a unified mathematical formalism for the dirac

815

0015-9018/02/0600-0815/0 © 2002 Plenum Publishing Corporation

Foundations of Physics, Vol. 32, No. 6, June 2002 (© 2002)

A Unified Mathematical Formalism for the DiracFormulation of Quantum Mechanics

M. Gadella1 and F. Gómez2

1 Departamento de Física Teórica, Facultad de Ciencias, c. Real de Burgos, s.n., 47011Valladolid, Spain; e-mail: [email protected]

2 Departamento de Análisis Matemático, Facultad de Ciencias, c. Real de Burgos, s.n., 47011Valladolid, Spain; e-mail: [email protected]

Received January 9, 2002; revised March 18, 2002

We revise the mathematical implementation of the Dirac formulation of quantummechanics, presenting a rigorous framework that unifies most of versions of thisimplementation.

1. INTRODUCTION

The mathematical basis of the Dirac formulation of Quantum Mechanicshas been established in the late sixties and early seventies by severalauthors. (1,2,3,4,5) These mathematical analysis of the Dirac formalism are notall equivalent, although its most popular version is based in the idea ofrigged Hilbert spaces also called Gelfand triplets (A monograph on thesubject was published more recently, (6) but it uses the complete differentapproach of trajectory spaces that we shall not discuss in here.) The aim ofthis paper is to make a revision of this mathematical formalism in order tounify and complete it. Thus, we intend to construct a unified frameworkfor the von Neumann–Marlow (1) interpretation of the Dirac formulation ofQuantum Mechanics and the construction by Bohm, (4) Antoine, (3)

Roberts, (2) and Melsheimer (5) based in the structure of rigged Hilbert spa-ce. (7) It is interesting to note that a previous attempt in this direction wasmade in Ref. 8.

Mathematical Physicists are used to working in the von Neumannmathematical formalism for Quantum Mechanics based in Hilbert space.

However, it is well known that von Neumann mathematics do not fulfill acrucial requirement of Dirac: that any observable A had a complete set ofeigenvectors, with real eigenvalues on the spectrum of A (the possiblevalues of a measurement of A) so that any pure state be a linear combina-tion (finite or infinite) of these eigenvectors. Von Neumann approach isonly good if the system under study has a purely discrete spectrum, as ithappens with atomic systems. However, for systems allowing observableswith continuous spectrum, like those involved in scattering processes, vonNeumann theory does not justify the Dirac formulation.

In an attempt to make von Neumann and Dirac compatible, Marlow (1)

proposes a construction of the Hilbert space of states as a direct integral ofHilbert spaces, so that

H=Fs(A)Hl dm(l).

The measure m is supported on the spectrum of A, s(A).The construction of this integral decomposition, and therewith of the

Hilbert spaces Hl, depends on the observable A (or equivalently, thecomplete system of compatible observables) under consideration. Marlowargues that the complete system of eigenvectors of the Dirac formulationmay be replaced by any of the measurable orthogonal basis {ej(l)} thatspan H. Thus, we have a direct integral decomposition for any vectork ¥H in the spirit of Dirac. Thus, if k, j ¥H with integral decompositions

k=Fs(A)

k(L) dm(l) and j=Fs(A)

j(l) dm(l)

we have that

(k, j)=Fs(A)

Cdim Hl

j=1(k(l), e(l))l (ej(l), j(l))l dm(l) (1)

in agreement with the Dirac formula.Nevertheless, the construction of Marlow is not good enough. In fact,

if l belongs to the continuous spectrum of A, neither the scalar products ofthe form (ej(l), j(l))l are defined in H (since then Hl is not a subspaceof H), nor the mapping j W (ej(l), j(l))l is a continuous linear functionalon H.

Shortly after the paper by Marlow was published, A. Bohm (4)

proposed to rig the Hilbert space H with two locally convex spaces F andf× with

F …H … F×. (2)

816 Gadella and Gómez

Here, F is a locally convex space endowed with a topology finer than thetopology in H and F× its antidual (space of continuous antilinear func-tionals on F). The space F is dense in H. If the topology on F is nuclear,based on results by Gelfand (7) and Maurin, (9) we can show that for a givenobservable A (represented as a self adjoint operator on H) there is a riggedHilbert space of the form (2) such that:

(i) AF … F and A is continuous on F.

(ii) The operator A can be extended to F× by means of the dualityformula

OAj | FP=Oj | A×FP, -j ¥ F, F ¥ F×

where Oj | FP denotes the action of the functional F ¥ F× on thevector j ¥ F. The operator A× is the extension of A into F×.

(iii) There exists a measure m on the spectrum of the operator A,s(A), such that for almost all l ¥ s(A), with respect to m, thereexists a Fl ¥ F× such that A×Fl=lFl.3

3 In this case, we obviously have that OAj | FlP=lOj | FlP for all j ¥ Y. We say that Fl is ageneralized eigenvector of A with generalized eigenvalue l.

(iv) For each pair k, j ¥ Y, we have the Dirac completeness relation

(j, k)=Fs(A)

Oj | FlPOFl | kP dm(l)

with OFl | jP :=Oj | FlP* and the star means complex conjugate.

In a series of two papers published in 1969, (3) Antoine critisizes thework of Marlow to conclude that one needs to extend the Hilbert space inorder to implement the Dirac formalism. Once again, he uses rigged Hilbertspaces in order to implement this extension. His construction relies again inthe von Neumann direct integrals of Hilbert spaces. Later, he proposesdifferent structures (10) (Lattices of Banach Spaces and Partial Inner ProductSpaces) to extend quantum mechanics beyond Hilbert spaces. Antoine alsointroduces a new formulation of symmetries in Quantum Mechanics usingrigged Hilbert spaces and discuss the Dirac representation formula forcontinuous operators from F into F×. (3)

Earlier in the decade, Foias published a series of papers with a differ-ent point of view. (11) Unlike Marlow and Antoine, Foias does not make useof the Direct intergrals of Hilbert spaces, but instead on the so calledintegral decompositions of F with respect the self adjoint operator A with

A Unified Mathematical Formalism 817

AF … F. These decompositions are described by means of the eigenopera-tors c(l) of A. These are linear continuous operators from F into F×

b (thespace F× endowed with the strong topology with respect to the dual pair(F, F×) (12)). These eigenoperators are defined such that they fulfill thefollowing identity:

A×c(l)=c(l) A=lc(l)

Then, Foias defines the eigenforms of A with eigenvalue l to be the vectorsof F× of the form f× :=c(l) f, where f is an arbitrary vector in F. Each ofthese f× has the property that A×f×=lf×, since for arbitrary j ¥ F wehave that

OAj | f×P=Oj | A×f×P=lOj | f×P, -j ¥ F.

The integral decomposition of A is then defined as a triplet (s(A), m, c(l)),where m is a positive regular Borel measure on s(A) and, for almost all l

with respect to the measure m, c(l) is an eigenoperator of A with eigenvaluel. For any pair j, k ¥ Y, the mapping l W Of | c(l) jP is m-integrable andthe following equation holds:

(f, j)H=Fs(A)

Of | c(l) jP dm(l).

The ideas of Foias will be developed by Roberts in 1966 (2) andMelsheimer in 1974. (5) These authors made use of direct integral decompo-sitions to show that the techniques of Gelfand–Maurin and Foias areequivalent under the strong assumption of the nuclearity of the space F.The most relevant results of these works are:

(i) The construction of rigged Hilbert spaces for given operatoralgebras.

(ii) The determination of necessary and sufficient conditions for anoperator A to have a spectral decomposition on F (its closure Amust be subnormal and formally normal). (2)

(iii) The representation of the eigenoperators c(l) of A in terms ofthe eigenforms.

Finally, in the framework of the mathematical theory of scattering, (13, 14) thestationary methods which were proposed independently by Howland (15) andRetjo (16) have in common the use of an auxiliary Banach space F, a part ofwhich is identified with a subspace dense in H. This allows to Howlandhimself (17) to construct a theory of eigenfunction expansions. Like in the


Table 1. The above half summarizes the status of Dirac kets in the conventionalvon Neumann–Mackey formulation of Quantum Mechanics. The half below

summarizes the same notion on auxiliary topological vector spaces

DIRAC KETS


constructions of Gelfand and Foias, such eigenfunctions are vectors in theantidual space F× of F. Kato and Kuroda (18) introduce these ideas in amore general framework, as a perturbation problem, with a direct andsimple method to obtain the direct integral of the spectral decomposition.In the Kato Kuroda theory is not necessary that the auxiliary space F beneither normed nor dense in H nor that the topology on F be compatiblewith the topology inherited by F from H. It is sufficient that F be agenerator with respect the spectral measure (we shall explain its precisemeaning later). The eigefunction expansion is similar to the given in Ref. 7and 19. Under certain conditions, (18) an explicit relation among completesystems of eigenfunctions of the Lippman–Schwinger equations. In partic-ular, Howland (17) considers the eigenfunction expansion in the time inde-pendent scattering theory and obtains the classical Lippmann–Schwingerequations in terms of the resolvents.

The situation described in this part of the Introduction is summarizedin Table I.

1.1. Locally Convex Riggings

The point of departure of the present article is Marlow interpretationof the Dirac formalism. From the Marlow point of view it is necessary toknow the explicit form of the direct integral decomposition of the Hilbertspace H, something which is missed in the work of Marlow. The task is notsimple and very technical and is presented in Sec. 3. This identificationallows us to obtain explicitly the family of eigenoperators in the Foiasdecomposition. We shall show in Sec. 6 that Foias eigenoperators andeigenforms can be written in terms of the generalized eigenvectors found inSec. 3.

On the other hand, if a priori we know a complete system of general-ized eigenvectors (that we write in the form {|lkP}) defined as functionalson a dense subspace F of the Hilbert space H, then, these functionalsdefine a locally convex topology on F by means of the family of seminormsgiven by the mappings f W |Olk | fP|. For the case that F had a topologymaking these seminorms continuous and only in this case, the generalizedeigenvectors {|lkP} will be vectors in the antidual space F×. This meansthat the topology on F should be equal or stronger than the topologyproduced on F by this family of seminorms if we want that {|lkP} … F×.This allows us to construct triplets F …H … F× that satisfy all Diracrequirements for a given observable A, without the hypothesis of nuclearityneeded for the Gelfand–Maurin theorem. Such a possibility has alreadybeen studied by Antoniou and Suchanecki. (20)


This allows us to construct minimal riggings with respect to a partialorder in the riggings of H as we shall describe in Sec. 4. These minimalriggings are tight riggings in the sense that the set of generalized eigenvaluesof A coincides with the spectrum of A. This fact may be desirable in theDirac formulation of Quantum Mechanics since the spectrum of the exten-sion of A and the spectrum of A are the same. However, this property isnot good when we use rigged Hilbert spaces for other purposes, like todescribe resonance phenomena. (21) A tight rigging of particular interest wasconstructed in Ref. 22.

Along the minimal riggings we have other of particular interest inwhich F is either a nuclear space or a countable inductive limit of theHilbert Schmidt type. (1) These are universal riggings, in the sense that theirantiduals always contain complete systems of generalized eigenvectors forany Vitali spectral measure (see Sec. 4).

Table 2. The column of the left lists the equipments of a Hilbert space here studied.To the right, we present their respective applications.

LOCALLY CONVEX EQUIPMENTS


The Dirac formalism can be also implemented in the Kato Kurodasense. (18) This opens possibilities that shall be explored in the future.

We summarize all these comments on Table 2.This paper is organized as follows: In Sec. 2, we introduce a brief

summary of the mathematical tools that are necessary in order to under-stand the next section of the paper. We assume, however, here that thereader knows the notion of direct integral of Hilbert spaces, which can befound for instance in Ref. 19 and 23. Sec. 3 is the core of our paper. Weconstruct there the eigenprojections that were missing in the Marlow for-malism and settle the basis of our unifying formalization. In Sec. 4, wepresent some results concerning minimal riggings. Universal riggings arediscussed in Sec. 5, where we give new versions of the nuclear and theinductive spectral theorems. Latter, in Sec. 6, we calculate explicitly theeigenoperators and eigenfunctions for the Foias–Roberts–Melsheimerformalism and discuss some of their properties. Finally in Sec. 7, we use theKato–Kuroda eigenfunction expansions as a new mathematical basis forthe Dirac formalism.

