Related papers: Dilation theory: a guided tour
In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and…
The design of fixed point algorithms is at the heart of monotone operator theory, convex analysis, and of many modern optimization problems arising in machine learning and control. This tutorial reviews recent advances in understanding the…
To generalise evolution families we consider systems of contractions $\{\varphi(u, v)\}_{(u, v) \in E}$ defined on the edges of a graph $\mathcal{G} = (\Omega, E)$. In this setup the Markov property, or \emph{divisibility}, can be modelled…
Researchers have identified complex matrices $A$ such that a bounded linear operator $B$ acting on a Hilbert space will admit a dilation of the form $A \otimes I$ whenever the numerical range inclusion relation $W(B) \subseteq W(A)$ holds.…
The dilatation generator measures the scaling dimensions of local operators in a conformal field theory. In this thesis we consider the example of maximally supersymmetric gauge theory in four dimensions and develop and extend techniques to…
Based on explicit computations, various concepts of discrete time scattering theory are reviewed, discussed, and illustrated. The dynamics are taking place on a discrete half-space. All operators are represented graphically. The expressions…
We show that (for the weak operator topology) the set of unitary operators on a separable infinite-dimensional Hilbert space is residual in the set of all contractions. The analogous result holds for isometries and the strong operator…
Using works of T.~Ando and L.~Gurvits, the well-known theorem of P.R.~Halmos concerning the existence of unitary dilations for contractive linear operators acting on Hilbert spaces recast as a result for $d$-tuples of contractive Hilbert…
Density operators are one of the key ingredients of quantum theory. They can be constructed in two ways: via a convex sum of `doubled kets' (i.e. mixing), and by tracing out part of a `doubled' two-system ket (i.e. dilation). Both…
We introduce the notion of $Q$-commuting operators which is a generalization of commuting operators. We prove a generalized version of commutant lifting theorem and Ando's dilation theorem in the context of $Q$-commuting operators.
We prove the decomposition of arbitrary diagonal operators into tensor and matrix products of smaller matrices, focusing on the analytic structure of the resulting formulas and their inherent symmetries. Diagrammatic representations are…
This paper examines actions of right LCM semigroups by endomorphisms of C*-algebras that encode an additional structure of the right LCM semigroup. We define contractive covariant representations for these semigroup dynamical systems and…
We study the (generalized) semi-Weyl commutation relations $$ U_gAU_g^*=g(A) \quad \text{ on }\quad \Dom(A), $$ where $A$ is a densely defined operator and $G\ni g\mapsto U_g$ is a unitary representation of the subgroup $G$ of the affine…
In previous work we established a multilinear duality and factorisation theory for norm inequalities for pointwise weighted geometric means of positive linear operators defined on normed lattices. In this paper we extend the reach of the…
Some connections between operator theory and wavelet analysis: Since the mid eighties, it has become clear that key tools in wavelet analysis rely crucially on operator theory. While isolated variations of wavelets, and wavelet…
Koopman operator theory, a powerful framework for discovering the underlying dynamics of nonlinear dynamical systems, was recently shown to be intimately connected with neural network training. In this work, we take the first steps in…
Density operators are one of the key ingredients of quantum theory. They can be constructed in two ways: via a convex sum of 'doubled kets' (i.e. mixing), and by tracing out part of a 'doubled' two-system ket (i.e. dilation). Both…
In dispersive media, dissipation appears in the Schr\"odinger representation of classical Maxwell equations as a sparse diagonal operator occupying an $r$-dimensional subspace. A first order Suzuki-Trotter approximation for the evolution…
We present a generalized version of the discretization-invariant neural operator and prove that the network is a universal approximation in the operator sense. Moreover, by incorporating additional terms in the architecture, we establish a…
Ando's theorem states that any pair of commuting contractions on a Hilbert space can be dilated to a pair of commuting unitaries. Parrott presented an example showing that an analogous result does not hold for a triple of pairwise commuting…