On Pairs of 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 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.

In this chapter we shall prove the spectral theorem for self-adjoint (unbounded) operators A in Hilbert spaces. It says that for each A there exists a unique spectral measure E on a(A), so that A = ∫ t dE. In the sequel H is a complex Hilbert space.

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.

The rigged hilbert spaces approach in singular perturbation theory

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

On normal operator logarithms

Parabolic Itô equations with monotone nonlinearities

In this chapter, the spectral flow of paths of self-adjoint unbounded Fredholm operators is introduced like in [31] as the spectral flow of unitary operators obtained from the unbounded operators via the Cayley transform. As such, the spectral flow of unbounded self-adjoint Fredholm operators inherits many natural properties which are also listed in Section 7.1. In the next Section 7.2, the paths are in the subset of Fredholm operators with compact resolvent and satisfy certain summability conditions which then allow connecting them to so-called η-invariants. Section 7.3 is an application of spectral flow of unbounded self-adjoint Fredholm operators to certain paths arising from Hamiltonian systems. The chapter is concluded by Section 7.4 which shows that for certain paths of unbounded self-adjoint Fredholm operators the spectral flow is still given as the index of an (unbounded) Fredholm operator. In this section the notion of spectral flow is generalized to gap continuous paths of possibly unbounded Fredholm operators. As in [31], this will be achieved by taking the Cayley transform of the path to use the spectral flow of the resulting path of unitaries. More precisely, let t ∈ [0, 1] 󳨃 → H t ∈ 𝔽 sa (H) be continuous with respect to the gap metric. Then by Theorem 6.3.14, the path t ∈ [0, 1] 󳨃 → C(H t ) ∈ 𝔽𝕌 0 (H) is norm-continuous. Therefore its spectral flow is well defined in the sense of Section 4.5, and one can define the spectral flow of t ∈ [0, 1] 󳨃 → H t in a similar manner as in Proposition 4.6.2. Definition 7.1.1. Let t ∈ [0, 1] 󳨃 → H t ∈ 𝔽 sa (H) be continuous with respect to the gap metric. Then the spectral flow of this path is defined by The first result shows that for a path of bounded self-adjoint Fredholm operators this definition coincides with Definition 4.1.2. This implies that all examples of paths with nontrivial spectral flow from Chapter 4 also provide examples of nontrivial spectral flow in the sense of Definition 7.1.1. A path of truly unbounded operators with nonvanishing spectral flow will be given in Example 7.1.4 below. Proposition 7.1.2. Let t ∈ [0, 1] 󳨃 → H t ∈ 𝔽𝔹 sa (H) be a norm-continuous path of selfadjoint Fredholm operators. Its spectral flow defined by (4.4) fulfils Sf(t ∈ [0, 1] 󳨃 → H t ) = Sf(t ∈ [0, 1] 󳨃 → C(H t )).

The mathematical structure of Quantum Mechanics is usually introduced as a calculus of non-commuting self-adjoint (unbounded) operators, the “observables,” on a Hilbert space of “states” (cf. [15]). There is no doubt that Quantum Mechanics is consistent and describes correctly many experiments, but we are supposed to renounce completely the visualization of quantum phenomena in space-time.

We explore connections between Krein's spectral shift function ζ(λ,H 0, H) associated with the pair of self-adjoint operators (H 0, H),H=H 0+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K *(H 0−λ−i0)−1 K) associated with the operator-valued Herglotz functionJ+K *(H 0−z)−1 K, Im(z)>0 inH, whereV=KJK * andJ=sgn(V). Our principal results include a new representation for ζ(λ,H 0,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E J+A(λ)+tB(λ)(−∞, 0)),E J((−∞, 0))), ℝ, whereA(λ)=Re(K *(H 0−λ−i0−1 K),B(λ)=Im(K *(H 0−λ-i0)−1 K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H 0, H) coincides with the trindex associated with the pair (Ξ(J+K *(H 0−λ−i0)K), Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ operators and the Fredholm determinant of the abstract scattering matrix. We also provide a generalization of the classical Birman—Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.

Let A 1 {A_1} and A 2 {A_2} be (unbounded) selfadjoint operators on a Hilbert space H \mathcal {H} which commute on a dense linear subspace of H \mathcal {H} . To conclude that A 1 {A_1} and A 2 {A_2} strongly commute, additional assumptions are necessary. Two propositions which contain such additional conditions are proved in §1. In §2 we define different commutants of unbounded operator algebras (form commutant, weak unbounded commutant, strong unbounded commutant) and we discuss the relations between them and their bounded parts. In §3 we construct a selfadjoint ∗ {*} -representation of the polynomial algebra in two variables for which the form commutant is different from the weak unbounded commutant.

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.

A new characterization of anticommutativity of (unbounded) self-adjoint operators is presented in connection with Clifford algebra. Some consequences of the characterization and applications are discussed.