arXiv subjects

Raj Dahya

Publications and source records attributed to Raj Dahya.

9 recordsLinked to original sources

arXiv2026

The Choi-Cholesky algorithm for completely positive maps

We establish explicit means via which natural dilations of completely positive (CP) maps can be constructed \`a la Kraus's IInd representation theorem. To obtain this, we rely on the Choi-Jamio{\l}kowski correspondence and develop a Cholesky algorithm for bi-partite systems. This enables a canonical construction of adjoint actions which recover the behaviour of the original CP-maps. Our results hold under separability assumptions and the requirement that the maps are completely bounded and preserve the subideal of finite rank operators.

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arXiv2025

Dilations of non-Markovian dynamical systems on graphs

To generalise evolution families we consider systems of contractions $\{φ(u, v)\}_{(u, v) \in E}$ defined on the edges of a graph $\mathcal{G} = (Ω, E)$. In this setup the Markov property, or \emph{divisibility}, can be modelled via $φ(u, v)φ(v, w) = φ(u, w)$ for edges $(u, v), (v, w), (u, w) \in E$. We obtain results in three settings: 1) contractive Banach space operators; 2) positive unital maps on $C^{\ast}$-algebras; and 3) CPTP-maps on trace class operators on a Hilbert space. In the discrete setting, we are able to dilate possibly indivisible families of contractions to divisible families of operators with 'nice' properties (viz. surjective isometries resp. $C^{\ast}$-algebraic automorphisms resp. unitary representations). In the special case of linearly ordered graphs equipped with the order topology, we establish sufficient conditions for strongly continuous dilations of possibly indivisible families in the Banach space and $C^{\ast}$-algebra contexts. To achieve these results we work with string-rewriting systems, and make use of and extend dilation theorems of Stroescu [44], Kraus [23, 24], and vom Ende--Dirr [50].

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arXiv2025

Free dilations of families of $\mathcal{C}_{0}$-semigroups and applications to evolution families

Commuting families of contractions or contractive $\mathcal{C}_{0}$-semigroups on Hilbert spaces often fail to admit power dilations resp, simultaneous unitary dilations which are themselves commutative (see [45, 13, 15]). In the \emph{non-commutative} setting, Sz.-Nagy [60] and Bo\.{z}ejko [5] provided means to dilate arbitrary families of contractions. The present work extends these discrete-time results to families $\{T_{i}\}_{i \in I}$ of contractive $\mathcal{C}_{0}$-semigroups. We refer to these dilations as continuous-time \emph{free unitary dilations} and present three distinct approaches to obtain them: 1) An explicit derivation applicable to semigroups that arise as interpolations; 2) A full proof with an explicit construction, via the theory of co-generators \`{a} la S{\l}oci\'{n}ski [54, 55]; and 3) A second full proof based on the abstract structure of semigroups, which admits a natural reformulation to semigroups defined over topological free products of $\mathbb{R}_{\geq 0}$ and leads to various residuality results. In 2) a II\textsuperscript{nd} free dilation theorem for topologised index sets is developed via a reformulation of the Trotter--Kato theorem for co-generators. As an application of this we demonstrate how evolution families can be reduced to continuously monitored processes subject to temporal change, \`{a} la the quantum Zeno effect [22, 23, 24, 30, 37].

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arXiv2024

Interpolation and non-dilatable families of $\mathcal{C}_{0}$-semigroups

We generalise a technique of Bhat and Skeide (2015) to interpolate commuting families $\{S_{i}\}_{i \in \mathcal{I}}$ of contractions on a Hilbert space $\mathcal{H}$, to commuting families $\{T_{i}\}_{i \in \mathcal{I}}$ of contractive $\mathcal{C}_{0}$-semigroups on $L^{2}(\prod_{i \in \mathcal{I}}\mathbb{T}) \otimes \mathcal{H}$. As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott's construction (1970), we then demonstrate for $d \in \mathbb{N}$ with $d \geq 3$ the existence of commuting families $\{T_{i}\}_{i=1}^{d}$ of contractive $\mathcal{C}_{0}$-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt. the topology of uniform wot-convergence on compact subsets of $\mathbb{R}_{\geq 0}^{d}$ of non-unitarily dilatable and non-unitarily approximable $d$-parameter contractive $\mathcal{C}_{0}$-semigroups on separable infinite-dimensional Hilbert spaces for each $d \geq 3$. Similar results are also developed for $d$-tuples of commuting contractions. And by building on the counter-examples of Varopoulos--Kaijser (1973--74), a 0--1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, \textit{viz.} that `typical' pairs of commuting operators can be simultaneously embedded into commuting pairs of $\mathcal{C}_{0}$-semigroups, which extends results of Eisner (2009--10).

