Spectral properties in a class of operators and group representations in nested Hilbert spaces

REPORTS

Vol. 11 (1977)

ON MATHEMATICAL

PHYSICS

No. 3

SPECTRAL PROPERTIES IN A CLASS OF OPERATORS AND GROUP REPRESENTATIONS IN NESTED HILBERT SPACES

Fr. DEBACKER-MATHOT Institut

de Physique TMorique, (Received

U.C.L., August

Louvain-La-Neuve,

Belgium

16, 1976)

We characterize the spectrum of the elements of some operator algebras acting in a nested Hilbert space. In one of those algebras we prove the spectral theorem for Hermitian operators and the SNAG theorem for unitary representations of locally compact Abelian groups. Then we extend the SNAG theorem to representations in three more general classes of operators.

Introduction

In a previous paper [l] we studied three algebras of operators acting in an arbitrary nested Hilbert space (NHS) [2], [3], which were denoted &, B and %?(~2 2 ?8 2 %‘). %7is a von Neumann algebra, which guarantees existence of projections in any NHS. The elements of B and % were related to operators acting in a certain Hilbert space. We will pursue this study in the present paper.’ After recalling some definitions we characterize in Section 2 the spectrum of an operator belonging to SQ, respectively to B or %. Those three different notions of spectrum are compared to each other and also with the usual Hilbert space spectrum. In Section 3, we prove the spectral theorem for Hermitian elements of the class %?.From there follows the existence of spectral families of projections and thus an abundant supply of nested subspaces (in a NHS not every vector subspace is a nested subspace [4]). In the same class V, we study unitary representations of Abelian locally compact groups (Section 4) and we prove the Stone-Naimark-Ambrose-Godement (SNAG) theorem which asserts the existence of (nested) projection-valued measure on the character group. An example of application is the representation of an Abelian Lie group in the scale of Hilbert spaces associated with the infinitesimal generators by Nelson’s construction [5], [6]. In that NHS the representation operators belong to the class %. So, given a Hilbert space representation of an Abeiian Lie group, one can always build an NHS so that the representation can be extended to the whole NHS and decomposed by the SNAG theorem. 1 The material of this paper is a part of the author’s doctoralldissertation December 1975). (3611

(University

of Louvain,

FR. DEBACKER-MATHOT

362

But, if for some reasons, one does not want to build the Nelson scale and if one prefers to consider representations in another NHS (for instance, if one needs nuclearity) it is useful to extend the SNAG theorem to more general classes of representations than those in the class V. Using a result of R. Hirschfeld [7] (SNAG theorem in projective limits of Hilbert spaces) we prove in Section 5 the SNAG theorem for three different types of representations

:

a) Representations which are not everywhere defined in the NHS, not necessarily unitary but which are equicontinuous when restricted to some dense domain W (see below). In the central Hilbert space HO of the NHS, that type of representations may give rise to a representation of the group by unbounded densely defined operators. b) Representations which are the (nested) adjoint of those considered in a). They are everywhere defined in the NHS but need not be unitary. Such a representation may be realized in HO by very singular objects with a non-dense or even zero domain. c) Unitary representations for which the restriction to W is equicontinuous, (this is the intersection of the two preceeding classes); those induce on HO representations by unitary operators. So the SNAG theorem in that case is the extension to the whole NHS of the usual one in the central Hilbert space HO. For those three types of representations, the SNAG theorem no longer gives projectionvalued spectral measures (as in +?‘>but only operator-valued spectral measures on the character group and the properties of those spectral measures reflect the properties of the considered representation (equicontinuity, everywhere de6ned or not . ..). Finally, we show that those spectral measures are systems of imprimitivity for the respective representations. 1. Preliminaries In order to make this paper reasonably self-contained we will recall in this section some definitions and results from [l], [2] and [3]-[4] that will be needed in the sequel. 1.1. A partial

inner product

space is a complex

vector

space

V together

with:

1” A family { V,jr E Z} of vector subspaces of V which covers V. The family is ordered by inclusion and admits an order-reversing involution V,. c+ V;. 2” An

Hermitian

positive

definite

form ( * j * ) defined

exactly

on

such that the subspace

lJ V, x V; rs1

and

Y* = (7 V, separates vectors of V. Thus for each r E Z, the form TEI ( . [ . ) puts the pair (V,, V; ) in duality; in particular (V#, V) is a dual pair. The partial inner product space V is a nested Hilbert space (NHS) if it satisfies the additional conditions: 3” For each r E Z, V, is a Hilbert 4” There is a unique 5” The family

element

{Vl} is stable

space (denoted

by H, from now on),

0 E Z such that HO = H,. under

finite intersections.