2. MATHEMATICAL TOOLS

In this section, we present the most important mathematical tools thatshall be used along this paper. The first one is the concept of direct integralof Hilbert spaces. The second one is the notion of locally convex topologi-cal vector spaces of which Banach spaces and therefore Hilbert spaces arecommon examples. However, we need a more general type of locallyconvex spaces to construct rigged Hilbert spaces that are used in physics toimplement the Dirac formulation of quantum mechanics, define vectorstates for resonances, obtain general spectral decompositions for unitaryand self adjoint operators including resonance states and define an arrowof time. (4, 21, 24–28) The third one is the crucial notion of spectral measurespace. (23)

We do not intend to define here neither the direct integral of Hilbertspaces nor locally convex topological vector spaces. Both concepts areextensively presented in the standard literature. For instance, see for directintegrals of Hilbert spaces Refs. 23, 19 and 29 and for locally convex spacesRefs. 7, 12, 30–32, etc.

However, a presentation of the notation and mathematical terms to beused along this paper is important in order to fully understand its contents.Here is a list of these terms:

Let H be a Hilbert space and A a self-adjoint operator on H. Theclassical version of the spectral theorem (33) associates to the self-adjoint


operator A a spectral measure space (L, A,H, P). Here, L=s(A), thespectrum of A, A is the class of Borel sets of L and P(E) is the orthogonalprojection corresponding to E ¥A.

Nevertheless, we shall consider here a more general situation, whichhas been introduced in Ref. 23. Here, the Hilbert space H is separable (weare considered always separable Hilbert spaces only) and any measurespace (L, A, m) to be considered in this paper is s-finite and has acountable basis (the measure space (L, A, m) has a countable basis ifthere exists a sequence {En} of measurable sets in A such that for anyE ¥A and for all e > 0, there exists an Ek, from the sequence {En}, suchthat m[(E0Ek) 2 (Ek 0E)] < e). (23)

Let g be an element of H. By Hg we denote the closure of the space

Hg={f ¥H : f=P(E) g},

where E runs out the Borel sets in L. We say that the sequence of vectors{gj}

mj=1, m=1, 2, ..., . in H is a generating system of H if

H=Âm

j=1Hgj

.

If f, g ¥H, mf, g(E) :=(f, P(E) g) when E … L is a complex measureon L. If f=g, we write mf :=mf, f and therefore, mf(E)=(f, P(E) f).The support of mg is L(g), so that, L(g) … L. The type of the measure m onL is the equivelance class of all measures that are equivalent with m and isdenoted by [m] (two measures, on the same measurable space W, m and n

are equivalent if m is absolutely continuous with respect to n and viceversa).If the measure m is absolutely continuous with respect to the measure n, wewrite [n] P [m].

A nonzero vector g ¥H is of maximal type with respect to the spectralmeasure P if for each f ¥H, [mg] P [mf]. In this case, g is called amaximal vector. Such maximal vectors always exist, provided that H beseparable (23) and we consider here separable Hilbert spaces only. The type[mg] of a maximal vector is called the spectral type of P and denoted by[P].

Let (L, A,H, P) be a spectral measure space. Thus, there exists agenerating system {gj}

mj=1, where 1 [ m [ ., such that: (23)

[P]=[g1] P [g2] P · · · ,

where [gi]=[mgi], i=1, 2, ... . This sequence of types is a characteristic of

the spectral measure P and does not depend on the choice of the generatingsystem {gj}

mj=1.


Then, take {gj}mj=1 and the following subsets of L

Lk=L(gk)0L(gk+1), -k < m

Lm= 3k ¥ [1, mP

L(gk)

The symbol [1, mP denotes the set of numbers {1, 2, ..., m}, where m canbe either finite or infinite. The family {Lk}m

k=1 is a partition of L. Wedefine the multiplicity function Np of P as

NP(l)=k, if l ¥ Lk

NP depends on P only. Therefore, it does not depend on {gj}mj=1.

Let(L, A,H, P) and (L, A,HŒ, PŒ) be two spectral measure spaces onthe same measurable space L, A). We say that (L, A,H, P) and(L, A,HŒ, PŒ) are structurally isomorphic if there exists a unitary operatorV:HWHŒ such that

VP(E)=PŒ(E) V; -E ¥A (f)

What we call structural isomorphism is often called in the literatureunitary equivalence. We have chosen the term structural isomorphismbecause the identity (f) does not hold for any unitary mapping between Hand HŒ and therefore, we are here using an idea more restricted thanunitary equivalence.

Two spectral measure spaces (L, A,H, P) and (L, A,HŒ, PŒ) arestructurally isomorphic if and only if [P]=[PŒ] and NP=NPŒ, [P] almosteverywhere (recall that H and HŒ are both infinite dimensional and sepa-rable).

Let (L, A,H, P) be a spectral measure on the Hilbert space H andS(L, P) the set of measurable functions on L with values in C (the set ofcomplex numbers), which are finite almost everywhere with respect to themeasures mg(E)=(g, P(E) g) (we say that these functions are finite almosteverywhere with respect to P or that are finite a.e. with respect to P). Foreach f ¥ S(L, P), the operator

Jf :=FL

f dP

is well defined with domain

Df=3f ¥H : FL

|f|2 dmf < .4


If L is a closed subset of R, then (L, A,H, P) gives a unique selfadjoint operator A, according to von Neumann theorem, (33) with L=s(A).The operators Jf commute with the self adjoint operator A determined bythe spectral measure space (L, A,H, P).

A list of the main properties of Jf, with f ¥ S(L, P), is the following:(i) (f, Jf g)=>L f dmf, g.

(ii) (f, Jf f)=>L f dmf.(iii) ||Jf f||2=>L |f|2 dmf.(iv) ||Jf ||=ess sup |f| a.e. in P.(v) Let g ¥H. Then, g ¥ Df if and only if g ¥ L2(L2, mg).

(vi) Let g ¥H. Then Hg reduces Jf, i.e.

Jf(Hg 5 Df) …Hg and Jf(H+g 5 Df) …H+

g

(vii) Let g ¥H. Then, the mapping

Vg: L2(L, mg) WHg, f W Jf g

is unitary.(viii) For any g ¥ Df, the space Hg has the form

Hg={Jf g: f ¥ S(L, P)}

Now, let us consider a direct integral of Hilbert spaces:

Hm, N=FL

Hl dm(l)

The subindex N denotes the function N(l) :=dimHl. A typical example ofdirect integral of Hilbert spaces is given by

L2(R, dx)=FR

Cl dl

where Cl are identical copies of the complex plane C. We have one for eachl ¥ R. Note that each of the Cl is not subspace of L2(R, dx). Here, each Cl0

can be viewed as the set of complex functions on R that vanish on R0{l0}(l0 ¥ R).

The functional version of the von Neumann spectral theorem (33) estab-lish that for each self adjoint operator A, there exists a Borel measure m ons(A) and a direct integral of Hilbert spaces

Hm, N=Fs(A)Hl dm(l) (3)


such that there exists a unitary operator V:

V:HWHm, N (4)

with the property that for

Hm, N ¦ g=Fs(A)

g(l) dm(l)

we have

[VAV−1 g](l)=lg(l) (5)

This means that the form of the operator VAV−1 is diagonal on Hm, N:

VAV−1=Fs(A)

lIl dm(l) (6)

where Il is the identity on the space Hl.The unitary mapping V transforms the spectral measure space of A,

(L, A,H, P) into the spectral measure space for VAV− which is(L, A,Hm, N, P) where P(E)=VP(E) V−1=qE, the multiplication operatorby characteristic function of E, qE(l) (qE(l) is one if l ¥ E and zerootherwise). Thus,

(P(E) g)(l)=qE(l) g(l), (E ¥A, g ¥Hm, N)

In (3) the essential supreme of N(l), with respect to [P], is here denoted bym. Obviously, m can be either finite or infinite. Again, by [1, mP, wedenote the set of numbers {1, 2, ..., m}, where now m=ess sup N(l)).

For the operator Jf, we have the following relation:

Qf :=VJfV−1=FL

f(l) Il dm(l) (7)

We have a structural isomorphism between two direct integrals ofHilbert spaces defined on the same measurable space (L, A):

Hm, N :=FL

H(l) dm(l) and Hn, NŒ :=FL

HŒ(l) dn(l)

if there exists a unitary operator V:Hm, N WHn, NŒ such that for any E ¥A,

VP(E)=PŒ(E) V Z VQf=Q −

fV, -f ¥ S(L, m)=S(L, n)


We determine the operator V through the measure family of unitary(almost elsewhere with respect [m]=[n]) operators V(l):H(l) WHŒ(l),with

h(l) W (Vh)(l) :=p(l) V(l) h(l)

The function p(l) is given by the square root of the Random Nikodymderivative of m with respect to n:

p(l)==dm

dn(l), l ¥ L

We have defined structural isomorphisms between spectral measurespaces and direct integrals of Hilbert spaces. Now, if (L, A,H, P) is aspectral measure space and Hm, N a direct integral of Hilbert spaces on themeasurable space (L, A), such that

[P]=[m] and NP=N(l), a.e.

then, there exists a structural isomorphism between (L, A,H, P) and Hm, N

in the sense that there exists a unitary operator V:HWHm, N such that forany E ¥A:

VP(E)=P(E) V Z VJf=QfV, -f ¥ S(L, P)=S(L, m)

The mapping V will be given in Proposition 1, Sec. 3. The operator V willplay an important role in order to identify the Dirac kets, the self operatorsand the self forms of the spectral measure. This will be seen in the follow-ing sections.

3. DIRAC FORMULATION AND DIRECT INTEGRALS

In the Dirac formulation of Quantum Mechanics (34) each observable Ahas a complete systems of eigenkets {|lP} whose respective eigenvaluescover the spectrum s(A) of A, which is the set of all possible results of ameasure of the observable A. The completeness of the system {|lP} meansthat there exists a measure on s(A) such that for each ket |fP and each braOj| we have the following Parseval type identity

Oj | fP=Fs(A)

Oj | lPOl | fP dm(l)


with

A |lP=l |lP

for almost all l ¥ s(A) with respect to the measure m and

Oj | lP :=Oj | lP*

where the star denotes complex conjugate. If we omit the arbitrary bra Oj|,we have

|fP=Fs(A)

|lPOl | fP dm(l) (8)

The operator A admits the following integral form

A=Fs(A)

l |lPOl| dm(l)

that should be interpreted in the sense that for suitable kets |fP and brasOj|, one has

Oj | A | fP=Fs(A)

lOj | lPOl | fP dm(l) (9)

For each measurable Borel function f(l): s(A) W C, the following opera-tor can be defined

f(A)=Fs(A)

f(l) |lPOl| dm(l) (10)

which should also be interpreted in the sense of (9), provided that the cor-responding integral converges. This formula is called the Dirac functionalcalculus formula.

Later, von Neumann (33) identified the observable A with a self adjointoperator on a separable Hilbert space H and the spectrum of A as theHilbert space spectrum s(A) of the operator A. In the von Neumann for-mulation of Quantum Mechanics in Hilbert Space the above formulas onlymake sense if the spectum A is purely discrete (as it happens in the harmo-nic oscillator for example). In this case, if s(A)={lk} and Ak fk=lk fk forf=1, 2, ... , we have that instead of (8) and (9)

f=Ck

(f, fk) fk, Af=Ck

lk(f, fk) fk (11)


However, most self adjoint operators representing quantum observableshave a continuous spectrum for which (11) is not valid. Therefore, theDirac formalism cannot be implemented in the Hilbert space H.

