Maria Tjani

Compact composition operators on Besov spaces

We give a Carleson measure characterization of the compact composition operators on Besov spaces. We use this characterization to show that every compact composition operator on a Besov space is compact on the Bloch space. Finally we give conditions that guarantee that the converse holds.

COMPOSITION OPERATORS ON SOME MOBIUS INVARIANT BANACH SPACES

We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

Isometries among composition operators on Besov type spaces

Given , , let denote the Besov type space of analytic functions on the unit disk . Allen, Heller and Pons have shown that the isometries among composition operators on certain Besov spaces, , are induced by rotations. We extend this to all Besov spaces and in fact to all Besov type spaces . We show that in every Besov type space, except on , rotations are the only symbols inducing isometries. We show that this is the case in every weighted Dirichlet space , as well.

Weighted Composition Operators from the Analytic Besov Spaces to BMOA

Let ψ and \( \varphi \) be analytic functions on the open unit disk \( (\mathbb{D}) \) with \( \varphi(\mathbb{D})\sqsubseteq\mathbb{D}\, {\rm and\, let }\, 1\leq p<\infty \). We characterize the bounded and the compact weighted composition operators \( {W_{\psi\varphi}} \) from the analytic Besov space B p into BMOA and into VMOA. We also show that there are no isometries among the composition operators.