An extension of the Euclidean Berezin number

Nooshin Eslami Mahdiabadi ; Mojtaba Bakherad

Filomat, Tome 37 (2023) no. 24, p. 8377

Résumé

The Berezin transform ̃ A of an operator A, acting on the reproducing kernel Hilbert space H = H(Θ) over some (non-empty) set Θ, is defined by ̃ A(λ) = 〈Aˆk〈Aˆ 〈Aˆk λ , ˆ k λ 〉 (λ ∈ Θ), wherê k λ = k λ ∥k λ ∥ is the normalized reproducing kernel of H. The Berezin number of an operator A is defined by ber(A) = sup λ∈Θ ∣ ̃ A(λ)∣ = sup λ∈Θ ∣〈Aˆk∣〈Aˆ ∣〈Aˆk λ , ˆ k λ 〉∣. In this paper, by using the definition of-generalized Euclidean Berezin number, we obtain some possible relations and inequalities. It is shown, among other inequalities, that if A i ∈ L(H(Θ)) (i = 1,. .. , n), then ber(A 1 , ..., An) ≤ −1 (n ∑ i=1 (ber(A i))) ≤ n ∑ i=1 ber(A i), in which ∶ [0, ∞) → [0, ∞) is a continuous increasing convex function such that (0) = 0.

Détail

DOI : 10.2298/FIL2324377E

Classification : 47A63, 15A18, 15A45
Keywords: Berezin number, Berezin set, Berezin symbol, Euclidean Berezin number