arXiv subjects

Aashish Kumar Maurya

Publications and source records attributed to Aashish Kumar Maurya.

3 recordsLinked to original sources

arXiv2026

Regularity of the Sz.-Nagy and Foiaş Factorization of Characteristic Functions and Its Multivariable Analogue

Given a contraction with an invariant subspace and a row contraction with a joint invariant subspace, a factorization of their characteristic functions was obtained by Sz.-Nagy and Foiaş, and by Haria, Maji, and Sarkar, respectively. In this article, we investigate the regularity of this factorization in both the single-variable and multivariable cases. We construct examples of a contraction $T$ with an invariant subspace such that this factorization is not regular in general. If $T$ is completely non-unitary, we prove that this factorization is either regular or strange. Furthermore, we obtain a characterization of the regularity of this factorization. Using this characterization, we identify several classes of contractions and row contractions for which this factorization is regular. Among these classes, the most notable classes are pure contractions and pure row contractions. Additionally, for any integer $k>2$, we introduce the concept of $k$-strange factorizations for contractive analytic functions, which extend the concept of strange ($2$-strange) factorizations introduced by Sz.-Nagy and Foiaş. Finally, we prove that if a contraction is pure or is a completely non-unitary contraction for which $Δ_{Θ_T}(t)$ has finite rank almost everywhere, then its characteristic function does not admit any $k$-strange factorization.

math.FA↗

arXiv2026

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

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 Foiaş, 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.

math.OA↗

arXiv2026

$k$-Regular Factorizations and Joint Invariant Subspaces of Completely Non-Coisometric Row Contractions

This article investigates $k$-regular factorizations of characteristic functions associated with completely non-coisometric row contractions. In this setting, a one-to-one correspondence is established between chains of joint invariant subspaces \[ \mathcal{M}_1 \subseteq \cdots \subseteq \mathcal{M}_{k-1} \] and $k$-regular factorizations of the characteristic function of a completely non-coisometric row contraction. A functional model corresponding to a given $k$-regular factorization of a purely contractive multi-analytic operator satisfying the Szegő condition is further constructed, and the associated chain of joint invariant subspaces is characterized in terms of the underlying multi-analytic factors. Finally, it is shown that any such chain of joint invariant subspaces induces a block upper-triangular decomposition of the underlying row contraction, and that the characteristic function of each diagonal block coincides with the purely contractive part of the corresponding factor in the $k$-regular factorization.

math.FA↗