On a class of unitary operators on weighted Bergman spaces

Namita Das ; Swarupa Roy

Filomat, Tome 37 (2023) no. 7, p. 2013

Résumé

In this paper we consider a class of weighted composition operators defined on the weighted Bergman spaces L 2 a (dA α) where D is the open unit disk in C and dA α (z) = (α + 1)(1 − |z| 2) α dA(z), α > −1 and dA(z) is the area measure on D. These operators are also self-adjoint and unitary. We establish here that a bounded linear operator S from L 2 a (dA α) into itself commutes with all the composition operators C (α) a , a ∈ D, if and only if B α S satisfies certain averaging condition. Here B α S denotes the generalized Berezin transform of the bounded linear operator S from L 2 a (dA α) into itself, C (α) a f = (f • ϕ a), f ∈ L 2 a (dA α) and ϕ ∈ Aut(D). Applications of the result are also discussed. Further, we have shown that if M is a subspace of L ∞ (D) and if for ϕ ∈ M, the Toeplitz operator T (α) ϕ represents a multiplication operator on a closed subspace S ⊂ L 2 a (dA α), then ϕ is bounded analytic on D. Similarly if q ∈ L ∞ (D) and B n is a finite Blaschke product and M (α) q Rane C (α) Bn ⊂ L 2 a (dA α), then q ∈ H ∞ (D). Further, we have shown that if ψ ∈ Aut(D), then N = q ∈ L 2 a (dA α) : M (α) q Rane C (α) ψ ⊂ L 2 a (dA α) = H ∞ (D) if and only if ψ is a finite Blaschke product. Here M (α) ϕ , T (α) ϕ , C (α) ϕ denote the multiplication operator, the Toeplitz operator and the composition operator defined on L 2 a (dA α) with symbol ϕ respectively.

Détail

DOI : 10.2298/FIL2307013D

Classification : 47B38, 47B32
Keywords: Weighted Bergman spaces, reproducing kernel, Toeplitz operator, little Hankel operator, weighted composition operator