@article{Aladsani2026, 
author = {Feryal Aladsani and Asmahan Alajyan and Salma Aljawi and Kais Feki},
title = {Novel Berezin number and norm inequalities for operator sums and products},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {3},
pages = {5738-5758},
keywords = {Berezin number, Berezin norm, reproducing kernel Hilbert space, operator inequalities, numerical radius, bounded linear operators},
url = {https://www.sciopen.com/article/10.3934/math.2026236},
doi = {10.3934/math.2026236},
abstract = {Let    (            X              F        ,  ⟨  ⋅  ,  ⋅  ⟩  ) be a reproducing kernel Hilbert space over a non-empty set        F  . Let                      u        ^              λ   and                      u        ^              μ   denote the normalized reproducing kernels of              X              F      . The Berezin number and the Berezin norm of a bounded linear operator        B   acting on              X              F       are, respectively, defined by         b    e    r    (      B    )  =      sup          λ      ∈              F                  |    ⟨      B                      u        ^              λ    ,                    u        ^              λ    ⟩      |      and    ‖      B        ‖                  b        e        r              =      sup          λ      ,      μ      ∈              F                  |    ⟨      B                      u        ^              λ    ,                    u        ^              μ    ⟩      |    .In this work, we establish new upper bounds for these two quantities. In particular, we derive bounds for their sums and obtain novel estimates for a specific type of product, namely        b    e    r    (            C        ∗        B    ), where              C        ∗   denotes the adjoint of        C  . Some of our results also involve another Berezin-type norm that is equivalent to the quantities mentioned above. Several applications and improvements of existing results in the literature are provided.}
}