The spectral theorem for unbounded self-adjoint operators

The unbounded Self-adjoint operators that strongly commute on a common dense subset of their domain commute pointwise. When the operators commute pointwise on the same dense subset, there is to guarantee that they will commute strongly. By imposing some conditions, we on the operators as well as the underlying space, we get pointwise commuting unbounded operators that commute strongly. This article shows that by suitably selecting two unbounded positive Self-adjoint operators with compact inverses we get a set of pointwise commuting self-adjoint operators that commute on common core. then prove that it strongly commutes on the same subspace. Keywords: Unbounded operators, Self-adjoint operators, Commutative operators DOI: 10.7176/MTM/10-8-03 Publication date: December 31 st 2020

In order to make this monograph self-contained, we summarize in this chapter some basic definitions and results for unbounded linear operators in a Hilbert space. In Section 1.1, we recall the definitions of C*-algebras and von Neumann algebras. In Section 1.2, we define and investigate the notion of closedness, the closure and the adjoint of an unbounded linear operator in a Hilbert space. Section 1.3 is devoted to the Cayley transform approach to the self-adjointness of a symmetric operator. Section 1.4 deals with the self-adjoint extendability of a symmetric operator with help of the deficiency spaces. In Section 1.5, we extend to unbounded self-adjoint operators the spectral theorem and the functional calculus theorem for bounded self-adjoint operators. Section 1.6 is devoted to Stone’s theorem. In Section 1.7, the polar decomposition of bounded linear operators is extended to closed linear operators. Section 1.8 introduces Nelson’s analytic vector theorem for the selfadjointness of closed symmetric operators. Section 1.9, finally, deals with the form representation theorem and the Friedrichs self-adjoint extension theorem.

Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert space

In the traditional Hilbert space quantum mechanics, (bounded or unbounded) selfadjoint operators are taken as physical observables. In the present work a more comprehensive approach is used: observables are represented as normalised positive operator measures. We develop the spectral representation theory of generally unbounded selfadjoint operators in this more general setting of positive operator measures, paving the way for important physical applications in the sequel.

On normal operator logarithms

This survey is based on the works of Kiev mathematicians and others in the structure theory of pairs of bounded and unbounded self-adjoint operators that satisfy an algebraic relation.

The function , where is Lerch's transcendent, satisfies the following two-dimensional formally self- adjoint second-order hyperbolic differential equation: where . The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space , where . We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of . We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.

We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L2-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).

New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form \({-Au \in \partial\varphi(u)}\) as well ass dissipative initial value evolutions of the form \({-\dot{u}(t) - Au(t) + \omega{u}(t) \in \partial\varphi(l,u(l))}\) where \({\varphi}\) is a convex potential on an infinite dimensional space, A is a linear operator and \({\omega}\) is any scalar. The framework developed in the above mentioned paper reformulates these problems as \({0\in \bar{\partial}L(u)}\) and \({\dot{u}(t) \in \bar{\partial}L(t,u(t))}\) respectively, where \({\bar{\partial}L}\) is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term.

We give a functional analytical definition of the Maslov index for continuous curves in the Fredholm-Lagrangian Grassmannian. Our definition does not require assumptions either at the endpoints or at the crossings of the curve with the Maslov cycle. We demonstrate an application of our definition by developing the symplectic geometry of self-adjoint extensions of unbounded symmetric operators. We discuss continuous variations of the form $A_D+C_t$, where $A_D$ is a fixed self-adjoint unbounded Fredholm operator and $\{C_t\}$ a family of bounded self-adjoint operators. We extend the definition of the spectral flow to such families of unbounded operators in a purely functional analytical way. We then prove that the spectral flow is equal to the Maslov index of the corresponding family of abstract Cauchy data spaces.

When the singular finite rank perturbations of an unbounded selfadjoint operator Ao in a Hilbert space c, formally defined by A(α) = A 0 + GαG*, are lifted to an exit Pontryagin space ℌ by means of an operator model, they become ordinary range perturbations of a self-adjoint operator H ∞ in ℌ ⊃ ℌ0:H T = H ∞-ΩT -1Ω*. Here G is a mapping from ℂd into some scale space ℌ (A 0), ℕ, of generalized elements associated with A 0, while Ω is a mapping from into ℂd the extended space ℌ, where H T , is defined. The connection between these two perturbation formulas is studied.KeywordsSingular finite rank perturbationextension theoryKrein’s formulaboundary tripletWeyl functiongeneralized Nevanlinna functionoperator model

