Images of Minimal-Vector Sequences Under Weighted Composition Operators on L2( $$ \mathbb{D} $$ )

Abstract

Translate Article

Let X be either the unit interval in ℝ or the unit disk \( \mathbb{D} \) in ℂ. Chalendar, Flattot, and Partington [2] study weighted composition operators Tw,ϒ on L2(X), where w ∈ L∞(X), ϒ : X → X is injective, and Tw,ϒf = wfoϒ for f ∈ L2(X). They introduce a (strict) partial order ≻ on X associated with Tw,ϒ and use it to obtain a sufficient condition for convergence of the sequence (Tn w,ϒyn) where (yn) is a backward minimal-vector sequence for Tw,ϒ. For the L2(\( \mathbb{D} \)) case, they give a detailed analysis of the situation where ϒ is linear-fractional. Through further study of the partial order ≻, we are able to generalize results from [2] that apply when ϒ is linear-fractional, replacing the linear-fractional hypotheses with univalence. In particular, our work yields generalizations of an invariant-subspace theorem in [2].