Elsevier

Indagationes Mathematicae

Volume 31, Issue 6, November 2020, Pages 1066-1098
Indagationes Mathematicae

The functional calculus approach to the spectral theorem

https://doi.org/10.1016/j.indag.2020.09.006Get rights and content

Abstract

A consistent functional calculus approach to the spectral theorem for strongly commuting normal operators on Hilbert spaces is presented. In contrast to the common approaches using projection-valued measures or multiplication operators, here the functional calculus is not treated as a subordinate but as the central concept.

Based on five simple axioms for a “measurable functional calculus”, the theory of such calculi is developed in detail, including spectral theory, uniqueness results and construction principles. Finally, the functional calculus form of the spectral theorem is stated and proved, with some proof variants being discussed.

Introduction

The spectral theorem (for normal or self-adjoint operators on a Hilbert space) is certainly one of the most important results of 20th century mathematics. It comes in different forms, two of which are the most widely used: the multiplication operator (MO) form and the one using projection-valued measures (PVMs). Associated with this variety is a discussion about “What does the spectral theorem say?” (Halmos [9]), where the pro’s and con’s of the different approaches are compared.

In this article, we would like to add a slightly different stance to this debate by advocating a consistent functional calculus approach to the spectral theorem. Since in any exposition of the spectral theorem one also will find results about functional calculus, some words of explanation are in order.

Let us start with the observation that whereas multiplication operators and projection-valued measures are well-defined mathematical objects, the concept of a functional calculus as used in the literature on the spectral theorem is usually defined only implicitly. One speaks of the functional calculus of a normal operator (that is, the mapping whose properties are listed in some particular theorem) rather than of a functional calculus as an abstract concept. As a result, such a concept remains a heuristic one, and the concrete calculus associated with the spectral theorem acquires and retains a subordinate status, being merely a derivate of the “main” formulations by multiplication operators or projection-valued measures. (At this point, we should emphasize that we have the full functional calculus in mind, comprising all measurable functions and not just bounded ones.) In practice, this expositional dependence implies that when using the functional calculus (and one wants to use it all the time) one always has to resort to one of its constructions.

In this respect, the multiplication operator version appears to have a slight advantage, since deriving functional calculus properties from facts about multiplication operators is comparatively simple. (This is probably the reason why eminent voices such as Halmos [9] and Reed–Simon [11, VII] prefer multiplication operators.) However, this advantage is only virtual, since the MO-version has two major drawbacks. Firstly, a MO-representation is not canonical and hence leads to the problem whether functional calculus constructions (square root, semigroup, logarithm etc.) are independent of the chosen MO-representation. Secondly (and somehow related to the first), the MO-version is hardly useful for anything other than for deriving functional calculus properties. (For example, it cannot be used in constructions, like that of a joint (product) functional calculus.)

In contrast, an associated PVM is canonical and PVMs are very good for constructions, but the description of the functional calculus, in particular for unbounded functions, is cumbersome. And since one needs the functional calculus eventually, every construction based on PVMs has, in order to be useful, to be backed up by results about the functional calculus associated with the new PVM.

With the present article we propose a “third way” of treating the spectral theorem, avoiding the drawbacks of either one of the other approaches. Instead of treating the functional calculus as a logically subordinate concept, we put it center stage and make it our main protagonist. Based on an axiomatic definition of a “measurable functional calculus”, we shall present a thorough development of the associated theory entailing, in particular:

  • general properties, constructions such as a pull-back and a push-forward calculus (Section 2);

  • projection-valued measures, the role of null sets, the concepts of concentration and support (Section 3);

  • spectral theory (Section 4);

  • uniqueness (and commutativity) properties (Section 5);

  • construction principles (Section 6).

Finally, in Section 7, we state and prove “our” version of the spectral theorem, which takes the following simple form (see Theorem 7.6).

Spectral Theorem: Let A1,,Ad be pairwise strongly commuting normal operators on a Hilbert space H. Then there is a unique Borel calculus (Φ,H) on d such that Φ(zj)=Aj for all j=1,,d.

