arXiv subjects

Shankar Veerabathiran

Publications and source records attributed to Shankar Veerabathiran.

9 recordsLinked to original sources

arXiv2025

Doubly Commuting Semigroups of Isometries

In this paper, we discuss the structure of doubly commuting semigroups of isometries. We record a new proof of Cooper's theorem in the Hilbert module setting. We discuss the Fell topology on the set of equivalence classes of irreducible, doubly commuting isometric representations of $\mathbb{R}_{+}^{d}$. We show that if $d$ is finite, the topology is $T_0$. We indicate the pathologies that occur when $d=\infty$. In particular, we show that Wold decomposition fails for isometric representations of $\mathbb{R}_{+}^{\infty}$ and prove that the Fell topology on the set of equivalence classes of irreducible, doubly commuting isometric representations of $\mathbb{R}_{+}^{\infty}$ is not $T_0$

math.OA↗

arXiv2024

Powers and roots of partial isometric covariant representations

Isometric covariant representations play an important role in the study of Cuntz-Pimsner algebras. In this article, we study partial isometric covariant representations and explore under what conditions powers and roots of partial isometric covariant representations are also partial isometric covariant representations.

math.OA↗

arXiv2024

A characterization of invariant subspaces for isometric representations of product system over $\mathbb{N}_0^{k}$

Using the Wold-von Neumann decomposition for the isometric covariant representations due to Muhly and Solel, we prove an explicit representation of the commutant of a doubly commuting pure isometric representation of the product system over $\mathbb{N}_0^{k}.$ As an application, we study a complete characterization of invariant subspaces for a doubly commuting pure isometric representation of the product system. This provides us a complete set of isomorphic invariants. Finally, we classify a large class of commuting isometric representations of the product system.

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arXiv2023

Berger-Coburn-Lebow representation for pure isometric representations of product system over $\mathbb N^2_0$

We obtain Berger-Coburn-Lebow (BCL)-representation for pure isometric covariant representation of product system over $\mathbb{N}_0^2$. Then the corresponding complete set of (joint) unitary invariants is studied, and the BCL- representations are compared with other canonical multi-analytic descriptions of the pure isometric covariant representation. We characterize the invariant subspaces for the pure isometric covariant representation. Also, we study the connection between the joint defect operators and Fringe operators, and the Fredholm index is introduced in this case. Finally, we introduce the notion of congruence relation to classify the isometric covariant representations of the product system over $\mathbb{N}_0^2$.

math.OA↗

arXiv2023

Beurling quotient subspaces for covariant representations of product systems

We characterize Beurling quotient subspaces for pure doubly commuting isometric representations of product systems. As a consequence, we derive a concrete regular dilation theorem for a pure completely contractive covariant representation which satisfies Brehmer-Solel condition and using it and the above characterization, we provide a necessary and sufficient condition that when a completely contractive covariant representation is unitarily equivalent to the compression of the induced representation on the Beurling quotient subspace. Further, we study the relation between Sz.Nagy-Foias type factorization of isometric multi-analytic operators and joint invariant subspaces.

math.OA↗

arXiv2022

Regular covariant representations and their Wold-type decomposition

Olofsson introduced a growth condition regarding elements of an orbit for an expansive operator and generalized Richter's wandering subspace theorem. Later on, using the Moore-Penrose inverse, Ezzahraoui, Mbekhta, and Zerouali extended the growth condition and obtained a Shimorin-Wold-type decomposition. Shimorin-Wold-type decomposition for completely bounded covariant representations, which are close to isometric representations, is obtained in \cite{HV19}. This paper extends this decomposition for regular, completely bounded covariant representation having reduced minimum modulus $\geq 1$ that satisfies the growth condition. To prove the decomposition, we introduce the terms regular, algebraic core, and reduced minimum modulus in the completely bounded covariant representation setting and work out several fundamental results. Consequently, we shall analyze the weighted unilateral shift introduced by Muhly and Solel and introduce and explore a non-commutative weighted bilateral shift.

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arXiv2020

Doubly commuting invariant subspaces for representations of product systems of $C^*$-correspondences

We obtain a Shimorin-Wold-type decomposition for a doubly commuting covariant representation of a product system of $C^*$-correspondences. This extends a recent Wold-type decomposition by Jeu and Pinto for a $q$-doubly commuting isometries. Application to the wandering subspaces of doubly commuting induced representations is explored, and a version of Mandrekar's Beurling type theorem is obtained to study doubly commuting invariant subspaces using Fock space approach due to Popescu.

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arXiv2019

Generating wandering subspaces for doubly commuting covariant representations

We obtain a Halmos-Richter-type wandering subspace theorem for covariant representations of C*-correspondences. Further the notion of Cauchy dual and a version of Shimorin's Wold-type decomposition for covariant representations of C*-correspondences is explored and as an application a wandering subspace theorem for doubly commuting covariant representations is derived. Using this wandering subspace theorem generating wandering subspaces are characterized for covariant representations of product systems in terms of the doubly commutativity condition.

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arXiv2018

Covariant representations of subproduct systems: Invariant subspaces and curvature

Let $X=(X(n))_{n \in \mathbb{Z_+}}$ be a standard subproduct system of $C^*$-correspondences over a $C^*$-algebra $\mathcal M.$ Assume $T=(T_n)_{n \in \mathbb{Z_+}}$ to be a pure completely contractive, covariant representation of $X$ on a Hilbert space $\mathcal H,$ and $\mathcal S$ to be a non-trivial closed subspace of $\mathcal H.$ Then $\mathcal{S}$ is invariant for $T$ if and only if there exist a Hilbert space $\mathcal{D},$ a representation $π$ of $\mathcal M$ on $\mathcal D,$ and a partial isometry $Π: \mathcal{F}_X\bigotimes_π\mathcal{D}\to \mathcal{H} $ such that $$Π(S_n(ζ)\otimes I_{\mathcal{D}})=T_n(ζ)Π~\mbox{whenever}~ζ\in X(n), ~n\in \mathbb{Z_+},~\mbox{and}$$ $\mathcal S$ is the range of $Π,$ or equivalently, $P_{\mathcal S}=ΠΠ^*.$ This result leads us to many important consequences including Beurling type theorem and other general observations on wandering subspaces. We extend the notion of curvature for completely contractive, covariant representations and analyze it in terms of the above results.

math.OA↗