A possible way out was proposed by Marlow (1) using the vonNeumann representation theorem discussed in the previous section. Foreach self adjoint operator A, there exists a Borel measure m on s(A) and adirect integral of Hilbert spaces

Hm, N=Fs(A)Hl dm(l)

and a unitary operator V:

V:HWHm, N

with the properties descrived in Sec. 2.If f1, 2 :=V−1g1, 2, we have that

(f1, Af2)H=(Vf1, VAV−1Vf2)Hm, N=F

s(A)l(g1(l), g2(l))l dm(l) (12)

where (—, —)H, (—, —)Hm, Nand (—, —)l represent the scalar product in

H,Hm, N and Hl respectively. We recall that for each pair

Hm, N ¦ g :=Fs(A)

g(l) dm(l) and Hm, N ¦ h :=Fs(A)

h(l) dm(l)

we have that

(g h)Hm, N=F

s(A)(g(l), h(l))l dm(l)

In addition, if {ek(l)}N(l)k=1 is an orthonormal measurable basis of Hl (with

dimHl=N(l)), we have that

(g, h)Hm, N=F

s(A)C

N(l)

k=1(g(l), ek(l))l (ek(l), h(l))l dm(l) (13)

In Sec. 2, it was established that VAV−1 admits the following integralform

VAV−1=Fs(A)

lIl dm(l) (14)


where Il is the identity on Hl. Then, if f :=V−1f, w :=v−1h, (5) and (13)give

(f, Ah)H=(f, VAV−1h)Hm, N=F

s(A)C

N(l)

k=1l(f(l), eh(l))l (ek(l), h(l))l dm(l)

15

Formula (15) is quite similar to (9), except that in (15) we have con-sidered the possibility of degeneracy of the l. If f(l) is a given measurablefunction on s(A), we have the following functional calculus formula:

(f, f(A) h)=Fs(A)

CN(l)

k=1f(l)(f(l), ek(l))l (ek(l), h(l))l dm(l) (16)

where the operator f(A) is defined on a certain domain Df …H. Formula(16) is sometimes written as:

f(A)=Fs(A)

CN(l)

k=1f(l) |ek(l))(ek(l)| dm(l) (17)

Compare (17) to (10) omitting the sum in k in (17) for simplicity. Bothformulas are identical if we write |e(l))=|lP. We have to note that |e(l)) isnot an eigenvector of VAV− with eigenvalue l as Hl is not a subspace ofHm, N, although obviously, A(l) |lP=l |lP with

A(l)=lIl

(see (14)).The main results of this section are presented in Theorem 1 and in

Theorem 2. Both results are a consequence of Proposition 1, which hasbeen given in Ref. 23.

Proposition 1. There exists a generating system {gk}mk=1 in H, with

respect to P, such that:

(i) [P] P [g1] P [g2] P · · · . We recall that for simplicity in thenotation, we write [g]=[mg] (see Sec. 2).

(ii) If 1 [ j [ k [ m (m=ess sup N(l)), then the measure mgkcoin-

cides with mkkon its support L(gk), which means that

mgj|L(gk)=mgk


(iii) If {ej(l)}mj=1 is a measurable orthonormal basis on Hm, N, then,

for all h ¥Hm, N, we have that:

V−1h=Âm

j=1

1Fs(A)

1ej(l), = dm

dmg1

(l) h(l)2l

dP(l)2 gj (18)

Theorem 1. Under the conditions of Proposition 1, for eachf, h ¥H, we have

(f, P(E) h)= Cm

j=1F

E([Vf](l), ej(l))l (ej(l), [Vh](l))l dm(l) (19)

Proof. Let E be a Borel set of s(A) and qE(l) its characteristic func-tion. For each k ¥ |1, mP, we have:

(ej(l), qE(l) · ek(l))=30, if j ] kqE 5 L(gk)(l) if j=k

(20)

From (20), Proposition 1 and the identity P(L(g)) g=g, we obtainthe following:

V−1 1=dmg1

dm· qE · ek

2=1FL

qE(l) · qL(gk)(l) dP(l)2 gk

=P(E) P(L(gk)) gk=P(E) gk.

Since V is a unitary operator, for each h ¥H, E ¥A and k ¥ |1, mP, wehave:

mh, gk(E)=(P(E) gk, h)H=(VP(E) gk, Vh)Hm, N

=FL

1=dmg1

dm(l) · qE(l) · ek(l), [Vh](l))l dm(l)

=FE

=dmg1

dm(l) (ek(l), [Vh](l))l dm(l) (21)

The generating system [gj]mj=1 decomposes the Hilbert space H into

the following orthogonal sum:

H=Âm

j=1Hgj

(22)


which allows us to write each h ¥H uniquely as the orthogonal sumh=Ám

j=1 hj. Due to the isomorphism between Hgjand L2(s(A), mg), for

each hj ¥Hj, there exists a unique function hj ¥ L2(s(A), mgj) such that

hj=Jhjgj

where Jf has been defined in Sec. 2.Since we are assuming (ii) from Proposition 1 and the fact that the

domain of the function on l given by (ek(l), [Vh](l))l is included into1l \ k Ll=L(gk), we can write (21) as

(gk, P(E) h)H=(gk, P(E) hk)H

=FE 5 L(gk)

=dmg1

dm(l) (ek(l), [Vh](l))l dm(l)

=FE

= dm

dmgk

(l) (ek(l), [Vh](l))l dmgk(l) (23)

On the other hand, we have by the von Neumann theorem that

(gk, P(E) h)H=1gk, P(E)5Âm

j=1Jhj

gj62H

= Cm

j=1(gk, JqEJhj

gj)H

=FE

hk(l) dmgk(l) (24)

Formulas (23) and (24) give us the following identity:

FE

= dm

dmgk

(l) (ek(l), [Vh](l))l dmgk(l)=F

Ehk(l) dmgk

(l) (25)

for all Borel set E in s(A). Therefore, the functions under the integral signmust coincide almost everywhere with respect to mgk

on L(gk) (the supportof mgk

). If we assign this integrand the value 0 on s(A)0L(gk), then for allh ¥H we have that

= dm

dmgk

(l) (ek(l), [Vh](l))l=hk(l), m a.e. (26)

Take now f ¥H let f=Ámj=1 fj be its decomposition as in (22) where

fj ¥Hgjfor all j ¥ [1, mP. Then, from (26) and the properties of Jf, we

obtain the following identities:


(f, P(E) h)H=1P(E) 5Âm

j=1Jfj

gj6 , 5Â

m

j=1Jhj

gj62H

= Cm

j=1(gj, P(E) Jfg

jJhj

gj)H

= Cm

j=1F

Efg

j (l) hj(l) dmgj(l)

= Cm

j=1F

E

= dm

dmgj

(l) (Vf(l), ej(l))l

×= dm

dmgj

(l) (ej(l), [Vh](l))l dmgj(l)

= Cm

j=1F

E(Vf(l), ej(l))l (ej(l), [Vh](l))l dm(l)

and this concludes our proof.

Theorem 2. Under the same conditions than in Theorem 1, for eachh ¥H and k ¥ |1, mP (see Sec. 2 for a definition of |1, mP, we have thefollowing identities:

(ek(l), [Vh](l))l== dm

dmg1

(l)dmgk, h

dm(l)

==dmg1

dm(l)

dmgk, h

dmg1

(l)

==dmgk

dm(l)

dmgk, h

dmgk

(l) (27)

Proof. For each h ¥H, E ¥A and k ¥ |1, mP, we have obtained therelation (21) in the proof of Theorem 1. This is

mh, gk(E)=F

E

=dmg1

dm(l) (ek(l), [Vh](l))l dm. (21Œ)

Since mh, gkO mgk

O m, from (21Œ) we obtain that the Random Nikodymderivative of mh, gk

with respect to m coincides with the function under theintegral sign in (21Œ) save for a sset of m zero measure. Therefore, for eachh ¥ H and k ¥ |1, mP, we have


(ek(l), [Vh](l))l== dm

dmg1

(l)dmgk, h

dm(l)

==dmg1

dm(l)

dmgk, h

dmg1

(l). (28)

Since mgk, h O mgk, the support of mgk, h is included in L(gk). Then, if we use

(ii) in Proposition 1, we can write (28) as

(ek(l), [Vh](l))l==dmgk

dm(l)

dmgk, h

dmgk

(l). (29)

where we are assuming that the Random–Nikodym derivativedmgk, h

dmgkvanishes

outside L(mgk), the support of mgk

. This completes the proof.Note that Theorem 2 gives us the form of PlV. Let us consider h ¥H.

Then, since {ei(l)}ki=1 is an orthogonal basis in Hl, we have

Vh=FL

[Vh](l) dm(l)=FL

3 Ck

i=1(ei(l), [Vh](l)l) ei(l)4 dm(l)

Then,

PlVh= Ck

i=1(ei(l), [Vh](l)l) ei(l)

is given explicitly by (29).

Example. Let us consider the position operator on L2(R, dx) definedas usual by

Q: DQ Q L2(R, dx)

f W x · f(x)

with domain

DQ=3f ¥ L2(R, dx) : FR

|xf(x)|2 dx < .4 .

The operator Q is self adjoint on this domain with spectral measure givenby

(P(E) f)(l)=qE(l) · f(l), -f ¥ L2(R, dx), -E ¥ B.


where B is the class of Borel sets in the real line. The spectral measurespace for Q is given by

(R, B, L2(R, dx), P)

For, any f ¥ L2(R, dx) which is different from zero almost elsewhere withrespect to the Lebesgue measure on R is a generating vector. The directintegral given by the von Neumann theorem

Hm, N=FRHx dm(x)

is equal to L2(R, dx). Therefore, N(l)=1, Hx=C ( for all x ¥ R) anddm(x)=dx. The measurable orthonormal basis e(x) in our case are mea-surable functions e(x): R W C such that

|e(x)|=1

almost everywhere with respect to the Lebesgue measure. The unitarymapping V is given by

V−1: L2(R, dx) Q L2(R, dx)

Vh W1F

Re*(x) = dm

dmg(x) [Vh](x) dP(x)2 g.

Thus,

(e(x), [Vh](x))x==dmg

dm(x)

dmg, h

dmg(x). (30)

The measures on (30) are given by

mg, h(E)=(g, P(E) h)=FE

g*(x) h(x) dx, -E ¥ B,

mg, g(E)=(g, P(E) g)=FE

g*(x) g(x) dx, -E ¥ B.

From these identities, we easily obtain the Radon–Nikodym derivatives ofthe measures mh, g and mg with respect to the Lebesgue measure on R. TheseRadon–Nikodym derivatives are:

dmg, h

dx(x)=g*(x) h(x) and

dmg

dx(x)=g*(x) g(x). (31)


As the generating vector g(x) is a.e. different from zero, we can divide thefirst identity in (31) by the second to obtain

dmg, h

dmg(x)=

h(x)g(x)

. (32)

If we replace (32) in (30), we conclude that the action of the measur-able orthonormal basis on each Hx is proportional to the action of theDirac delta d(x):

(e(x), [Vh](x))x==dmg

dm(x)

h(x)g(x)

, m a.e. (33)

In particular, if m=mg, we have

h(x)g(x)

=1

g(x)dq(h),

almost elsewhere with respect to m.

Concluding remarks

1. It is possible to construct a spectral decomposition a la Dirac interms of the whole spectrum of the observable A as in (9) by using directintegrals of Hilbert spaces. This fact was already shown by Marlow. (1)

However, the Dirac requirement that each observable should have acomplete set of eigenvectors is not fulfilled. If l ¥ s(A) is in the continuousspectrum of A, the Hilbert space Hl is not a subspace of the direct integralHm, N=>s(A) Hl dm(l), and therefore, we cannot find on Hl an orthogonalbasis of eigenvectors of A (or rather of VAV−). From this point of view, themost we can do, is the following: If f ¥H is in the domain of the observ-able A and Vf=>s(A) f(l) dm(l), it is clear that

PlVAf=lf(l)

so that l ¥ s(A) would be an eigenvector of A in some generalized sense,a.e. in m(l). However, if l is in the continuous spectrum of A, then, Pl isnot even continuous.

