$k$-Regular Factorizations and Invariant Subspaces of Completely Non-Unitary Contractions

Abstract

We introduce the notion of $k$-regular factorizations for contractions into $k$ factors, generalizing the classical notion of regular factorization due to Sz.-Nagy and Foia\c{s}, and develop a systematic framework for their analysis. Using this concept, a one-to-one correspondence is established between chains of invariant subspaces \[ \mathcal{M}_1 \subseteq \cdots \subseteq \mathcal{M}_{k-1}, \] associated with a completely non-unitary contraction and the class of all $k$-regular factorizations of its characteristic function. An explicit functional model for the corresponding completely non-unitary contraction is constructed, and the associated functional model representations of the chain of invariant subspaces are obtained. Finally, examples illustrating the applicability of these results are provided. Furthermore, we introduce symmetric $k$-regular tuples for commuting $k$-contractions, proving this property holds when the product of contractions has a finite-dimensional defect space and is $k$-regular under at least one permutation. Importantly, we demonstrate that the classical counterexamples for commuting $3$-tuples provided by Parrott, Crabb-Davie, and Kaijser-Varopoulos fail to be symmetric $3$-regular tuples. This structural failure highlights the significance of symmetric $k$-regularity and offers a promising framework that encourages further research into this property and the commutative dilation theory of commuting $k$-contractions.