Here, we use a notion of strong commutativity which is formally different from that used by Schmüdgen in [15], but is more suitable for our approach. In a final section we then show that both notions are equivalent.

In order to advertise our approach, let us point out some of its “special features”. Firstly, the axioms for a measurable calculus are few and simple, and hence easy to verify. Restricted to bounded functions they are just what one expects, but the main point is that these axioms work for all measurable functions.

Secondly, the aforementioned axioms are complete in the sense that each functional calculus property which can be derived with the help of a MO-representation can also be derived directly, and practically with the same effort, from the axioms.1 This is of course not a rigorous (meta)mathematical theorem, but a heuristic statement stipulated by the exhaustive exposition we give. In particular, we demonstrate that many properties of multiplication operators (for example its spectral properties) are consequences of the general theory, simply because the multiplication operator calculus satisfies the axioms of a measurable calculus (Theorem 2.9 and Corollary 4.6).

Thirdly, the abstract functional calculus approach leads to a simple method for extending a calculus from bounded to unbounded measurable functions (Theorem 6.1). This method, known as “algebraic extension” or “extension by (multiplicative) regularization”, is well-established in general functional calculus theory for unbounded operators such as sectorial operators or semigroup generators. (See [6], [8] and the references therein.) It has the enormous advantage that it is elegant and perspicuous, and that it avoids cumbersome arguments with domains of operators, which are omnipresent in the PVM-approach (cf. Rudin’s exposition in [14]).

We shall work generically over the scalar field K{R,}. The letters H,K usually denote Hilbert spaces, the space of bounded linear operators from H to K is denoted by L(H;K), and L(H) if H=K.

A (closed) linear subspace of HK is called a (closed) linear relation. Linear relations are called multi-valued operators in [6, Appendix A], and we use freely the definitions and results from that reference. In particular, we say that a bounded operator TL(H) commutes with a linear relation A if TAAT, which is equivalent to (x,y)A(Tx,Ty)A.A linear relation is called an operator if it is functional, i.e., it satisfies (x,y),(x,z)Ay=z.The set of all closed linear operators is C(H;K){AHKA is closed and an operator},with C(H)C(H;H).

For the spectral theory of linear relations, we refer to [6, Appendix A]. For a closed linear relation A in H we denote by σ(A),σp(A),σap(A),ρ(A) the spectrum, point spectrum, approximate point spectrum and resolvent set, respectively. The resolvent of A at λρ(A) is R(λ,A)(λIA)1.

A measurable space is a pair (X,Σ), where X is a set and Σ is a σ-algebra of subsets of X. A function f:XK is said to be measurable if it is Σ-to-Borel measurable in the sense of measure theory. We abbreviate M(X,Σ){f:XKf measurable}andMb(X,Σ){fM(X,Σ)f bounded}. These sets are unital algebras with respect to the pointwise operations and unit element 1 (the function which is constantly equal to 1).

Note that Mb(X,Σ) is closed under bp-convergence, by which we mean that if a sequence (fn)n in Mb(X,Σ) converges boundedly (i.e., with supnfn<) and pointwise to a function f, then fMb(X,Σ) as well.

If the σ-algebra Σ is understood, we simply write M(X) and Mb(X). If X is a separable metric space, then by default we take Σ=Bo(X), the Borel σ-algebra on X generated by the family of open subsets (equivalently: closed subsets, open/closed balls).

Let (Ω,F,μ) be a measure space. A null set is any subset AΩ such that there is NF with AN and μ(N)=0. A mapping a:dom(a)X, where dom(a)Ω and (X,Σ) is any measurable set, is called almost everywhere defined if Ωdom(a) is a null set. And it is called essentially measurable, if it is almost everywhere defined and there is a measurable function b:ΩX such that {xdom(a)a(x)b(x)} is a null set.