SPECTRAL PROPERTIES IN A CLASS OF OPERATORS

363

When V is an NHS we shall denote it by HI as in [2]. HI can be considered as the algebraic inductive limit of the spaces H,, r E I w.r. to the canonical embeddings E,: H, + H,, (when H, E HS). Those embeddings are linear, bounded, injective, have dense range and satisfy the following two conditions [2] : a) For each r E I, E,, is the identity map on H,. b) If H, c H, c H,, then E,, = ErSES,. For each r E I, H, and H; are Hilbert spaces, dual of each other. So there exist unitary maps u;, . H, -+ H; and u,: H; -+ H, adjoint to each other (by the Riesz lemma). The partial inner product can be related to the individual scalar products of the H,.‘s by means of those maps u,;: = (%F_C9gr>r = cc 3w?r)i and it is independent

of r (provided the r.h.s. is well defined).

1.2. A map A from HI into itself is called an operator if: 10 A is defined on a subset 9 of HI of the form 9 = is a non-empty

U H, ED(A)

(where

D(A) c r

set of indices).

2O For each H, E 52, AIH, is a linear bounded map from H, into some H, . 3” A has no proper extension satisfying lo and 20, i.e. A is maximal. This definition particularizes to the NHS case that of an operator in a partial inner product space [3] and it contains the previous one of [2], where an operator is defined as an inductive limit of bounded operators between Hilbert spaces. The restriction of A to H,, mapping it continuously into H, will be denoted by A, and called the representative of A between H, and H, (thus a representative is always a bounded operator between Hilbert spaces). The set J(A) = ((r, s) E Ix Z(A, exists} is called the domain of A and is maximal by condition 3”. This implies, in particular, that an operator is fully determined by any one of its representatives. The set of representatives satisfies the following coherence conditions : A s’r’ = E,,sAwE,,., when H,, c H, and H, c H,,. For every operator A in HI, there exists a uniquely defined operator A*, called its adj,int, such that A** = A. The domain of A’$ is: J(A*) = {(i, 7) E 1x11 (r, S)E .7(A)} and A” is defined by: (A*);; = M~&G

= ‘(A,,)

(where + denotes the Hilbertian adjoint and t the transposed map). An operator A such that A = A* is called Hermitian. 1.3. Amongst the set of all operators of an NHS we have introduced in [l] three distinguished algebras that we shall need in the sequel: a) The *-algebra & consisting of those operators which map each H, continuously into itself, i.e..: ~4 = {Al Vr E I, (r, r) E J(A)}.

FR. DEBACKER-MATHOT

364

b) The subalgebra &I of .&:

where I] * IIr denotes the usual norm of B(H,), the set of bounded operators in H,. ~49is a Banach algebra. c) The subalgebra $F?of ~43:

which is a von Neumann algebra. Simultaneously with the NHS HI (which is the inductive limit of all the spaces H,, r E I) we introduced the direct sum A? of the H,‘s, which is itself a Hilbert space: x

= 0 H, = {f=

(.%~I

rel

fro KVr

and c IILIIB

rEI

< ~1.

We consider the following sets of operators in X: a) 9’ = {Pr: 3’ -+ H,I r E Z}, the set of the orthogonal projections of X on individual subspaces H, . b) B = {&I (r, s) E I x I, H, G H,), the set of all canonical embeddings of H, into H, (when H, E HJ both considered as subspaces of Z. c) The operator U defined by: ((Uf)JrEI

= (%Ek)rEl.

