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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]

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]

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]

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]

The existence of a non-trivial singular positive bilinear form Simon (J. Funct. Analysis 28, 377-385 ()) yields that on an infinite-dimensional complex Hilbert space ${\mathcal {H}}$ the set of bilinear forms ${\mathcal {F}}(\mathcal {H})$ is richer than the set of linear operators ${\mathcal {V}}(\mathcal {H})$. We show that there exists an structure preserving embedding of partially ordered groups from the abelian po-group ${\mathcal {S}}_{D}(\mathcal {H})$ of symmetric bilinear forms with a fixed domain D on a Hilbert space ${\mathcal {H}}$ into the po-group of linear symmetric operators on a dense linear subspace of an infinite dimensional complex Hilbert space l( M). Moreover, if we restrict ourselves to the positive parts of the above mentioned po-groups, we can embed positive bilinear forms into corresponding positive linear operators. [ABSTRACT FROM AUTHOR]

Let L be a double triangle lattice of projections in a finite von Neumann algebra acting on a separable and complex Hilbert space K . We show that every derivation from the reflexive algebra determined by L into B ( K ) is quasi-spatial and automatically continuous. We also obtain that every local derivation on the reflexive algebra is a derivation. [ABSTRACT FROM AUTHOR]

The spectral properties and the asymptotic behaviour of the discrete spectrum for a special class of infinite tridiagonal matrices are given. We derive the asymptotic formulae for eigenvalues of unbounded complex Jacobi matrices acting in l² (∕). [ABSTRACT FROM AUTHOR]

We study positive bilinear forms on a Hilbert space which are not necessarily bounded nor induced by some positive operator. We show when different families of bilinear forms can be described as a generalized effect algebra. In addition, we present families which are or are not monotone downwards (Dedekind upwards) σ-complete generalized effect algebras. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present the solution to Stechkin’s problem for differential operators and functionals of first and second orders on the class of functions that are defined on a finite interval and have bounded third derivative. Two related problems are discussed: (1) finding sharp constants in the Landau–Kolmogorov inequalities, and (2) optimal recovery of an operator with the help of a set of linear operators (or arbitrary single-valued mappings) on elements of some set that are given with an error. [Copyright &y& Elsevier]

Abstract: Given two linear relations and we characterize the existence of a linear relation (operator) such that , respectively . These factorizations extend and improve well-known results by Douglas and Sebestyén. [Copyright &y& Elsevier]

Abstract: Hilbert -modules are the analogues of Hilbert spaces where a -algebra plays the role of the scalar field. With the advent of Kasparovʼs celebrated KK-theory they became a standard tool in the theory of operator algebras. While the elementary properties of Hilbert -modules can be derived basically in parallel to Hilbert space theory the lack of an analogue of the Projection Theorem soon leads to serious obstructions and difficulties. In particular the theory of unbounded operators is notoriously more complicated due to the additional axiom of regularity which is not easy to check. In this paper we present a new criterion for regularity in terms of the Hilbert space localizations of an unbounded operator. We discuss several examples which show that the criterion can easily be checked and that it leads to nontrivial regularity results. [Copyright &y& Elsevier]

Abstract: The explicit representation for the Moore–Penrose inverse of an operator T between Hilbert spaces has been given by C.W. Groetsch (1975) . We obtain his formula for the Moore–Penrose inverse of an unbounded operator between Hilbert -modules. Ill-posed problems with unbounded operator between Hilbert -modules are also discussed. [Copyright &y& Elsevier]

The classical Hamiltonian 1/2(-d2/dx2+x2) of the very classical quantum harmonic oscillator, which is regarded as a germ of the most of what comes about in quantum mechanics, can be sublimed to an abstract operator in a separable Hilbert space. Having this done one may ask for a condition which would allow it to be identified among operators of a suitable class. This class is that corresponding to three diagonal matrices and the property which makes the action successful is a kind of diagonal invariance (up to change of basis) within the class in question. [Copyright &y& Elsevier]

In this paper the quantum GL(2, C) group constructed in [4] is investigated. The deformation parameter q = eSHAPE="CASE" ALIGN="C" STYLE="S">2πeN, where N is an even natural number. It is shown that this group can be obtained by the double group construction applied to the quantum ‘az + b’ group. The latter is the quantum deformation of the group of affine transformations of complex plane introduced in [9]. [Copyright &y& Elsevier]

We study operators with located graph in Bishop-style constructive mathematics. It is shown that a bounded operator has an adjoint if and only if its graph is located. Locatedness of the graph is a necessary and sufficient condition for an unbounded normal operator to have a spectral decomposition. These results suggest that located operators are the right generalization of bounded operators with an adjoint. [ABSTRACT FROM AUTHOR]

We classify spectrum preserving additive bijections which map the set of linear (possibly unbounded) maps on a complex Banach space X to the set of linear (possibly unbounded) maps on a complex Banach space Y. [ABSTRACT FROM AUTHOR]

The article presents the theorems of the Haar system.