Berezin number inequalities in terms of Specht's

Abstract

Smooth functions are associated with operators on Hilbert spaces of analytic functions through the Berezin transform. The Berezin symbol and the Berezin number of an operator A on the Hilbert functional space H(Ω) over some set Ω with the reproducing kernel are defined, respectively, by 
 A ̃(μ)=〈A K_μ/K_μ ,K_μ/K_μ 〉,μ∈Ω and ber(A)=sup┬(μ∈Ω)⁡|A ̃(μ)|. 
 By using this bounded function A ̃, we present some new Berezin number inequalities of Hilbert functional space operators. Some inequalities with respect to Specht's ratio are improved and generalized. Using these modifications, we also establish various new inequalities for the Berezin radius and Berezin norm of operators.

More Correct Berezin Symbol Inequalities

“…Berezin number of the operator 𝑇 are defined by 𝐵𝑒𝑟(𝑇) = {𝑇 (𝜉): 𝜉 ∈ Ω} 𝑎𝑛𝑑 𝑏𝑒𝑟(𝑇) = 𝑠𝑢𝑝 𝑇 (𝜉) : 𝜉 ∈ Ω , respectively (see, (Karaev,2006;Karaev, 2013). In some recent works, several Berezin radius inequalities have been examined by authors (Başaran and Gürdal, 2021), (Başaran and Gürdal, 2023a), (Başaran and Gürdal, 2023b), (Chalender et al, 2012), (Garayev et al, 2020), (Garayev and Alomari, 2021), , (Gürdal and Başaran, 2022).…”