2. We have explicitly written the l-components of Vh in terms ofRadom–Nikodym derivatives and hence completed the work of Marlow. (1)


4. LOCALLY CONVEX EQUIPMENTS OF A SPECTRAL MEASURE

We have seen in the previous section that the attempt to implementthe Dirac formulation of quantum mechanics by means of direct integralsof Hilbert spaces has failed because the nonexistence of eigenvectors of theobservable A where their corresponding eigenvalues are in the continuousspectrum of A. The next attempt would be looking for these eigenvalues oncertain extensions of the Hilbert space of states H. These extensions arelocally convex topological vector spaces (tvs) with weaker topologies thatthe Hilbert space topology, so that closed unbounded operators on Hcould be extended to continuous operators on the extension of H.

We start with a tvs (F, yF), where F denotes a vector space on thecomplex plane C and yF a locally convex topology on F. Let us considerthe space F× of continuous antilinear mappings (functionals) from F intoC. This is a vector space and we can endow it with the weak topology: foreach f ¥ F, we define on F× the following seminorm: pf(F) :=|F(f)|,-F ¥ F×. This weak topology on F× will be henceforth denoted by y×

F. Thevector space F× with the topology y×

F (on the pair (F×, y×F)) is called the

antidual of F.From now on, we shall denote the action of F ¥ F× into f ¥ F as

Of | FP. This action is linear to the right and antilinear to the left, just asthe scalar product of Hilbert spaces.

Let us consider the dual pair (F, F×) and assume that F is a propersubspace4 of F×. Now, assume that one of the seminorms on F is a norm

4 This means that F is algebraically isomorphic to a proper subspace of F× that we shall iden-tify with F.

and that it fulfills the parallelogram law. (35) Then, this norm comes from ascalar product. The closure of F with respect this norm is a Hilbert spaceH. If the topology on F is spanned by an infinite set of inequivalenceseminorms, the topology on F is stronger (contains more open sets) thanthe topology that F has as a subspace of H. In this case, we have:

F …H … F× (34)

Here, the mapping F WH is continuous. A structure like (34) is called arigged Hilbert space.

The next definition intends to relate relation (19) with eigenvectors ofa self adjoint operator with eigenvalues in the continuous spectrum. Ofcourse these eigenvectors must exist in a suitable extension of the Hilbertspace H.


Definition. The tvs (F, yF) rigs or equips the spectral measure space(L, A,H, P) if and only if the following conditions hold:

(i) There exists a one to one linear mapping I: F WH with range(image in H of F by I) dense in H. If we identify each f ¥ F with its image,I(f) in H, we can assume that F …H is a dense subspace of H and I thecanonical injection from F into H.

(ii) (First version). There exists a s-finite measure m on (L, A), a setL0 … L with zero m measure and a family of vectors in F× of the form

{|lk×P ¥ F× : l ¥ L0L0, k ¥ [1, mP}, (35)

where m ¥ {., 1, 2, ...}, such that

(f, P(E) j)H=Fm

k=1Cm

k=1Of | lk×POj | lk×P* dm(l),

-f, j ¥ F, -E ¥A. (36)

In particular, if E=L, then, P(E)=IH, the identity on H and

(f, j)H=FL

Cm

k=1Of | lk×POj | lk×P* dm(l),-f, j ¥ F.

(ii) (Second version) There exists a s-finite measure on (L, A), a setL0 … L with zero m measure and a mapping:

(L0L0) × [1, mP Q F×

l × k W |l×P,(37)

such that for each f ¥ F, the complex function f | lk×P belongs toL2(L × [1, mP, m × d), where d is the discreet measure on |1, mP, i.e.,

FL

Cm

k=1|Of | lk×P|2 dm(l) < ., -f ¥ F,

and relations (36) hold.

Each family of the form (35) or (37) satisfying (36) is called a completesystem of Dirac kets (also called generalized eigenvectors) of the spectralmeasure space (L, A,H, P) on (F, yF). The quadruple (F, yF, m, |lk×P) is arigging of (L, A,H, P).

Formula (36) determines the complete system of Dirac kets of thespectral measure uniquely except for:


(i) The m-zero measure set L0.

(ii) The order of the vectors |l1×P, |l2×P, ..., LN(l)×P.

(iii) A phase factor for each |lk×P. If |lk×P is a complete systems ofDirac kets, then also is e if(lk) |lk×P, f being a measurable func-tion from L × N into R.

Consider the complex function depending on the variables l ¥ L0L0

and k ¥ |1, mP:

j(l, k) :=Oj | lk×P*

The function j(l, k) is defined almost everywhere with respect to themeasure m × d on L × |1, mP.

The mapping U: F W L2(L × |1, mP, m × d) given by

j W j(l, k)=Oj | lk×P*=Uj

is obviously unitary. This means that it is one to one and that its range UF

is dense in L2(L × |1, mP, m × d).The next result shows that the spectral measure must be absolutely

continuous with respect to the measure m on the previous definition. By[P] and [m], we shall denote the types of the measures P and m (seeSec. 2).

Lemma. Let (F, yF, m, |lk×P) be a rigging of the spectral measurespace (L, A,H, P). Then, [m] P [P], i.e., P is either of the same type thanm or is absolutely continuous with respect to m.

Proof. Let us assume that [m] O [P]. Then, there exists a measurableset l1 … L such that P(L1) ] 0 and m(L1)=0. As we know, P(L1)H is a(different from zero) closed subspace of H. Let h ] 0 with h ¥ P(L1)H. SinceF is dense in H, there exists a sequence {fn} … F converging to h in H. Wealso have that (P(L1) fn, fn)H Q ||h||2 ] 0 and >L1

Ofn | lk×POfn | lk×P* dm(l)=0 for all k ¥ [1, mP and all n ¥ N, which contradicts (36). This concludesthe proof.

Now assume that [m] P [P]. Then exists L2 … L with P(L2)=0 andm(L2) ] 0. Then, for each f, j ¥ F, we have that (P(L2) f, j)=0 andhence,

Cm

k=1F

L2

Of | lk×POj | lk×P* dm(l)=0.


from which we obtain that |lk×P is the zero vector a.e. on m in L2 and therestriction to the measure on L2 does not play any role. Thus, we mayassume without loss of generality that

[P]=[m]

4.1. Minimal riggings

We can define the following partial order in the class of the riggings ofa spectral measure space:

Definition. Let (F, yF) and (F, yF) and (Y, yY) be two riggings of thespectral measure space (L, A,H, P). We say that (F, yF) is finer than(Y, yY) and we write (F, yF) \ (Y, yY), if F ı Y and yF \ yY. In particular,if yF and yY are finer than the topology inherited of H, we have

F ı Y ıH ı Y× ı F×.

This is a partial ordering in the class of riggings of (L, A,H, P).The next result shows the existence of minimal riggings.

Theorem 3. Let (L, A,H, P) be a spectral measure space. Eachdirect integral of the form Hm, N associated to (L, A,H, P), by the vonNeumann theorem, along with one of its measurable orthonormal basis{ek(l)}N(l)

k=1, or equivalently, each generating system {gk}mk=1 in H with

respect to P such that

(a) [P]=[g1]P P [g2]P P · · · ,(b) if 1 [ j [ k [ m, then mgj

| L(gk)=mgk,

provide a rigging (F, yF, m, |lk×P). This rigging is characterized by thefollowing properties:

(i) The subspace F is dense in H and is given by

F=3f ¥H : existsdmf, gk

dmgk

(l) < ., -l ¥ L0L0, -k ¥ [1, N(l)P4 ,

where L0 is a subset of L with m zero measure (or equivalently, Pzero measure).

(ii) The complete family of antilinear functionals on F, fulfilling(36), is of the form

{|lk×P: l ¥ L0, k ¥ [1, N(l)P},


where we define each |lk×P in terms of the isomorphish V inTheorem 2:

Of | lk×P=([Vf](l), ek(l))Hl==dmgk

dm(l)

dmf, gk

dmgk

(l),

-f ¥ F. (38)

(iii) yF is the weak topology s(F, F×), i.e., the coarsest compatiblewith the dual pair (F, F×). The topological dual F× is the vectorspace spanned by the set |lk×P.

The topology yF is produced by the following family of seminorms:

f W |Of | lk×P|, l ¥ L0L0, k ¥ [1, N(l)P.

Then, the rigging (F, yF, m, |lk×P) is minimal. This means that notopology on F coarser than y (except for the indeterminacy that producesthe choice of the zero m measure set L0) can rig the spectral measure space(L, A,H, P).

Proof. After Theorem 2, for each h ¥H the identity

([Vh](l), ek(l))Hl==dmgk

dm(l)

dmh, gk

dmgk

(l)

is valid except on a set Lh of zero m measure.Since the Hilbert space H is separable, there exists a sequence {hj} of

vectors dense in H. The set L0=1j Lhjhas m zero measure and the

hypothesis of Theorem 3 hold for all l ¥ L0L0. Now, define

F=3f ¥H : existsdmf, g

dmg(l) < ., -l ¥ L0L0

4 .

The vector space F is a subspace of H that contains the dense sequence{hj} and is, therefore, dense in H.

Consider now, the set of linear mappings on F given by |lk×P, asdefined in (ii). This set spans a vector space, Y, of linear functionals on F.Relations (19) in Theorem 1 shows that Y separates points of F. Then, theinitial topology y induced by Y into F is locally convex and the topologicaldual of (F, y) is precisely Y. Therefore, y is the weak topology s(F, Y), thecoarsest compatible with the dual pair (F, Y).

If we change the generating system {gk}, we just multiply each vectorin the family |lk×P by a complex number. Therefore, the topology y doesnot change if we change the generating system {gk}.


For any other topology on F coarser that y (note that there is anindetermination in the definition of y due to the indetermination of thechoice of the zero m measure set L0), the topological dual F× becomessmaller and, therefore, F× cannot contain a complete family of vectors|lk×P fulfilling (36). On the other hand, for any locally convex topology onF stronger than y, the topological dual contains all the vectors of thefamily |lk×P and rig the spectral measure space (L, A,H, P). This conclu-des the proof.

Note that the proof of Theorem 3 is based on the following idea: if wea priori know the antilinear functionals |lk×P on F, these vectors define theseminorms f W |Of | lk×P|. Then |lk×P ¥ F× if and only if the topology onF makes these seminorms continuous. In other words, the topology on F

must be either equal to y or stronger in order that |lk×P ¥ F×.

Concluding remarks

1. Let A be a self adjoint operator and let (L, A,H, P) its corre-sponding spectral measure space. Then, according to Theorem 3, there is arigging (F, yF, m, |lk×P) such that (36) holds. Thus,

(Af, j)H=FL

l d(P(l) f, j)H=FL

Cm

k=1lOf | lk×POj | lk×P* dm(l)

(39)

with f, j ¥ F. If F is reduced by A, i.e., AF … F, we have that

(Af, j)H=FL

Cm

k=1OAf | lk×POj | lk×P* dm(l) (40)

Since the functions of the form j(l, k) are dense inL2(L × |1, mP, m × d), and (40) imply that

lOf | lk×POj | lk×P*=Of | lk×POj | Ok×P*

or

lOf | lk×P=OAf | lk×P (41)

If we extend A into F× by the duality formula

OAf | FP=Of | AFP, -f ¥ F, -F ¥ F×,

we have that (41) implies that

lOf | lk×P=Of | A | lk×P


for all f ¥ F. Therefore,

A |lk×P=l |lk×P (42)

almost elsewhere with respect to the measure m. Note that N(l) representsthe degeneracy of the eigenvalue l. Observe that this type of riggings fulfillthe Dirac requirements.

