On a class of Toeplitz and little Hankel operators on L2 a (U + )

Namita Das ; Sworup Kumar Das

Filomat, Tome 36 (2022) no. 13, p. 4397

Résumé

In this paper we establish certain algebraic properties of Toeplitz operators and a class of little Hankel operators defined on the Bergman space of the upper half plane. We show that if K is a compact operator on L 2 a (U +), M(s) = i−s i+s , τ a (s) = (c−1)+sd (1+c)s−d where a = c + id ∈ D, s ∈ U + and J f (s) = f (−s) then lim |a|→1 − ||K − T J(M • τa) KT * M • τa || = 0 and for φ, ψ ∈ h ∞ (D), if ℏ αs(ψ • M) T φ • M − T φ • M ℏ αs(ψ • M) is compact, then lim w=x+iy y→0 ||c([ℏ αs(ψ • M) d w ] ⊗ [ℏ * φ • M d w ]) + c([ℏ J(φ • M) d w ] ⊗ [ℏ * αs(ψ • M) d w ])|| = 0, where d w (s) = 1 √ π w + i w − i (−2i)Im w (s + w) 2 , w ∈ U + , ℏ φ is the little Hankel operator on L 2 a (U +) with symbol φ and α s is a function defined on U + with |α s | = 1, for all s ∈ U +. Applications of these results are also obtained.

Détail

DOI : 10.2298/FIL2213397D

Classification : 30H20, 47B35
Keywords: Bergman space, Upper half plane, Toeplitz operators, Little Hankel operators, Weighted composition operators