Let N denote the set of all positive integers and N0=N∪{0}. For m∈N, let Bm={z∈Cm:|z|<1} be the open unit ball in the m−dimensional Euclidean space Cm. Let H(Bm) be the space of all analytic functions on Bm. For an analytic self map ξ=(ξ1,ξ2,…,ξm) on Bm and ϕ1,ϕ2,ϕ3∈H(Bm), we have a product type operator Tϕ1,ϕ2,ϕ3,ξ which is basically a combination of three other operators namely composition operator Cξ, multiplication operator Mϕ and radial derivative operator R. We study the boundedness and compactness of this operator mapping from weighted Bergman–Orlicz space AσΨ into weighted type spaces Hω∞ and Hω,0∞.
Consider an open unit disk
D
=
z
∈
ℂ
:
z
<
1
in the complex plane
ℂ
,
ξ
a holomorphic function on
D
, and
ψ
a holomorphic self-map of
D
. For an analytic function
f
, the weighted composition operator is denoted and defined as follows:
W
ξ
,
ψ
f
z
=
ξ
z
f
ψ
z
. We estimate the essential norm of this operator from Dirichlet-type spaces to Bers-type spaces and Bloch-type spaces.
LetD = {z ? C : |z| < 1} be the open unit disk in the complex plane C. By
H(D), denote the space of all holomorphic functions on D. For an analytic
self map ? on D and u, v ? H(D), we have a product type operator Tu,v,?
defined by Tu,v,? f (z) = u(z) f (?(z)) + v(z) f ?(?(z)), f ? H(D), z ? D,
This operator is basically a combination of three other operators namely
composition operator, multiplication operator and differentiation operator.
We study the boundedness and compactness of this operator from
Dirichlet-type spaces to Zygmund-type spaces.
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