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Azad Rohilla

Publications and source records attributed to Azad Rohilla.

6 recordsLinked to original sources
arXiv2026

Wold-type decomposition and Beurling-Type Theorem for Covariant Representations

Using operator inequalities, we study a Wold-type decomposition of covariant representations. Building on this decomposition, we prove a Beurling-type theorem showing that every nonzero invariant subspace is uniquely determined by its wandering subspace. Our results extend classical theorems of Beurling and subsequent developments for left-invertible operators to the setting of covariant representations of $C^*$-correspondences, providing a unified framework for invariant subspace theory under operator inequalities.

math.OA↗
arXiv2026

Wold-type decomposition for doubly twisted left-invertible covariant representations

In this article, we have introduced the notion of a near-isometric covariant representation of a $C^*$-correspondence. The other objective is to provide a unified approach to several known results for a large class of left-invertible covariant representations of a product system and prove Wold-type decomposition for the case of doubly twisted left-invertible covariant representations and study some applications.

math.OA↗
arXiv2024

Quantum $U$-channels on $S$-spaces

If the symmetry, (an operator $J$ satisfying $J=J^*=J^{-1}$) which defines the Krein space, is replaced by a (not necessarily self-adjoint) unitary, then we have the notion of an $S$-space which was introduced by Szafraniec. In this paper, we consider $S$-spaces and study the structure of completely $U$-positive maps between the algebras of bounded linear operators. We first give a Stinespring-type representation for a completely $U$-positive map. On the other hand, we introduce Choi $U$-matrix of a linear map and establish the equivalence of the Kraus $U$-decompositions and Choi $U$-matrices. Then we study properties of nilpotent completely $U$-positive maps. We develop the $U$-PPT criterion for separability of quantum $U$-states and discuss the entanglement breaking condition of quantum $U$-channels and explore $U$-PPT squared conjecture. Finally, we give concrete examples of completely $U$-positive maps and examples of $3 \otimes 3$ quantum $U$-states which are $U$-entangled and $U$-separable.

math.FA↗
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.

math.OA↗