Let $B$ be a finite Blaschke product and $K_B = H^2 \ominus BH^2$ the associated finite-dimensional model space. For a scale of function spaces, including Hardy, Bergman, and Dirichlet, we have decompositions of the form

\[ X = K_B \oplus BK_B \oplus B^2 K_B \oplus \ldots \]

with appropriate norm estimates. The applications include weighted composition operators [1], commutants of multipliers [2], and wandering subspaces [3].

This is joint work with Eva Gallardo-Gutiérrez, with additional material from Isabelle Chalendar and Daniel Seco.

[1] I. Chalendar, E.A. Gallardo-Gutiérrez and J.R. Partington, Weighted composition operators on the Dirichlet space: boundedness and spectral properties. Math. Annalen 363 (2015), 1265--1279.

[2] E.A. Gallardo-Gutiérrez and J.R. Partington, Multiplication by a finite Blaschke product on weighted Bergman spaces: commutant and reducing subspaces. Preprint (2021), https://arxiv.org/abs/2105.07760.

[3] E.A. Gallardo-Gutiérrez, J.R. Partington and D. Seco, On the wandering property in Dirichlet spaces. Integral Equations and Operator Theory 92 (2020), paper no. 16.