2. Minimal riggings are tight riggins (see Sec. 6).

5. INDUCTIVE AND NUCLEAR VERSIONS OF THE SPECTRALTHEOREM

5.1. Vitali Systems

Definition. Let (L, A, m) be a measure space such that if l ¥ L, thethe set {l} is measurale and m(l)=0. A Vitali system is a family V ofmeasurable set in L such that: (36)

(i) Being given a measurable set A ¥A and e > 0, there exist acountable family of Vitali sets A1, A2, ... such that

E … 3.

n=1An, m 1 3

.

n=1An

2 < m(E)+e

(ii) Each A ¥A has a border, i.e., a zero m-measure set “E such that

(a) If l ¥ E\(E 5 “E), then any Vitali set containing l withmeasure sufficiently small is contained in E 5 “E.

(b) If x ¨ E 5 “E, then any Vitali set containing l with measuresufficiently small has no common point with E 5 “E.

(iii) Let E … L a set admitting a covering by Vitaly sets B … V suchthat for each l ¥ E and each e > 0, there exists a set Ae(x) ¥ B withm[Ae(x)] < e and l ¥ Ae(x). Then, E has a covering, except for a set of m

zero measure, but a countable set of mutually disjoint sets in B, i.e.,

E … 3.

j=1Aj, Ai 5 Aj=” if Aj ¥ B

Definition. A sequence {E1, E2, ...} of measurable sets in A admits acontraction to a point l0 ¥ L if


(i) For each En in the sequence, there is a Vitali set An such thatl0 ¥ An and limn Q . m(An)=0

(ii) There exists a positive constant c(c < 0) such that

m(En) \ cm(An), - n ¥ N

Definition. Let V be a function from A into R, countably additive.Then, the derivative of n at the point l0 with respect to the Vitali system Vis given by

dn(l0)=lime Q 0

n[Ae(l0)]m[Ae(l0)]

(provided that the limit exists), where Ae(l0) is any Vitali set with m

measure smaller than e containing l0.

Theorem (Vitali-Lebesgue). Let (L , A, m) be a measure space, V aVitali system on L and n a mapping from A into R with the property ofbeing absolutely continuous with respect to m. Then, the derivative of n

with respect to V exists, save for a set of m zero measure. This is given ateach point l0 by

Dn(l0)= limn Q .

n(En)m(En)

where E1, E2, ... is a sequence of measurable sets admitting a contraction tol0. This derivative coincides with the Radon–Nikodym derivative of n withrespect to m, dn

dm.(36)

Example. The Lebesgue measure on (R, B), where B is the Borels-algebra has a particular Vitali system: the set of all open balls

5.2. Vitali Spectral Measures: The Inductive Version of the SpectralTheorem

Definition. The spectral measure space given by (L,A,H, P) is a Vitalispectral measure space if being given its continuous part (L,A,Hc, Pc), thereexists a measure m on L with [m]=[Pc], such that the measure space(L,A, m) admits a Vitali system.


If (L, A,H, P) is a Vitali spectral measure space, the Vitali Lebesguetheorem guarantees that if g, h ¥Hc, for almost all l ¥ L the Radon–Nikodym derivative dmh, g

dmg(l) exists and is equal to llimn Q 0

mh, g(En)mg(En) for any

sequence of set En admitting a contraction to l.

Definition. A rigged Hilbert space F …H … F× is a universal rigging ifF× contains a complete family of generalized eigenvectors for any Vitalispectral measure space on H.

For the case that m be a Vitali measure, its Radon–Nikodym derivativecoincides with its derivative with respect to its corresponding Vitali system,on all points at which both derivatives exists, in accordance to the VitaliLebesgue Theorem.

On the other hand, it is possible to extend the notion of Radon–Nikodym derivative to non-Vitali measures. Here two approaches are pos-sible: the distributional and the martingale approach (20) (and referencesquoted therein). These approaches may be useful to extend the ideas of thepresent section to non Vitali singular spectral measures.

The inductive limit (Hn, In) (n ¥ N) of a countable system where Hn areseparable Hilbert spaces (where Hn …H) such that the identity mappingsIn:HWH are Hilbert–Schmidt is a universal rigging.

Theorem 4. Let H be a separable Hilbert space and

(Hn, In) (n ¥ N)

an inductive system for which each Hn is a separable Hilbert space and theidentity mappings In:Hn WH are Hilbert–Schmidt for all n ¥ N. If

F=span3 0n ¥ N

R(In)4

is dense in H and yl is the inductive topology produced by the system(Hn, In) (n ¥ N) on F, then, (F, yl) rigs any Vitali spectral measure space(L, A,H, P). In particular (F, yl) rigs the absolutely continuous anddiscrete parts of any spectral measure space of the form (C, B,H, P).

Proof. For each n ¥ N, the mapping In:HWH is Hilbert–Schmidt.Therefore, there exits two orthogonal sequences {gn

j } …Hn and {hnj } …H

with ||gnj ||Hn

=1(j ¥ N), ; j ||hnj ||2 < . and

In:Hn 0H: h - Cj

(h, gnj )Hn

hnj


Let us assume that P is simple and let g be a generator of H=Hg. Letr be a positive measure on (L, A) such that r ’ ug. Then, the Radon–Nikodym derivative dmg

dr is a.e. positive, i.e., dmgdr > 0 except in the r zero

measure set Lg. Let us define the measure m as

m(E)=FE

=dmg

drdr, E ¥A

Then, the measure m is equivalent to mg and if h ¥Hn, then, mInh, g O m. Foreach E ¥A with m(E) ] 0, we have

:mInh, g(E)m(E)

:=:; j(h, gnj )Hn

(hnj , P(E) g)H

m(E):

[ ||h||Hn

(; j ||hnj ||2H)1/2 ||P(E) g||Hm(E)

[ ||h||Hn|In |H.S.

1FE

dmg

drdr2

1/2

FE=dmg

drdr

, (43)

where |In |H.S.=(; j ||hnj ||2H)1/2 is the Hilbert–Schmidt norm of In and

||P(E) g||H=(g, P(E) g)H=mg(E)1/2

The mapping x W `x is a convex function on (0, .). By the Jenseninequality, we have that

1FE

dmg

drdr2

1/2;FE

=dmg

drdr [ 1

Therefore, for each EA with m(E) ] 0, we have the following inequality:

:mInh, g(E)m(E)

: [ ||h||Hn|In |H.S., - h ¥Hn (44)

From Theorem 3, we obtain that the spectral measure has a completefamily of functionals {|l×P} such that

O” | l×P==dmg

dm(l)

dm”, g

dmg(l)== dm

dmg(l)

dm”, g

dm(l), ” ¥ F (45)


Since (L, A,H, P) is a Vitali spectral measure space, for each l ¥ L,save for a zero measure set, there exists a sequence of sets {Em} in A withpositive m measure that admits a contradiction to l. We have that

limm Q .

mInh, g(Em)m(Em)

=dmInh, g

dm(l) (46)

From (44), (45) and (46), we obtain for each l ¥ L\L0 that

|OIn(h) |l× |P| [ ||h||Hn|In |H.S.

= dm

dmg(l)

Thus, for each n ¥ N and if n ¥ N, the linear form given by Fl p In is con-tinuous on Hn and its norm is smaller than |In |H.S. `(dm)/(dmg) (l).

Now, for each n ¥ N, let {hnk} be a dense sequence in Hn. Then, the set

L0, n=1k Lhnk

has [P] zero Lebesgue measure as well as L0=1n L0, n. Foreach n ¥ N and for each l ¥ L>\L0, Fl p In is continuous on Hn. Therefore,the complete family {Fl: l ¥ L\L0} associated to the spectral family is inthe dual space F×.

If P is not simple, we choose a generating system {gk} as in Theorem 1and Theorem 2 and we apply the above reasoning for each cyclic subspaceHgk

. This concludes the proof.

5.3. Nuclear Version of the Spectral Theorem

The orginal version of the Gelfand-Maurin nuclear spectraltheorem (19, 9) assumed that F is endowed with a nuclear topology, thecanonical injection I: F WH is continuous, and therefore nuclear and Areduces F(AF … F) and is continuus on F.

The present version of the nuclear spectral theorem uses the relationbetween the spectral measures and the direct integral of Hilbert spaces.

To begin with, let us write the following Lemma, due to Roberts: (2)

Lemma. Let f be a locally convex topological vector space and let Tbe a nuclear operator T: F WH, where H is a Hilbert space. Then, thereexists a separable Banach space X and two operators T1: F W X andT2: F WH such that T1 is continuous, T2 is nuclear and T=T2 p T1.

Our version of the nuclear spectral theorem is the following:

Theorem 5. LetH be a Hilbert space, F a dense subspace in H and yF atopology on F such that the canonical injection I: F WH is continuous.


Then, (F, yF) rigs any Vitali spectral measure space (L,A,H, P). In particular(F, yF) rigs the absolutely continuous and discrete parts of any spectralmeasure space of the form (C, B,H, P).

Proof. Since I: F WH is continuous, then there exists a positiveconstant C > 0 and and absolutely continuous 0-neighborhood of F suchthat for the seminorm qU on F, it holds that

||Ij ||H [ CqU(j) (47)

Let NU={j ¥ F : qu(j)=0}. On the factor space F/NU, we define thenorm ||x||U=qU(x) and we denote by F(U) the normed space [F/NU, || · ||U]and by fU: F W F(U) the canonical map, which is continuous. After (47),Ij+0 if j ¥ NU. We can define the mapping IU: F(U) QH, which is con-tinuous due to (47). In addition, I: IU p fU. Let V be another absolutelycontinuous 0-neighborhood in F such that V … U. Then, qU [ qV. Dueto(47), there also exists the continuous mapping IV: F(V) WH such that I=IV p fV. Furthermore, IV=IU p fUV, where fUV: F(V) Q F(U) is the canonicalmaping of the projective system.

The topological space (f, yF) is nuclear and, in addition, a space Np forall 1 [ p < .. Therefore, there exists a 0-neighborhood V in F such thatthe canonical mapping fUV: F(V) Q F(U) belongs to Np(F(V), F(U)). Then themapping IV=IU p fUV is in Np(FV,H) and there exists a orthonormalbasis {hj} in H and a sequence {f −

j} in (F(V)) − with (; j ||f −

j ||p−V)1/2 < . such

that

IV(j)=IU p fUVj=CjOj −

j | jP hj

Now, let us assume that P is simple. Let g be a generating vector ofH=Hg. As in Theorem 4, let r be a measure defined on (L, A) suchthatr ’ ug. Then, the Radon–Nikodym derivative dmg

dr > 0, i.e., is positive(save for in the set Lg of zero r measure). We define the measure m by:

m(E)=FE

=dmg

drdr, E ¥A

The measure m is equivalent to mg. If j ¥ F(V), then mIVj, gO m and for each

E ¥ A with m(E) ] 0, we have

:mf, g(E)m(E)

:=:; jOj −

j | jP hj, P(E) g)Hm(E)

:

[ 1Cj

|Oj −

j | jP|221/2 (>Edmgdr dr)1/2

>E `dmgdr dr

(48)


Again, the convexity of the function x W `x on (0, .) and Jenseninequality show that the last quotient in (48) is [ 1. Therefore, if wechoose p=2, for each E ¥A with m(E) ] 0, we have that

:mIVj, g(E)

m(E): [ 1C

j|Oj −

j | jP|221/2

[ 1Cj

||j||2V ||j −

j ||2−V21/2

[ ||j||V 1Cj

||j −

j ||2−V21/2

(49)

This inequality shows that the family of linear forms FVE, on the normed

space F(V) of the form

FVE: F(V) QHQ C : j W IVj Q

UIVj, g(E)

m(E),

where E runs over all sets in A with positive m measure, is equiboundedwith common bound not bigger than (; j ||j −

j ||2−V)1/2. Thus, the family of

mappings FVE p fV is equicontinuous on F because we define the topology

on F so that fV be continuous.Since I=IV p fV, the mappings FV

E p fV just coincide to

FE: F QHQ C : f W If W

mIf, g(E)

m(E)

The mappings FE are also an equicontinuous family in F −, as E runs outthe sets in A with positive m measure.