Section snippets

Definition

A measurable (functional) calculus on a measurable space (X,Σ) is a pair (Φ,H) where H is a Hilbert space and Φ:M(X,Σ)C(H)is a mapping with the following properties (f,gM(X,Σ),λK):

  • (MFC1)

    Φ(1)=I;

  • (MFC2)

    Φ(f)+Φ(g)Φ(f+g) and λΦ(f)Φ(λf);

  • (MFC3)

    Φ(f)Φ(g)Φ(fg) and dom(Φ(f)Φ(g))=dom(Φ(g))dom(Φ(fg));

  • (MFC4)

    Φ(f)L(H) and Φ(f)=Φ(f¯) if f is bounded2 ;

  • (MFC5)

    If fnf pointwise and boundedly, then Φ(fn)Φ(f) weakly.

Property (MFC5) is called the weak bp-continuity of the mapping Φ. We shall see below,

Projection-valued measures and null sets

If (Φ,H) is a measurable functional calculus on a measurable space (X,Σ), then the mapping EΦ:ΣL(H),EΦ(B)Φ(1B)L(H)(BΣ)is a projection-valued measure. This means that EEΦ has the following, easy-to-check properties:

  • (PVM1)

    E(B) is an orthogonal projection on H for each BΣ.

  • (PVM2)

    E(X)=I.

  • (PVM3)

    If B=n=1Bn with all BnΣ then n=1E(Bn)=E(B) in the strong (equivalently: weak) operator topology.

(The equivalence of convergence in weak and strong operator topology in (3) is shown similarly as (f) in Theorem 2.1.) A

Spectral theory

In this section we shall see that a measurable calculus (Φ,H) contains the complete information about the spectrum of each operator Φ(f). To this end, define the Φ-essential range of fM(X,Σ) by essranΦ(f){λKε>0:|fλ|εNΦ}.Then we have the following important result.

Theorem 4.1

Let (Φ,H) be a measurable functional calculus on (X,Σ), let fM(X,Σ), c0 and λK. Then the following assertions hold:

  • (a)

    σ(Φ(f))=σap(Φ(f))=essranΦ(f).

  • (b)

    fessranΦ(f)   Φ-almost everywhere.

  • (c)

    Φ(f)L(H),Φ(f)c|f|c Φ-almost everywhere.

  • (d)

    Φ(

Uniqueness for measurable calculi

In this section we shall establish several properties that determine a measurable functional calculus uniquely. The first one has already been mentioned in Section 2.4.

Lemma 5.1

Let (Φ,H) and (Ψ,H) be two measurable calculi on (X,Σ) such that Φ(f)=Ψ(f) for all bounded functions f. Then Φ=Ψ.

Lemma 5.1 can be easily refined.

Proposition 5.2

Let (Φ,H) and (Ψ,H) be two measurable calculi on (X,Σ). If the corresponding projection-valued measures coincide, i.e., if EΦ=EΨ, then Φ=Ψ.

Proof

It follows from the hypothesis that Φ and Ψ

Construction of measurable calculi

In this section we describe different steps that lead to the construction of a measurable functional calculus. In the results we have in mind one starts with a “partial calculus” so to speak. That is, one is given a subset MM(X,Σ), in the following called our set of departure, and a mapping Φ:MC(H) that has the properties of a restriction of a measurable calculus. And one aims at asserting that this partial calculus is in fact such a restriction, that is, can be extended (uniquely, if

The spectral theorem

Finally, we shall state and prove “our” version(s) of the spectral theorem.

Acknowledgments

In preliminary form, parts of this work were included in the lecture notes for the 21st International Internet Seminar on “Functional Calculus” during the academic year 2017/2018. I am indebted to the participating students and colleagues, in particular to Jan van Neerven (Delft), Hendrik Vogt (Bremen) and, above all, to Jürgen Voigt (Dresden) for valuable remarks and discussions.

This work was completed while I was spending a research sabbatical at UNSW in Sydney. I am grateful to Fedor

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