A strongly continuous semigroup \( \mathbb{T} \) is called a contraction semigroup if % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaacY % hatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-nj8 % ujab-bW90naaBaaaleaacGaxOniDaaqabaGccaGG8bGaaiiFaGqaai % aa+bcacqWFKjcHcaGFXaGae8ha3hdaaa!4E6D! \[ ||\mathbb{T}_t || \leqslant 1 \] for all t ≥0. This chapter is a continuation of the previous one: we present basic facts about unbounded operators and strongly continuous semigroups on Hilbert spaces, but now the emphasis is on contraction semigroups and their generators, which are called m-dissipative operators. We also discuss other important classes of operators (self-adjoint, positive and skew-adjoint operators) that arise as generators or as ingredients of generators of contraction semigroups.We also investigate some classes of self-adjoint differential operators: Sturm-Liouville operators and the Dirichlet Laplacian on various domains in ℝn.

The monogenic functional calculus is a means of constructing functions of a finite system of bounded or unbounded operators. For bounded noncommuting operators, a polynomial function produces a polynomial of the operators in which all possible operator orderings are equally weighted. For example, for two bounded selfadjoint operators A 1, A 2, the operator \(\frac{1} {2}(A_{1}A_{2} + A_{2}A_{1})\) is associated with polynomial \(z\mapsto z_{1}z_{2}\) in two variables by the monogenic functional calculus. The same formula applies just when the spectrum \(\sigma (\xi _{1}A_{1} +\xi _{2}A_{2})\) of a finite linear combination \(\xi _{1}A_{1} +\xi _{2}A_{2}\) of A 1 and A 2 is a subset of the real numbers for any \(\xi _{1},\xi _{2} \in \mathbb{R}\). The article begins with a discussion of Clifford algebras and Clifford analysis and points out the connection with Weyl’s functional calculus for a finite system of selfadjoint operators in Hilbert space. The construction of the Cauchy kernel in the monogenic functional calculus is achieved with the plane wave decomposition of the Cauchy kernel in Clifford analysis. The formulation applies to any finite system \((A_{1},\ldots ,A_{n})\) of closed unbounded operators such that the spectrum \(\sigma \left (\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\right )\) of the closed operator \(\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\) is contained in a two-sided sector in \(\mathbb{C}\) for almost all \(\xi \in \mathbb{R}^{n}\). The connection with harmonic analysis and irregular boundary value problems is emphasized. The monogenic functional calculus may be applied to the finite commuting system \(D_{\Sigma } = i(D_{1},\ldots ,D_{n})\) of differentiating operators on a Lipschitz surface \(\Sigma \) so that an H ∞-functional calculus \(f\mapsto f(D_{\Sigma })\) is obtained for functions f uniformly bounded and left monogenic in a sector containing almost all tangent planes to \(\Sigma \).KeywordsBounded Linear OperatorClifford AlgebraFunctional CalculusSelfadjoint OperatorMonogenic FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The instability index of a linear operator with point spectrum is defined to be the total multiplicity of its eigenvalues with positive real parts. Under certain conditions the computation of the instability index of an unbounded nonsymmetric operator acting in a Hilbert space can be reduced to an analogous problem for a certain selfadjoint operator. It is shown that if the original operator is a differential operator acting in a space of vector-valued functions with a single variable, then the selfadjoint operator corresponding to it can be found in the form of an integro-differential operator which can be realized constructively by solving a special elliptic boundary value problem. Analogues of known theorems of Morse on the connection between the instability index and the number of conjugate points are established for this integro-differential operator.Bibliography: 26 titles.

Results of Schoenberg and others on limits of periodic splines as their order,m, goes to infinity are extended to sequences of D m -splines determined by the powers of an unbounded non-negative self-adjoint operator D on a Hilbert space,H, and an evaluation mapL fromH toR n . All such limits lie in the lowest frequencyn-dimensional invariant subspace for D,T* n . When each term in the sequence is the D m -spline whose image underL matches a fixed vector,y, (anL-interpolant), then the limit is theL-interpolant toy fromT* n . When the terms aresmoothing splines derived fromy then the limit exists when the smoothing parameter goes to 0 ast −m . Ift is not an eigenvalue,α i, of D, the limit is theL-least squares best fit toy fromT* l ,l=card {j: α j<t}.