From (ii) in Theorem 3, we conclude that the spectral measure has acomplete family of gnerealized eigenvectors, |l×P, which is defined, save aset of zero mu measure |L0P, by

Of | l×P==dmg

dm(l)

dmf, g

dmg(l)== dm

dmg(l)

dmf, g

dm(l), f ¥ F (50)

Since the spectral measure space (L, A,H, P) is a Vitali spectralmeasure space, for each l ¥ L, not in a set of zero measure, and for eachsequence of sets {Em} in A, with positive mg measure which admits a con-traction to l, we have

limm Q .

FEm(f)= lim

m Q .

FVEm

p fV(f)=dmIf, g

dm(l), f ¥ F (51)


From (49), (50) and (51), we obtain the following inequality

|Of | l×P| [ ||fV(f)||V 1Cj

||j −

j ||2−V21/2 = dm

dmg(l)

valid save for a set, Lf, with m measure equal to zero.Now, if the space F is separable, then there exists a sequence {fn}

which is dense in F such that the set L0=1nLfnhas zero m measure and

the family {|l×}: l ¥ L0L0} is in F t.On the other hand, if F is not separable, the Lemma shows that the

canonical injection I: F WH is a product of two mappings with an inter-mediate separable Banach space X. Then, we replace F(V) by X in the pre-ceding proof to extend it to this case.

If P were not simple, then we choose a generating system {ck} and usethen above proof for each cyclic subspace Hgk

the proof now is complete.

6.. RIGGED HILBERT SPACES

A rigged Hilbert space, also called Gelfand triplet, is a trinity of spaces

F …H … F×

where: (i) H is a separable Hilbert space; (ii) F is a dense subspace of Hwith its own locally convex topology yf which is finer than the Hilbertspace topology yH inherited from H, so tht the canonical injection I: F QH

is continuous; (iii) F× is the antidual of F

Usually, one demands that, in addition, the canonical mapping I benuclear. We are going to use this requirement in what follows.

6.1. The Adjoint Operator

We shall denote the action of F ¥ F× on j ¥ F by Oj | FP. We defineOF | jP :=Oj | FPg. We shall denote the scalar product of two vectorsj, f ¥H as (j, f).

Definition. Let A: F WH be a continuous operator (with thetopologies (yf, yH)). The adjoint operator, A×, of A, A× :HQ F× is definedby the relation:

(Af, f)=Of | A×fP, -f ¥H, -f ¥ F,

The operator A× is well defined and is weakly continuous.


Definition. Let L(F) be the space of continuous linear operators onF. Let A ¥ L(F). A has a conjugate Ac, if there exists a Ac ¥ L(F) such that(j, Af)=(Acj, f), for all j, f ¥ F. The space of operators having a con-jugate is denoted by Lc(F). The operator A ¥ Lc(F) is real if A=Ac.

For any A ¥ Lc(F), we may consider its adjoint

A×: F×Q F×

which is weakly (and also strongly) continuous on A×. For any real opera-tor A, A× extends A because

Of | A×I×IjP=(IAf, Ij)=(If, IAcj)=Of | I×IAcjP, -f, j ¥ F

Therefore, A×I×I=I×IAc. Analogously, we have that Ac×I×I=I×IA. There-fore, Ac× is an extension of A, which is weakly and strongly continuous.

6.2. Integral Decompositions

Let us denote by F×b the TVS F× with the strong topology b(F×, F),

with respect to the antidual pair (F, F×) and let c be an arbitrary mappingin L(F, F×

b ).

Definition.

(i) The operator c is self adjoint if Oj | cfP=Of | cjPg for any pairf, j ¥ F.

(ii) c is positive if Of | cfP > 0, -f ¥ F.

(iii) c is real if Of | cfP ¥ R, -f ¥ F.

Definition. An integral decomposition of F is a triplet (L, m, c(l)),with the following properties:

(i) The set L is a locally compact Hausdorf topological space and m

a regular positive measure on the Borel s-algebra B of L.

(ii) c(l) ¥ L(F, F×b ) and c(l) is positive for almost all l with respect

to m.

(iii) For each j, f ¥ F the function l W Of | c(l) jP is m integrableand

(f, j)=FL

Of | c(l) jP dm(l)


The following result is due to Roberts: (2)

Proposition 2. Let F be a locally convex topological vector space andlet (L, m, c(l)) an integral decomposition of F. Then, there exists a uniquemeasure v on L with values on L(F, F×) such that

Of | v(E) jP=FEOf | c(l) jP dm(l), -f, j ¥ F, -E ¥ B

Definition. Let L be a locally compact Hausdorf topological spaceand B the Borel s-algebra on L. A measure S on (L, B) with values inL(H) is semispectral if for each E ¥ B, the operator S(E) is symmetric,verifies 0 [ S(E) [ 1 and, for each h, g ¥H, (h, S(E) g) is a measure on(L, B).

The next result is also due to Roberts: (2)

Proposition 3. Let F be a locally convex topological vector spacesuch that F is a dense subspace of the Hilbert space H. Let also assumethat the canonical injection I: F WH is continuous. If (L, m, c(l)) is anintegral decomposition of F, then there exists a unique semispectralmeasure S on L with values in L(H) such that:

(If, S(E) Ij)=FEOf | c(l) jP dm(l), -f, j ¥ F, -E ¥ B

Definition.

(i) Let A ¥ Lc(F). A functional j× ¥ F× is a eigenform of with eigen-value of l ¥ C if

OAcf | j×P=Of | Ac×j×P=lOf | j×P, -f ¥ F

This means that the extension, Ac×, of A into F× fulfills the relationAc×j×=lj×.

(ii) A positive operator c ¥ L(F, F×) is an eigenoperator of A witheigenvalue l if

Ac×c=cA=lc

If c is an eigenoperator of A, then it is straightforward thatA×c=cAc=lgc. Also, if c is an eigenoperator of A with eigenvalue l and iffor a given j ¥ F we have that cj ] 0, the, the functional j×=cj is a


eigenform of A with eigenvalue l. The proof of this statement is verysimple:

OAcf | j×P=Of | Ac×j×P=lOf | j×P, -f ¥ F

Analogously, j×=cj is also an eigenform of Ac with eigenvalue lg.

Definition. Being given A ¥ Lc(F), we say that (L, m, c(l)) is anA-integral decomposition of f if for each l ¥ L, the operator c(l) is eitherthe zero operator or an eigenoperator of A. This A-decomposition is real ifL ı R.

Definition. A closed operator A on H is formally normal if D(A) …

D(A†) and ||Ah||=||A†h|| -h ¥H. The operator A is subnormal if there existsa normal extension A of A on a Hilbert space H2 that contains to H.

It is shown in Ref. 37 that any semispectral measure E W S(E) can beextended to a spectral measure H2 with H …H2 . From this fact, Roberts (2)

obtains the following result:

Proposition 4. Let A ¥ Lc(F). Under the conditions of the previousporposition if F has an A-integral decomposition, then the closure A of Ais a subnormal operator as well as formally normal on H.

The reciprocal of this result was also proven by Roberts, (2) using aresult by Gardin (38) and Maurin (39)

Proposition 5. Let H be a Hilbert space and let F …H be a dense inH, locally convex topological vector space which is nuclear, being thecanonical injection I: F WH continuous. Let >L Hldm(L) be a decomposi-tion of H as a direct integral of Hilbert spaces. Then there exists nuclearmappings m-almost everywhere on L, I(l): F WHl, such that

(If, h)=FL

(I(l) f, h(l))l dm(l), f ¥ F, h ¥H (52)

Then, the reciprocal of Proposition 4 can be presented as follows:

Proposition 6. Let H and F be as in the previous proposition. LetA ¥ Lc(F) such that the closure of IAI−1 on H is subnormal and formallynormal. Then, F admits an A-integral decomposition with the followingeigenoperators:

c(l)=I×(l) I(l)


where the mappings I(l) fulfill the relation (52) for each direct integral ofHilbert spaces associated to any of the normal extensions of A by means ofthe spectral theorem. Moreover, the A-decomposition of F is real if andonly if A is real.

Definition. Two integral decompositions (L, m1, c1(l)) and (L, m2, c2(l))are equivalent if the measures m1 and m2 are equivalent (belong to the same type[m1]=[m2]) and, save for a set of measure, we have that

c1(l)=dm2

dm1(l) c2(l)

where dm2dm1

is the Randon-Nikodym derivative of m2 with respect to m1.

A real operator A ¥ Lc(F) has an A-decomposition unique save forequivalence if and only if A is essentially self adjoint on H. In this case, thesemispectral measure of Proposition 6 is a spectral measure on H. Thisspectral measure is unitarily equivalent to the spectral measure P providedb the spectral representation of A, the closure of A. Therefore if (R, m, c) isan A-decomposition of F, we have that

(f, P(E) j)=FEOf | c(l) jP dm(l)=F

E(I(l) f, I(l) j)l dm(l), f, j ¥ F

6.3. Representation of the Eigenoperators

If F …H … F× is a rigged Hilbert space for which the space F isnuclear and separable, O. Melsheimer (5) has found a representation of theeigenoperators of an A ¥ Lc(F), essentially self adjoint on H, in terms ofthe eigenforms of the adjoint of A, A×.

Let (R, B,H, P) be the spectral measure of the closure A of A. Foreach g ¥H,Hg is the closed subspace of H spanned by the vectors of theform P(E) g, with E ¥ B. The orthogonal projection Pg of H on Hg com-mutes with P(E) for all E ¥ B.

Then, there exists a unique (m almost everywhere) function f×g (l) ¥ F×

on R such that (5)

(If, PgP(E) Ij)=FEOf | f×

g (l)P Of | f×g (l)Pg dm(l)

for each f, j ¥ F and E ¥ B.


In the proof of this result, (5) the following identities (m a.e.) are shown:

Of | f×g (l)P=[Pgf] N (l), -f ¥ F

Of | I×[PgIj] N (l)P=[Pgf] N (l)g [Pgj] N (l)

It is shown in Ref. 5 that the mapping given by cg(l): j W I× [PgIj] N (l)(defined m a.e. in l) is an eigenoperator of A and

f×g (l)=

I×(l) g(l)g(l)

(53)

where g is the image of g by the isomorphism Hg ’ L2m.

In general if {gn} is a complete sequence of generators of H, i.e.,H=ÁHgn

, then the sequence {cgn(l)} converges absolutely (m a.e.) to a

c(l) in the strong topology on L(F, F×b ). Furthermore, c(l) f admits the

following representation:

c(l) f=CnOf | f×

gn(l)Pg f×

gn(l)

By

F×(l)

it is denoted (5) the closure of c(l) F with respect to the strong topologyb(F×, F) on F×. Also, the space of the eigenforms of A with eigenvalue l

for each l in the spectrum of A is denoted byFor each g ¥H, we have that I×(l) g(l) ¥ F×(l), m a.e.. In particular,

f×g (l) belongs to F×(l). The space f×

l is closed subspace of F× with theweak topology s(F×, F). Since the strong topology b(F×, F) is finer thans(F×, F) on F×, the subspace F×

l is also b-closed. For m almost all l ¥ R,we have that F×(l) … F×

l . In particular, for all h ¥H, I×l h(l) is m a.e.in F×

l .The subspace of F× given by F×

l 5 I×H is not zero vector if and only if

l is an eigenvalue of A on H. (5) In such a case, F×l 5 I×

H is the subspacespanned by the images by I× of the eigenvectors of A in H with eigenvaluel. It is also shown in Ref. 5 that the subspace of F× given by

c(l) F 5 I×H

is different from zero if and only if l is an eigenvalue of A in H. Then, if Pis the family of projections in the spectral decomposition of A on H, itholds that

c(l) F 5 I×H=I×P{l} IF 5 I×

H


In Ref. 5 the author does not determine in which cases the identityF×(l)=F×

l holds even when l belongs to the spectrum of A as operatoron H. Another problem which has not been solved in Ref. 5 is under whatconditions the spectrum of A as operator on H, s(A), coincides with thespectrum of A×, s(A×). In the most general case, we know that the latter isbigger than the former and examples can be easily constructed5.

