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Gaussian Processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP T u that is the image of another GP u under a linear transformation T acting on the sample paths of u are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when T is an unbounded operator such as a differential operator, which is common in many modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator T acting on the sample paths of a square-integrable (not necessarily Gaussian) stochastic process. Our proof technique relies upon Hille's theorem for the Bochner integral of a Banach-valued random variable. [ABSTRACT FROM AUTHOR]

Ando studied Lebesgue decomposition for (bounded) positive operators, where a notion known as the maximal absolutely continuous part plays a crucial role. Based on technique involving unbounded operators we study certain expressions for the maximal absolutely continuous part, Radon–Nikodym type results, behavior of the maximal absolutely continuous part under the tensor product operation and characterization of mutual absolute continuity and so on. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to develop the concepts and ideas used in the theory of weighted networks on infinite graphs. Our basic ingredients are a standard Borel space (V , B) and symmetric σ -finite measure ρ on (V × V , B × B) supported by a symmetric Borel subset E ⊂ V × V . This setting is analogous to that of weighted networks and are used in various areas of mathematics such as Dirichlet forms, graphons, Borel equivalence relations, Guassian processes, determinantal point processes, joinings, etc. We define and study the properties of a measurable analogue of the graph Laplacian, finite energy Hilbert space, dissipation Hilbert space, and reversible Markov processes associated directly with the measurable framework. In the settings of networks on infinite graphs and standard Borel space, our present focus will be on the following two dichotomies: (i) Discrete vs continuous (or rather the measurable category); and (ii) deterministic vs stochastic. Both (i) and (ii) will be made precise inside our paper; in fact, this part constitutes a separate issue of independent interest. A main question we shall address in connection with both (i) and (ii) is that of limits. For example, what are the useful notions, and results, which yield measurable limits; more precisely, Borel structures arising as “limits” of systems of graph networks? We shall study it both in the abstract, and via concrete examples, especially models built on Bratteli diagrams. The question of limits is of basic interest; both as part of the study of dynamical systems on path- space; and also with regards to numerical considerations; and design of simulations. The particular contexts for the study of this question entails a careful design of appropriate Hilbert spaces, as well as operators acting between them. Our main results include: an explicit spectral theory for measure theoretic graph-Laplace operators; new properties of Green’s functions and corresponding reproducing kernel Hilbert spaces; structure theorems of the set of harmonic functions in L 2 -spaces and finite energy Hilbert spaces; the theory of reproducing kernel Hilbert spaces related to Laplace operators; a rigorous analysis of the Laplacian on Borel equivalence relations; a new decomposition theory; irreducibility criteria; dynamical systems governed by endomorphisms and measurable fields; and path-space measures and induced dissipation Hilbert spaces. We discuss also several applications of our results to other fields such as machine learning problems, ergodic theory, and reproducing kernel Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the stabilization of a linearized viscous Saint-Venant system by constrained Dirichlet boundary control in infinite time horizon. We proved the well posedness of the considered stabilization problem. Also, using an augmented state method, we were able to determine the optimal control (the constrained control) as a feedback control law. Moreover, thanks to the feedback control law, we proved the exponential stability of the solution to the linearized viscous Saint-Venant system, (defined by an unbounded operator). Some numerical experiments are given to illustrate the efficiency of the constrained Dirichlet boundary control. [ABSTRACT FROM AUTHOR]

Under suitable assumptions the eigenvalues for an unbounded discrete operator \(A\) in \(l_2\), given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let \[\Lambda (A)=\{\lambda \in {\rm Lim}_{n\to \infty} \lambda _n : \lambda _n \text{ is an eigenvalue of } A_n \text{ for } n \geq 1 \},\] where \({\rm Lim}_{n\to \infty} \lambda_n\) is the set of all limit points of the sequence \((\lambda_n)\) and \(A_n\) is a finite dimensional orthogonal truncation of \(A\). The aim of this article is to provide the conditions that are sufficient for the relations \(\sigma(A) \subset \Lambda(A)\) or \(\Lambda (A) \subset \sigma (A)\) to be satisfied for the band operator \(A\).

In this paper, we study the questions of the existence of global weak solutions and local strong solutions of paired stochastic functional differential equations in a Hilbert space, one of which is an equation with an unbounded operator, and the other is an ordinary differential equation. We proved the existence and uniqueness theorems in the case of coefficients with polynomial growth.

