2014 | OriginalPaper | Chapter
Composition Operators on Large Fractional Cauchy Transform Spaces
Authors : Yusuf Abu Muhanna, El-Bachir Yallaoui
Published in: Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation
Publisher: Springer Basel
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For α >0 and
z
in the unit disk
D
the spaces of fractional Cauchy transforms
F
α
are known as the family of all functions
f
(
z
) such that
f
(
z
)=
$$\int_{T}[K(\overline{x}z)]^{\alpha}d\mu(x)$$
where
K
(
z
)=(1-
z
)
-1
is the Cauchy kernel,
T
is the unit circle and μ ∈
$$\mathcal{M}$$
the set of complex Borel measure on
T
. The Banach space
F
α
may be written as
F
α
=(
F
α
)
a
⊕ (
F
α
)
s
, where (
F
α
)
a
is isomorphic to a closed subspace of
$$\mathcal{M}_a$$
the subset of absolutely continuous measures of
$$\mathcal{M}$$
, and (
F
α
)
s
is isomorphic to
$$\mathcal{M}_s$$
the subspace of
$$\mathcal{M}$$
of singular measures. In this article we show that for α ≥1, the composition operator
C
φ
is compact on
K
α
C
φ
$$C_\varphi[K^{\alpha}(\overline{x}z)]\subset(F_{\alpha})_a$$
and in doing so, extend a result due to [1] who showed that
C
φ
is compact on
F
1
if and only if
C
φ
(
F
1
) ⊂ (
F
1
)
a
.