Journal of Operator Theory
Volume 88, Issue 2, Fall 2022 pp. 407-443.
Absolutely summing weighted composition operators on Bloch spaces
Authors:
Tonie Fares (1), Pascal Lefevre (2)
Author institution:(1) Univ. Artois, UR 2462, Laboratoire de Mathematiques de Lens (LML), F-62300 Lens, France
(2) Univ. Artois, UR 2462, Laboratoire de Mathematiques de Lens (LML), F-62300 Lens, France
Summary: We characterize $p$-summing composition operators
from a Bloch space $\mathcal{B}^{\mu}$ to another such space
$\mathcal{B}^{\beta}$, where $\mu,\beta>0$.
The corresponding result on little Bloch-type spaces is also proved.
We construct an example of a conformal mapping from $\mathbb{D}$
into itself which has a contact point with the unit circle $\mathbb{T}$,
and induces a compact composition operator, that fails to be $p$-summing
for any $p\geqslant 1$. We also detail the case of lens maps.
Moreover we explore the case of weighted composition operators and give
characterizations for a class of weights. We also show that compactness of a
composition operator on $\mathcal{B}^\beta$ and $\mathcal{B}^\beta_0$
implies its compactness on Bergman spaces.
DOI: http://dx.doi.org/10.7900/jot.2021oct20.2342
Keywords: composition operators, Bloch spaces, absolutely summing operators, lens map
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