Let A be an unbounded operator, particularly belonging to the class of densely defined closed operators on Hilbert spaces. The operator A ∗ A is called the Gram operator of A. In this manuscript, the cone nonnegativity of the Moore–Penrose inverse of unbounded Gram operators is characterized. This characterization extends the results of Sivakumar (Positivity 13:277–286, 2009) from bounded linear operators to unbounded linear operators. [ABSTRACT FROM AUTHOR]

We present an example of a densely defined, linear operator on the l 1 space with the property that each basis vector of the standard Schauder basis of l 1 does not belong to its domain. Our example is based on the construction of a Markov chain with all states instantaneous given by D. Blackwell in 1958. In addition, it turns out that the closure of this operator is the generator of a strongly continuous semigroup of Markov operators associated with Blackwell's chain. [ABSTRACT FROM AUTHOR]

Under suitable assumptions the eigenvalues for an unbounded discrete operator A in l2, given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let Λ(A) = {λ ∊ Limnλ=λn: λn is an eigenvalue of An for n = 1}, where Limnλ=λn is the set of all limit points of the sequence (λn) and An is a finite dimensional orthogonal truncation of A. The aim of this article is to provide the conditions that are sufficient for the relations s(A) σ(A) ⊂ Λ(A) or Λ(A) ⊂ σ(A) to be satisfied for the band operator A. [ABSTRACT FROM AUTHOR]

The Krylov subspace method has been investigated and refined for approximating the behaviors of finite or infinite dimensional linear operators. It has been used for approximating eigenvalues, solutions of linear equations, and operator functions acting on vectors. Recently, for time-series data analysis, much attention is being paid to the Krylov subspace method as a viable method for estimating the multiplications of a vector by an unknown linear operator referred to as a transfer operator. In this paper, we investigate a convergence analysis for Krylov subspace methods for estimating operator-vector multiplications. [ABSTRACT FROM AUTHOR]

In this work we investigate the spectral statistics of random Schrödinger operators Hω=−Δ+∑n∈Zd(1+|n|α)qn(ω)|δn⟩⟨δn|,α>0acting on ℓ2(Zd) where {qn}n∈Zd are i.i.d random variables distributed uniformly on [0,1]. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new definition of the pseudo-spectrum for operator pencils, which is associated with two bounded operators T and S defined in a Hilbert space. Unlike other definitions available in the literature, we prove, under specific conditions on T and S , that the pseudo-spectrum for operator pencils is equal to an 𝜖 -neighborhood of the generalized spectrum. Moreover, we demonstrate how to use this concept to redefine the pseudo-spectrum of an unbounded operator. We illustrate its usefulness through a numerical example dealing with the Schrödinger operator. [ABSTRACT FROM AUTHOR]

Let $G$ be an infinite graph on countably many vertices and let $\unicode[STIX]{x1D6EC}$ be a closed, infinite set of real numbers. We establish the existence of an unbounded self-adjoint operator whose graph is $G$ and whose spectrum is $\unicode[STIX]{x1D6EC}$. [ABSTRACT FROM AUTHOR]

Given a closed linear relation T between two Hilbert spaces H and K , the corresponding first and second coordinate projections P T and Q T are both linear contractions from T to H , and to K , respectively. In this paper we investigate the features of these graph contractions. We show among other things that P T P T ∗ = (I + T ∗ T) - 1 , and that Q T Q T ∗ = I - (I + T T ∗) - 1 . The ranges ran P T ∗ and ran Q T ∗ are proved to be closely related to the so called ‘regular part’ of T. The connection of the graph projections to Stone’s decomposition of a closed linear relation is also discussed. [ABSTRACT FROM AUTHOR]

The product of a matrix function and a vector is used to solve evolution equations numerically. Hashimoto and Nodera (ANZIAM J 58:C149–C161, 2016) proposed the Shift-invert Rational Krylov method for computing these products. However, since matrices produced by evolution equations behave like unbounded operators in infinite-dimensional spaces, an analysis with the unbounded operator is essential. In this paper, the Shift-invert Rational Krylov method is extended to be applied to unbounded operators. [ABSTRACT FROM AUTHOR]