Now, consider an operator A acting in the NHS HI. If A E 39, we can identify it with the family (Arr)rpr of its (r , r) representatives and that family belongs to B(m. Thus the elements of ~?3and in particular those of %‘,can be identified with bounded operators acting in the Hilbert space [email protected] have then the following characterization of LL~and V [I]: L%? = (9%&)’ where, for A c B(Z), elements of A.

and

d = (8&u

{U>)’

A’ denotes the set of elements of B(Z)

commuting

with all

2. Spectrum of operators in d, L-3?, %2 In an abstract algebra A! (with unit element 1) the spectrum of an arbitrary element A E .M is defined as follows [8]: Sp(A,A)

= {jlEC/ A-21$&r’}

where 4-l = {B E .M] 3C E A such that BC = CB = 11. The complement spectrum P(A, A) = C\Sp(A, 4) is called the resolvent set of A.

of the

SPECTRAL PROPERTIES IN A CLASS OF OPERATORS

365

In that way we can define the spectrum and the resolvent set of an element of &, 9, ‘8 respectively. Since d 1 9 2 %, we have\ a priori: Sp(A, &) E Sp(A, 4?) E Sp(A,%),

VA E%.

Now, using the identification of W with bounded operators in the direct sum % = @ H,, reI

it has been shown in [l] that %?is a C+-subalgebra of B(X). (+ is the Hilbertian involution: ((A+)&1 = ((4X)&.) Since the spectrum of an element does not increase when one passes from a C+-algebra to a C+-subalgebra [S], we have: 2.1. PROPOSITION.For every A E %, Sp(A , %T)= Sp(A , B(X)).

n

Thus the spectrum of the operator A acting in the NHS HI = lim ind. H, is simply rel its usual Hilbert space spectrum in the Hilbert space # = @ H,. ror Consider now %?.It is also a subalgebra of B(%) but not a C+-subalgebra. .G@ is not stable under the Hilbertian involution +, but rather under the involution * which consists in taking the adjoint in the NHS-sense. So we can only deduce: Sp(A, B(z))

c Sp(A, @),

VA E:99.

But in fact, in that case also we can prove the equality. 2.2. PROPOSITION.For every A E 8, Sp(A, W) = Sp(A, B(iZ?)) , Proof: Let A E P(A, B(S)) so that the resolvent operator (A- 21)-lbelongs to B(X) and commutes with all elements of B(X) commuting with A [9]. But we have seen before that 9f = (9~8) so (A- 11)-lcommutes also with 9 and 8 and thus belongs to L%J.So P(A, B(X)) c P(A, a), i.e. Sp(A, ~3) c Sp(A, B(m). n Next we examine the relation between the spectrum of an arbitrary element A in d and the spectra of its representatives A,,. 2.3. PROPOSITION.For every A E:d, Sp(A, J$) = ylSp(A,,, B(HJ). Proof a) Let 3, E P(A, d). The resolvent operator R(A, A) s (A- 11)-l belongs to d. Thus for every r E I, ((A- Id)-'),, E B(H,) and is in fact the inverse operator of A,, - %., . So

(W

> 4)w

= W,,

,4

f or every

b) Conversely, let A E cr P(A,,, B(H,)).

r E I and thus 1 E rz P(A,,, B(H,>).

For every r E I, R(A,,, 3L) belongs to B(H,)

and is the inverse operator of A,,- IE,, = (A- Al),,. So the set {R(A,,, A)>lrE I) defines a unique operator acting in the NHS: R(A , A) = (A- ill)-'. So iiE P(A, d). With a) and b) we have proved P(A, d) = 2I P(A,,, B(H,)), i.e.

FR. DEBACKER-MATHOT

366

Remark : For A E ~3, part a) of the preceeding proof remains valid; as for b), R(A , A) can be defined as an element of d but it does not necessarily belong to @. (It belongs to g iff sup IIR(A,,, n)llr < CO.) rsl This section can be summarized in the following way: 2.4. THEOREM.Let A be an operator in the NHS Hz and &,a, ators defined above. Then: 1” If A E -Qz,Sp(A, ~0 = ,c;‘,S~(4.m

%?the algebras of oper-

Wfr)).

2” flf~E,Sp(A,zzZ)

ES~(A,@

= Sp(A,B(%)).