5 For example, If P is the one dimensional momentum operator, Z is the Fourier transform ofthe space D(R) and D(R) is the space of all Schwartz functions with compact support and itsstandard limit inductive topology; (7) then, the spectrum of P× as an operator into D(R)coincides with all of C.

Definition. A rigging for which s(A)=s(A×) is called tight rigging.

We want to remark that any minimal rigging described in Theorem 3,fulfills these two properties: (i) For almost all l, with respect to m, we havethat F×(l)=F×

l and (ii) any minimal rigging is a tight rigging.Melsheimer (5) investigates the relation between the forms I×P(E) If

and the forms c(l) f, for each f ¥ F. To this end, a 0-neighborhood V of F

is selected and the following decomposition of the mappings I and I×.

F Q F(V) QHQ (F×)(V°) Q F×

is used. Then, the ideas of Foias (11) are used to show that if l0 is in theHilbert space spectrum of A, then, there exist two sequences {l −

n} and {l'

n }with l −

n ‘ l0 and l'

n a l0, such that for any s ¥ F, we have

limn Q .

I×P[l −

n, l'

n ] IfV

m[l −

n, l'

n ]=c(l0) F

and this limit is taken with respect to the strong topology b(F×, F) on F×.To give an end to this subsection, we note that the expression (53)

giving the eigenforms of A is nothing else than formula (27). In Ref. 5, theauthor cannot go further because the explicit form of the mappings I(l)and I×(l), in terms of the operator A or the associated spectral measure,were unknown. In Ref. 5, the existence of these two mappings was derivedof the nuclear spectral theorem. The proof of this theorem is based on thenuclearity of the canonical injection I: F QH, which has a representationof the form I( · )=; kO · | j×

k P hk6 were the knowledge of the explicit form

6 Here, {hk} is a boudned sequence in H, {jk} an equicontinuous sequence in the antidual F×

and {lk} a sequnece of complex numbers with ; |lk | < ..

of the sequence {hk}, {jk} and {lk} is not necessary.


6.4. Explicit Form of the EigenoperatorsThe following result perimts the indentification of the Foias operators

I(l) and I×(l):

Theorem 6. Let (L, A,H, P), {gk}mk=1,Hm, N, {ek(l)}N(l)

k=1, V, F and|lK×P as in Theorem 3. Then, the mappings

I(l): F Q Hl

f W Cm

k=1Of | lk×Pg ek(l)

= Cm

k=1(ek(l), [Vf](l))l ek(l)

= Cm

k=1

=dmgk

dm(l)

dmgk, f

dmgk

(l) ek(l) (54)

are well defined m almost everywhere on L. The following relation holds:

(P(E) If, h)=FE

(I(l) f, [Vh] (l))l dm(l) f ¥ F, h ¥H, E ¥A

(55)

In particular if E=L,

(If, h)=FL

(I(l) f, [Vh](l))l dm(l), f ¥ F, h ¥H (56)

Proof. In Theorem 3, we give the explicit form of F so that themappings |lk×P are defined are defined on F for m almost all l ¥ L. Rela-tions (56) coincide with relations (19) in Theorem 1.

The adjoint operator I×(l) is

I×(l):Hl Q F×

hl W I×(l) hl:F Q C

j W hl(I(l) j)where

hl(I(l) j)=1 Cm

k=1Oj | lk×Pg ek(l), hl2

Hl

=1 Cm

k=1(ek(l), [Vj](l))l ek(l), hl

2Hl

=1 Cm

k=1

=dmgk

dm(l)

dmgk, j

dmgk

(l) ek(l), hl2Hl


In particular if hl=ej(l), then I×(l) ej(l)=|lj×P

I×(l): Hl Q F×

ej(l) W I×(l) ej(l):F Q C

j W Oj | lj×P

The eigenoperators c(l)=I×(l) I(l) of the spectral measure are of theform

c(l)=I×(l) I(l):F Q F×

f W c(l) f):F Q C

j W [c(l) f](j)

where

[c(l) f](j)=1 Cm

k=1Of | lk×P ek(l), C

m

k=1Oj | lk×P ek(l)2

Hl

= Cm

k=1Of | lk×Pg Oj | lk×P

The eigenforms of the spectral measure f×=c(l) f ¥ F×, where f ¥ F aregiven by:

f×: F Q C

j W Cm

k=1Of | lk×Pg Oj | lk×P

In particular, if we choose f such that If ¥Hgj, then,

f×(j)=Of | lj×Pg Oj | lj×P, -j ¥ F

Once we have identified the eigenoperators and the eigenforms, thetopologies on F for which there exist an integral decomposition of thespectral measure can be explicitly dtermined:

Theorem 7. With the hypothesis of Theorem 6, we have that:

(i) A sufficient condition for the mappings I(l) in (54) be continuouswith respect to a topology y defined on F is that for l ¥ L (save for a setwith zero m measure) the family of antilinear forms {|lk×P: k=1, 2, ...} beequicontinuous with respect to y.


(ii) If m=m-sup N(l)=P-sup NP(l) is finite, then the mappingsI(l) are continuous for the minimal topology yF as defined in Theorem 3.

Proof. It is straightforward

The continouity of the eigenopeators and eigenforms can be derivedfrom the continuity of the mappings I(l) and the following Lemma:

Lemma. Under the conditions of Theorem 6, if we endow F with alocally convex topology for which I(l) is continuous and let F×

b the anti-dual space with the strong topology b(F×, F). Then,

(i) The mapping I×(l):H(l) Q F× is weakly and strongly continuous,i.e., it is s(H,H), s(F×, F)) and (|| · ||Hl

, b(F×, F)) continuous.(ii) The eigenoperator c(l)=I×(l) I(l): F Q F× belongs toL(F, F×

b ).

Proof.

(i) is a consequence of the properties of the adjoint operatordescribed in 6.1.

(ii) is straightforward.

Concluding remarks. We have found the explicit form of the Foiasoperators I(l) and give sufficient conditions for their continuity.

7. EIGENFUNCTION EXPANSIONS OF KATO-KURODA

Howland (17) and Kato–Kuroda (18) constructed a theory of eigenfunctionexpansions in which, as in the Gelfand and Foias theories, such eigenfunc-tions—the generalized eigenvectors—belong to the antidual space F× of anauxiliary tvs F. In this theory, F need not be a dense subset of H and theeigenfunction expansion is formulated in an abstract way analogous to thatgiven by Gelfand–Shilov (7) and Gelfand–Vilenkin. (19)

After a short review of the mathematical tools needed in the presentsection, we construct riggings in the spirit of the Kato Kuroda formalism.

7.1. Spectral forms

By a spectral system (L, A, m,H, P) we mean a spectral measure space(L, A,H, P) together with a s-finite nonnegative scalar measure m on(L, A).

By a standard process, P is decomposed into the absolutely continuouspart Pac and the singular part P s with respect to m. (23) The basic elements ofthe Kato–Kuroda theory are the spectral forms for such systems:


Definition. Let (L, A, m,H, P) be an spectral system. A spectral formfor this system is a complex function on L1 × F × F, where F is a subspaceof H and L1 ı L belongs to A, with the following properties:

(i) For each f, j ¥ F, l W s(l; j, f) is m-integrable in L1 and itsintegral on each E ı L1, E ¥A, is equal to (j, Pac(E) f), i.e.,

(j, Pac(E) f)=FE

s(l; j, f) dm(l) (57)

(ii) For each l ¥ L1, the function j, f W s(l; j, f) is nonnegativeHermitian form on F × F. We write s(l; f) for s(l; f, f).

The subspace F is usually called a spectral subspace and the subset L1

a spectral core of the spectral system.Since (f, Pac( · ) h) is a complex-valued m-absolutely continuous measure

for each f, h ¥H, the Radon-Nydodym derivative d(f, P(l) h)dm =dmf, h

dm (l) isdefined for m-a.e. l ¥ L and by (57) we have

s(l; f, h)=dmf, h

dm(l), m-a.e. (58)

The null set depending on f and h, it is in general difficult to chooses(l; f, h) as an Hermitian form in H for each l ¥ L. But it can be doneeasily if f, h and l are suitably restricted. This is what the last definition isconcerned with. When the spectral form s is that of (58) we refer to s as thecanonical spectral form.

Example. Let H=L2(R), L=R with the Lebesgue measure dl andlet P(E) be the operator of multiplication by the characteristic function ofE defined on the Borel s-algebra B of R. If F is the set of all continuousfunctions in L2(R), then s(l; j, f)=jg(l) f(l) defines a spectral form onR × F × F.

7.1.1. Approximate spectral forms

Let (L, A, m,H, P) be an spectral system with a spectral form(L1, F, s).

Definition. A function

(0, E0) × R × F × F Q C, (E0 > 0).

(E, l, j, f) W sE(l; j, f)


is called an approximate spectral form for (L1, F, s) if for each j, f ¥ F

and l ¥ L1, sE(l; j, f) is a Cauchy net when E Q 0 and its limit is equal to

limE Q 0

sE(l; j, f)=dmj, f

dm(l)

Example. Let H=L2=(R), m the lebesgue measure and let P(E) bethe multiplication by the characteristic function qE, E ¥ B. If z ¥ C0R, letR(z) :=J(l − z)−1 be the resolvent of P in z. The following result for thePoisson integral is well known: (13) For any f ¥H and for any l wheredmf(l)/dl exists, we have

d(f, P(l) f)dl

=limE Q 0

E

p(R(l+iE) f, R(l+iE) f)

=limE Q 0

E

pF

1(l − x)2+E2 dmf(x)

Let us consider the spectral form (R, F, s), where F is the space of all con-tinuous functions of L2(R) and s(l; j, f)=jg(l) f(l) is the canonicalspectral form. Then an approximate spectral form for (R, F, s) is given by

rE(l); j, f) :=(E/p)(R(l+iE) j, R(l+i) Ef (59)

Finally, we assume the spectral subspace F is a topological vectorspace (tvs) with its own topology. The following result is obvious.

Proposition 7. Let (L1, F, s) be a spectral form and letsE an approx-imate spectral form for it that is equicontinuous with respect to E > 0 foreach l ¥ L1. Then the spectral form s(l, · , · ) is continuous in F × F for eachl ¥ l1.

In the example considered above the approximate spectral form rE(l; j, f)is equicontinuous wiht respect to E > 0 if the topology in F is defined by||f||F :=supl ¥ R |f(l)|. In this case we have

E

p||R(l+ie) f||2=

E

pF

.

−.

|f(u)|2

(l − u)2+E2 du [ ||f||2F

7.2. Spectral representations

Let (L, A, m,H, P) be a spectral system with a spectral form(L1, F, s).


1. For each l ¥ L1, s(l; · ; · ) defines that a semi-inner product in F.LetNl be the set of all f with s(l; f)=0. Then the quotient space F/Nl isa pre-Hilbert space with the inner product induced by s(l, · , · ).

We denote by F1 l its completion, by ( · , · )l and || · ||l the inner productand norm F1 l and by ql the quotient map of F onto F/Nl … F1 l

2. Let us consider the product vector space F1 :=<l ¥ L1F2 l consisting

of all vector fields f={f(l)}l ¥ L1with f(l) ¥ F2 l, where we identify the

elements equal m-a.e. By a quasi-simple function f we mean a function ofthe form (finite sum)

f=C ak(l) fk, ak ¥ L.(m), fk ¥ F.

