Namita Das†, Swarupa Roy‡
Abstract: In this paper we have shown that if S ∈ L(L
2
a
(dAα)) and
Θ
(α)
S
(x, y)Θ
(α)
T
(x, y)(K(α)
(x, y))
2 ≈ Θ
(α)
ST
(x, y)(K(α)
(x, y))
2
for all x, y ∈
D and for all T ∈ L(L
2
a
(dAα)), then S = T
(α)
φ
for some φ ∈ H∞(D)
and the matrix of S is lower triangular, where Θ
(α)
S
(x, y) for S ∈
L(L
2
a
(dAα)) is a function on D × D meromorphic in x and conjugate
meromorphic in y. Further, we show that if ψ, φ ∈ L∞(D), R(α) ∈
L(L
2
a
(dAα)), then Θ
(α)
T
(α)
φ
(x, y)Θ
(α)
S
(α)
ψ
(x, y)(K(α)
(x, y))
2 ≈ Θ
(α)
R(α) (x, y)
·(K(α)
(x, y))
2 holds for all x, y ∈ D if and only if there exists β ∈ C
such that φ ≡ β and R(α) = S
(α)
βψ
.
MSC: 47B38, 47B32
keywords: Weighted Bergman spaces, reproducing kernel, Toeplitz operator, little Hankel operator, Berezin transform.
DOI 10.56082/annalsarscimath.2020.1-2.99
† namitadas440@yahoo.co.in, P.G. Dept. of Mathematics, Utkal University,Vani Vihar, Bhubaneswar- 751004, Odisha, India
‡ swarupa.roy@gmail.com, P.G. Dept. of Mathematics, Utkal University, Vani Vihar, Bhubaneswar- 751004, Odisha, India.
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application, Volume 12 no 1-2, 2020