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arXiv2023

Characterisations of dilations via approximants, expectations, and functional calculi

We consider characterisations of unitary dilations and approximations of irreversible classical dynamical systems on a Hilbert space. In the commutative case, building on the work in [9], one can express well known approximants (e.g. Hille- and Yosida-approximants) via expectations over certain stochastic processes. Using this, our first result characterises the simultaneous regular unitary dilatability of commuting families of $C_{0}$-semigroups via the dilatability of such approximants as well as via regular polynomial bounds. This extends the results in [13] to the unbounded setting. We secondly consider characterisations of unitary and regular unitary dilations via two distinct functional calculi. Applying these tools to a large class of classical dynamical systems, these two notions of dilation exactly characterise when a system admits unitary approximations under certain distinct notions of weak convergence. This establishes a sharp topological distinction between the two notions of unitary dilations. Our results are applicable to commutative systems as well as non-commutative systems satisfying the canonical commutation relations (CCR) in the Weyl form.

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arXiv2023

Dilations of commuting $C_{0}$-semigroups with bounded generators and the von Neumann polynomial inequality

Consider $d$ commuting $C_{0}$-semigroups (or equivalently: $d$-parameter $C_{0}$-semigroups) over a Hilbert space for $d \in \mathbb{N}$. In the literature (\textit{cf.} [29, 26, 27, 23, 18, 25]), conditions are provided to classify the existence of unitary and regular unitary dilations. Some of these conditions require inspecting values of the semigroups, some provide only sufficient conditions, and others involve verifying sophisticated properties of the generators. By focussing on semigroups with bounded generators, we establish a simple and natural condition on the generators, \textit{viz.} \emph{complete dissipativity}, which naturally extends the basic notion of the dissipativity of the generators. Using examples of non-doubly commuting semigroups, this property can be shown to be strictly stronger than dissipativity. As the first main result, we demonstrate that complete dissipativity completely characterises the existence of regular unitary dilations, and extend this to the case of arbitrarily many commuting $C_{0}$-semigroups. We furthermore show that all multi-parameter $C_{0}$-semigroups (with bounded generators) admit a weaker notion of regular unitary dilations, and provide simple sufficient norm criteria for complete dissipativity. The paper concludes with an application to the von Neumann polynomial inequality problem, which we formulate for the semigroup setting and solve negatively for all $d \geq 2$.

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arXiv2022

On the complete metrisability of spaces of contractive semigroups

The space of unitary $C_{0}$-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of $\mathbb{R}_{+}$, is known to admit various interesting residual subspaces. Before treating the contractive case, the problem of the complete metrisability of this space was raised in [Eisner, 2010]. Utilising Borel complexity computations and automatic continuity results for semigroups, we obtain a general result, which in particular implies that the one-/multiparameter contractive $C_{0}$-semigroups constitute Polish spaces and thus positively addresses the open problem.

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arXiv2021

The space of contractive $C_{0}$-semigroups is a Baire space

Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive $C_{0}$-semigroups under the topology of uniform weak operator convergence on compact subsets of $\mathbb{R}_{+}$. Eisner and Serény raised in 2009 the open problem: Does this space constitute a Baire space? Observing that the subspace of unitary semigroups is completely metrisable and appealing to known density results, we solve this problem positively by showing that certain topological properties can in general be transferred from dense subspaces to larger spaces. The transfer result in turn relies upon classification of topological properties via infinite games. Our approach is sufficiently general and can be applied to other contexts, e.g. the space of contractions under the pw-topology.

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arXiv2021

On the strong continuity of generalised semigroups

It is well known that weakly continuous semigroups defined over $\mathbb{R}_{+}$ are automatically strongly continuous. We extend this result to more generally defined semigroups, including multiparameter semigroups.

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