arXiv subjects

Kalpesh J. Haria

Publications and source records attributed to Kalpesh J. Haria.

10 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.

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arXiv2026

Characteristic Functions and Colligations

The characteristic function of row contractions and the characteristic function of liftings of row contractions are multi-analytic operators which are complete invariants up to unitary equivalence for row contractions and liftings of row contractions, respectively. We provide alternate proofs for these properties of characteristic functions using colligations. Co-isometric observable colligations with certain class of basic operators are characterized. Blaschke factor based transformations of the characteristic function of lifting are studied.

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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.

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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.

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arXiv2020

Commuting row contractions with polynomial characteristic functions

A characteristic function is a special operator-valued analytic function defined on the open unit ball of $\mathbb{C}^n$ associated with an $n$-tuple of commuting row contraction on some Hilbert space. In this paper, we continue our study of the representations of $n$-tuples of commuting row contractions on Hilbert spaces, which have polynomial characteristic functions. Gleason's problem plays an important role in the representations of row contractions. We further complement the representations of our row contractions by proving theorems concerning factorizations of characteristic functions. We also emphasize the importance and the role of the noncommutative operator theory and noncommutative varieties to the classification problem of polynomial characteristic functions.

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arXiv2018

Isometric dilations and von Neumann inequality for a class of tuples in the polydisc

The celebrated Sz.-Nagy and Foias and Ando theorems state that a single contraction, or a pair of commuting contractions, acting on a Hilbert space always possesses isometric dilation and subsequently satisfies the von Neumann inequality for polynomials in $\mathbb{C}[z]$ or $\mathbb{C}[z_1, z_2]$, respectively. However, in general, neither the existence of isometric dilation nor the von Neumann inequality holds for $n$-tuples, $n \geq 3$, of commuting contractions. The goal of this paper is to provide a taste of the isometric dilations, the von Neumann inequality and a sharper version of von Neumann inequality for a large class of $n$-tuples, $n \geq 3$, of commuting contractions.

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arXiv2016

Factorizations of Characteristic Functions

Let $A = (A_1, \ldots, A_n)$ and $B = (B_1, \ldots, B_n)$ be row contractions on $\mathcal{H}_1$ and $\mathcal{H}_2$, respectively, and $X$ be a row operator from $\oplus_{i=1}^n \mathcal{H}_2$ to $\mathcal{H}_1$. Let $D_{A^*} = (I - A A^*)^{\frac{1}{2}}$ and $D_{B} = (I - B^* B)^{\frac{1}{2}}$ and $Θ_T$ be the characteristic function of $T = \begin{bmatrix} A& D_{A^*}L D_B\\ 0 & B \end{bmatrix}$. Then $Θ_T$ coincides with the product of the characteristic function $Θ_A$ of $A$, the Julia-Halmos matrix corresponding to $L$ and the characteristic function $Θ_B$ of $B$. More precisely, $Θ_T$ coincides with \[ \begin{bmatrix} Θ_B & 0 \\ 0 & I \end{bmatrix} (I_Γ\otimes \begin{bmatrix} L^* & (I - L^* L)^{\frac{1}{2}} \\ (I - L L^*)^{\frac{1}{2}} & - L \end{bmatrix}) \begin{bmatrix} Θ_A & 0\\ 0& I\end{bmatrix}, \] where $Γ$ is the full Fock space. Similar results hold for constrained row contractions.

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arXiv2014

Functional Models and Minimal Contractive Liftings

Based on a careful analysis of functional models for contractive multi-analytic operators we establish a one-to-one correspondence between unitary equivalence classes of minimal contractive liftings of a row contraction and injective symbols of contractive multi-analytic operators. This allows an effective construction and classification of all such liftings with given defects. Popescu's theory of characteristic functions of completely non-coisometric row contractions is obtained as a special case satisfying a Szegö condition. In another special case of single contractions and defects equal to $1$ all non-zero Schur functions on the unit disk appear in the classification. It is also shown that the process of constructing liftings iteratively reflects itself in a factorization of the corresponding symbols.

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arXiv2014

Generalized repeated interaction model and transfer functions

Using a scheme involving a lifting of a row contraction we introduce a toy model of repeated interactions between quantum systems. In this model there is an outgoing Cuntz scattering system involving two wandering subspaces. We associate to this model an input/output linear system which leads to a transfer function. This transfer function is a multi-analytic operator, and we show that it is inner if we assume that the system is observable. Finally it is established that transfer functions coincide with characteristic functions of associated liftings.

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arXiv2012

Outgoing Cuntz Scattering System for a Coisometric Lifting and Transfer Function

We study a coisometry that intertwines Popescu's presentations of minimal isometric dilations of a given operator tuple and of a coisometric lifting of the tuple. Using this we develop an outgoing Cuntz scattering system which gives rise to an input-output formalism. A transfer function is introduced for the system. We also compare the transfer function and the characteristic function for the associated lifting.

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