Operator Integrals and Spectral Representations: The Bounded Case

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

2 - Traditional Quantum Mechanics

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.

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.

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

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.

On normal operator logarithms

The spectral representation of a symmetric nonmaximal operator transforms it into a multiplication operator by a complex variable acting in a linear space of pairs of vectorvalued functions holomorphic on the upper and lower half- planes. Each generalized spectral function of the same symmetric operator leads to its realization in form of a multiplication operator by a real variable in some Hilbert space. We study the connection between such realizations of the given operator and its spectral representation.

The rigged hilbert spaces approach in singular perturbation theory

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.

At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D + V with D having compact resolvent belonging to a general semifinite von Neumann algebra $${\mathcal{N}}$$ and the perturbation $$V \in {\mathcal{N}}$$ . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of semifinite spectral triples where D has compact resolvent belonging to $${\mathcal{N}}$$ and V is any bounded self-adjoint operator in $${\mathcal{N}}$$ . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8–10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].