@article{dragomir2026inequalities,
title = {{Inequalities for the Berezin Number and Norms of a Product of Three Operators}},
author = {Silvestru Sever Dragomir and Ghada N. Alnemer and Ahad Hamoud Alotaibi},
journal = {Axioms},
year = {2026},
month = nov,
doi = {10.3390/axioms15110817},
url = {https://www.mdpi.com/2075-1680/15/11/817},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Inequalities for the Berezin Number and Norms of a Product of Three Operators
AU - Silvestru Sever Dragomir
AU - Ghada N. Alnemer
AU - Ahad Hamoud Alotaibi
PY - 2026
DA - 2026/11/04
JO - Axioms
DO - 10.3390/axioms15110817
UR - https://www.mdpi.com/2075-1680/15/11/817
AB - 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.
SN - 2075-1680
ER -