The weighted Davis-Wielandt Berezin number for reproducing kernel Hilbert space operators - Eslami Mahdiabadi, Nooshin; Bakherad, Mojtaba; Hajmohamadi, Monire; Petrushka, Mykola (Journal of inequalities and applications, 2025)
A functional Hilbert space is the Hilbert space of complex-valued functions on some set Θ ⊆ C that the evaluation functionals φ λ ( f ) = f ( λ ) , λ ∈ Θ are continuous on H . Then, by the Riesz representation theorem, there is a unique element k λ ∈ H such that f ( λ ) = 〈 f , k λ 〉 for all f ∈ H and every λ ∈ Θ . The function k on Θ × Θ defined by k ( z , λ ) = k λ ( z ) is called the reproducing kernel of H . In this study, we defined the weighted Davis-Wielandt Berezin number, and then we obtained some related inequalities. It is shown, among other inequalities, that if X ∈ L ( H ) and ν ∈ [ 0 , 1 ] , then 1 2 ( ber 2 ( X ν + | X ν | 2 ) + c ber 2 ( X ν − | X ν | 2 ) ) ≤ d w ber ν 2 ( X ) ≤ 1 2 ( ber 2 ( X ν + | X ν | 2 ) + ber 2 ( X ν − | X ν | 2 ) ) , where X ν = ( 1 − 2 ν ) X ∗ + X . Some bounds for the weighted Davis-Wielandt Berezin number are also established.