Definition. f ¥ F2 is said to be S-measurable function if there is asequence {fk} of quasi-simple functions on L1 such that

limn Q .

||f(l) − qlfn(l)||l=0 for m-a.e. l ¥ L1.

3. We denote by Hm, F the set of all s-measurable elements f ¥ F2 suchthat ||f||2

Hm, F=>L1

||f(l)||2l dm(l) < .. Thus Hm, F is a Hilbert space with

inner product

(j, f)Hm, F:=F

L1

(j(l), f(l))l dm(l) < ..

Proposition 8. Quasi-simple functions on L1 to F are denselyembedded in Hm, F in the following sense:

(a) For any quasi-simple function f we have qf :={qlf(l)} ¥Hm, F.

(b) For each f ¥Hm, F and E > 0, there is a quasi-simple function j

such that ||j − f||Hm, F< E.

4. We denote by HF the smallest closed subspace of H containing F

a reducing P. HF is the closed span of the set of all vectors of the form[>L1

a(l) dP(l)] f with a ¥ L.(m) and f ¥ F. Hence HacF :=Pac

HF alsoreduces P and Hac

F =Hac if F generates H (i.e., if HF=H).The following theorem is the main result of this subsection [Ref. 18,

theorem 1.11]:

Theorem 8 (Kato Kuroda). There is a unitary operator V on HacF into

the direct integral Hm, F with the following properties:


(i) VJa(P) h=aVh={a(l) Vh(l)}l ¥ L1for each a ¥ L.(m) and

h ¥HacF .

(ii) VPac(L1) f={qlf}l ¥ L1for each f ¥ F.

7.3. Eigenfunction Expansions

Let (L, A, m,H, P) be a spectral system with a spectral form(L1, F, s). Kato and Kuroda introduced the following conditions, whichimply that the system has a representation in a somewhat more refinedsense:

(E1) There exist a s-finite measure space (C, B, r), a partial isometryW of H onto L2(C, r) with initial set Hac and a measurable functionw: C Q L such that

[WJah](t)=a(w(t))[Wh](X), r-a.e. t ¥ C, (60)

for each h ¥H and a ¥ L.(L, m). (The measurability of w means thatw−1(E) ¥ B whenever E ¥A. Thus a p w is r-measurable on C if a isA-measurable on L.)

(E2) There is a complex-valued function j on C × F such that foreach fixed t ¥ C, f W (t; f) is linear and for each fixed f ¥ F,

j(t; f)=[Wf](t), r-a.e. t ¥ C. (61)

(E3) F is a tvs and f W j(t; f) is continuous on F for each t ¥ C. Inthis case we write j(t; f)=Of | j×(t)P, where j×. Each j×(t) will be calledan eigenfunction of P.

Example. Let H=L2(R3), W the Fourier–Plancherel transformationof H onto H=L2(C, B, r), where C is another copy of R3, B the Borels-algebra on C and let r be the Lebesgue measure dt. If the map w of C

into L1=L=R+ is given by w(t)=|t|2=t21+t2

2+t23, A is the Borel

s-algebra of L, m the Lebesgue measure on L, P(E)=W−1P(E) W, whereP(E) is the operator of multiplication by qw−(E), and F=L1(R3) 5H withthe L1-topology, then j×(t) ¥ F× is given by the function (2p)−3/2 e it · l inthe sense that

Of | j×(t)P=(2p)−3/2 FR

3f*(l) e it · l, dl, f ¥ F, (62)


The spectral measure P is the one associated with the selfadjoint operator−D in H and the j×(t) are the eigenfunctions of this operator in the usualsense. In this case, a spectral form is

s(l; j, f)=(4p2)−1 FR

3 × R3

sin(l1/2 |x − y|)|x − y|

f(x) j*(y) dx dy,

l ¥ L1, j, f ¥ F. (63)

Proposition 9. Assume E1. We have w(t) ¥ L1 for r-a.e. t. If E ¥A

with m(E)=0, then r(w−1(E))=0.

Proposition 10. Assume E1 and E2. If f is a F-valued quasi-simplefunction on L1 and if h=V−1(q(l) f), then

[Wh](t)=j(t; f(w(t))) r-a.e. (64)

Proposition 11. Assume E1, E2, and E3. Let q(l): F Q F2 l be con-tinuous for each l. Let h ¥Hac and let {fn} be a sequence of strongly mea-surable functions on L1 to F such that for m-a.e. l ¥ L1, {fn(l)} is Cauchyand q(l) f(l) Q Vh(l) when n Q .. Then

[Wh](t)= limn Q .

Ofn(w(t)) | j×(t)P r-a.e. (65)

7.4. Kato–Kuroda Riggings

Now, we establish the connections between the direct integrals Hm, N

and Hm, F and the isomorphisms V of Proposition 1 (the construction ofBirman–Solomjak (23)) and Theorem 8 (the construction of Kato Kuro-da (18)).

In the following, we assume that the conditions in Ref. 23 are satisfied.In particular, every Hilbert space is separable, as assumed everywhere inthis paper, and every measure space have a numerable basis.

Let (L, A,H, P) be a spectral measure and let {gk}mk=1 be a generating

system, Hm, N a direct integral associated to P, {ej(l)}mj=1 an orthonormal

measurable basis and the operator V:HQHm, N as in Proposition 1. (As wehave seen, m is a s-finite nonnegative scalar measure on (L, A) of the sametype than [P].)

On the other hand, let us consider the spectral system (L, A, m,H, P),a spectral form (L1, F, s) for it (then, by the Theorem 8, F contains a gen-erating system of P since [m]=[P]), the direct integral Hm, F and anoperator VŒ:HQHm, F as in Theorem 8.


Since m(L0L1)=0, we can consider Hm, N defined on (L1, A). If{gk}m

k=1 … F, then, by (18), we have

[Vgk](l)==dmg1

dm(l) ek(l), l ¥ L1,

and, by (2) of Theorem 8,

[VŒgk](l)=ql gk, l ¥ L1.

In this case, HacF =H. After Sec. 3 (see (22) and the paragraph thereafter),

each h ¥H have a decomposition

h=Âm

j=1hj=Â

m

j=1Jhj

gj.

Then, by the properties of V described in Sec. 2,

[Vh](l)=Âm

j=1hj(l)[Vgj](l)=Â

m

j=1hj(l) =dmg1

dm(l) ej(l)

and, by the part (1) of Theorem 8,

[VŒh](l)=Âm

j=1hj(l)=Â

m

j=1hj(l) ql gj.

We recall that, by (26) and (27), we have m-a.e.

hk(l)== dm

dmgk

(l) (ek(l), Vh(l))l

=dmgk, h

dmgk

(l).

Now, we consider the eigenfunction expansions of Kato–Kuroda.Under the Birman–Solomjak conditions, we can always choose asL2(C, B, r) the space

L2(L1, m; l2) ’ L2(L1 × N, m × d),

where l2 denotes the Hilbert space of sequences of complex numbers (ck)such that ; |cn |2 < . and d denotes the discrete measure on N. (We con-sider the most general case, when m=ess sup N(l)=..) In this situationit is possible to obtain explicitly a complete system of eigenfunctions at thesame foot than the eigenvectors of Marlow and the eigenforms of Foias:


Theorem 9. Assume that the spectral system (L, A, m,H, P), thespectral form (L1, F, s), the generating system {gk}m

k=1, the direct integralHm, N, the orthonormal measurable basis {ej(l)}m

j=1 and the operatorV:HQHm, N are as above. Then, if F is a minimal rigging as in Theorem 3,L1=L0L0 and {gk}m

k=1 … F, we have a complete system of eigenfunctionsin L2(L1 × N, m × d) of the form

[j×(l × k)] f=(ek(l), [Vf](l))l

==dmgk

dm(l)

dmgk, f

dmgk

(l), (66)

for every f ¥ F, l ¥ L1 and k ¥ [1, mP. On the other hand, when simplyF=span{gj: j ¥ [1, mP} only the factors of normalization are relevant. Inthis case each f ¥ F is of the form (finite sum) f=; cj gj, where cj ¥ C,and then

[j×(l × k)] f=ck=dmgk

dm(l), l ¥ L1, k ¥ [1, mP. (67)

Proof. For each h ¥H and a ¥ L.(L, m) we have

VJah={a(l) Vh(l)}l ¥ L1.

By the definition of the orthonormal measurable basis {ej(l)}mj=1 and by

Theorem 2, we have m-a.e.

a(l)[Vh](l)=a(l) Âm

j=1(ej(l), [Vh](l))l ej(l)

=a(l) Âm

j=1

=dmgj

dm(l)

dmgj, h

dmgj

(l) ej(l).

To simplify the notation suppose that m=.. Then

Hm, N ’ L2(L1, m; l2) ’ L2(L1 × N, m × d).

These relations are of the following form:

a(l) Âm

j=1

=dmgj

dm(l)

dmgj, h

dmgj

(l) ej(l)

’ l W a(l) 1=dmgj

dm(l)

dmgj, h

dmgj

(l)2.

j=1

’ (l, k) W a(l) =dmgj

dm(l)

dmgj, h

dmgj

(l). (68)


Now, if in E1 we choose L2(L1 × N, m × d) as the space (C, B, r),

W:H Q L2(L1 × N, m × d)

h W5(l, k) W=dmgk

dm(l)

dmgk, h

dmgk

(l)6 ,

and

w: L1 × N Q L1

l × k W l,

then, by (68), the relations (60) are verified. The function j: L1 × N ×F Q C of E2 is of the form

j(l × k; f)=[Wf](l × k)

=(ek(l), [Vf](l))l

==dmgk

dm(l)

dmgk, f

dmgk

(l), (m × d)-a.e. l × k ¥ L1 × N.

Finally, for E3, if F is a minimal rigging as in Theorem 3 andL1=L0L0, then these equalities are satisfied for all l ¥ L1, the mappingf W j(t; f) is continuous on F for each l × k ¥ L1 × N and the eigenfunc-tions are of the form (66). In particular, if h=gj, then

[j×(l × k)] gj=dj, k=dmgk

dm(l), l ¥ L1, k ¥ [1, mP,

where dj, k is the function of Kronecker. The relations (67) can be deduceddirectly from these. This concludes the proof.

This theorem says to us that some families of eigenfunctions ofKato–Kuroda coincide with the families of eigenvectors or Dirac kets ofour locally convex equipments. Then we can use the powerful tool ofapproximate spectral forms to construct equipments. We will callKato–Kuroda riggings the equipments constructed in this form. They will bevery important to obtain generalized Lippmann–Schwinger equations inscattering theory.

8. CONCLUDING REMARKS

1. We have presented a unifying formalism that includes most ofversions of the mathematical theory for the Dirac formulation of quantummechanics.


2. We have found complete families of eigenvectors, in the sense ofMarlow, for a spectral measure in terms of a generating system of vectorsand a family of scalar measures associated to the spectral measure underconsideration.

3. We have defined the notion of minimal riggings for a certainspectral measure and shown that minimal riggings are tight riggings for theobservable corresponding to this spectral measure.

4. We have introduced the notion of universal riggings and shownthat universal riggings equip any Vitali measure. We have shown new ver-sions of the nuclear and inductive spectral theorems.

5. We have identified the eigenoperators and eigenforms in theFoias–Roberts–Melsheimer formalism.

6. We have introduced the Kato–Kuroda riggins and have identifiedsome families of eigenfunctions as did in 2 right above.

ACKNOWLEDGMENTS

We thank Drs. I. E. Antoniou, H. Baumgärtel, A. Bohm, F. LópezFernández-Asenjo, and R. de la Madrid for enlightening discussions.Partial financial support is acknowledged to DGICYT PB98-0370,DGICYT PB-98-0360 and the Junta de Castilla and León ProjectsPC02/99 and VA085/02.

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a unified mathematical formalism for the dirac

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