# 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