3O I~AEV,S~(A,JY)

_c Sp(A,@

= Sp(A,%‘) = S~(A,B(Z)).I

3. A spectral theorem for Hermitian elements of 9? 3.1. THEOREM. Let HI be an NHS and A = A* an Hermitian element of the algebra Gc: of operators defined above. There exists a spectral family of projections (E(n)/ il E R) in V such that

A = !;1 dE(A)

(1)

as a Stieltjes integral w.r. to the inductive limit topology on HI and also in the weak sense w.r. to. the partial inner product defined in Ht. Proof: Consider the realization of %? by bounded operators acting in the Hilbert direct sum %. By definition of 9, an Hermitian element (A = A*) is also a self-adjoint operator on Z(A = A+). By the usual spectral theorem in 2, there exists a family {E(A)} of projections in 3 such that (1) is valid in the strong operator topology of B(fl. Now, %Zbeing a von Neumann algebra [I], the spectral projections {E(n)) are still in Q? [IO] and thus are also projections in the NHS Hr. In each H,, the family {E(A),., = P,.E(1)Pr} is the spectral family associated with the representative A,., of A and the spectral decomposition of A,, holds in the strong operator topology of B(H,). As a consequence of this, the formula (1) holds on the whole HI, provided with the inductive limit topology of all the Hilbertian topologies of the H,.‘s. (f(“) + f in HI if there exists an HP such that f@) --f f in Hr). Moreover, by means of the representatives A,,, (1) is also valid in the weak sense w.r. to the partial inner product. Indeed we have for all f, h E HI such that (Af Ih) is defined: = (4,X-, u, h,_), = s ~d(E(&rfr,

ur; h& = i nd
n

3.2. The spectral family {E(I)} given by the preceeding theorem consists of orthogonal projections first in the Hilbert space X (on the subspace 0 E(L)),, Hr) and also in the IEI NHS HI (on the sub-NHS M,(A) = lim ind E(I),, H,). They are all totally orthogonal ret projections [I], i.e. E(L) = E(n)’ = E(I)* = E(A)+.

SPECTRAL PROPERTIES 1N A CLASS OF OPERATORS

367

In every NHS HI such that the algebra %?is nontrivial (see examples in [l] for sequences spaces and [l l] for scales of Hilbert spaces) we have thus obtained plenty ofprojections and of sub-NHS M,(A) satisfying: 1) For every 2, ii’ such that i < d’, MI(A) E M1(l’). 2) Al MI(J) = (01. 3) lJ M,(J) is dense in HI with the inductive limit topology. .kR

3.3. With help of Theorem 3.1 we can say a little more about the spectrum of Hermitian elements of 9. Let A = A* E:%. By the spectral theorem, there exists a family of spectral projections {E(Qj and families (EC&) associated to the representatives A,, for each r E I. Now, whether a point 3, E R belongs to the spectrum of an operator can be ascertained by the following criterion [9]: J E Sp(A,%‘) e For any E > 0, E(;I+e)-E(il-E)

# 0.

But this is equivalent to the following condition: for every r E Z and any E > 0, E(;i+e),, -E(1- E),, # 0, which in turn means that for every Ye I, il E Sp(A,, B(H,)). Finally, f or every YE I. In particular, the spectra of the representatives SP@ 3 +a = S&&r, w4)) A,, in the respective B(ZZ,) are all the same. Comparing this with Theorem 2.4, we see that for an Hermitian element of %, all notions of spectrum coi’ncide. VA = A* E%?Z:Sp(A, a> = Sp(A, S) = Sp(A,%) = SPM

B(K))

= Sp(A,,, B(H,))

(for every YE I).

4. SNAG theorem in ‘8. In a Hilbert space, the spectral theorem for self-adjoint operators can be extended to the classical theorem of Stone-Naimark-Ambrose-Godement (SNAG) which asserts that a strongly continuous unitary representation of a locally compact Abelian group can be decomposed into a direct integral over the character group by means of a projectionvalued measure. This theorem has been generalized by R. Hirschfeld [7] to the case of representations in a projective limit of Hilbert spaces. Here we extend it to representations in an NHS, unitary or not. We begin in this section with unitary representations in the class V. (Unitary operators in %?are an example of the unitary morphisms defined in [12].) 4.1. THEOREM. Let G be a locally compact Abelian group and U(G) u representation of G by unitary operators of the class V of an NHS HI such that the map g -+ U(g)f is continuorls when f E HF = n H, Vor the projective limit topology on HF). IFI

Then, there

368

FR. DEBACKER-MATHOT

exists a unique spectral measure E from the Bore1 sets of G (the character group of G) into the projections of GR,such that for any g E G : wz> = jmw)

(2)

as a Stieltjes integral w.r. to the inductive limit topology on Ht and in the weak sense w.r. to the partial inner product. G(g) denotes the value of the character $ at the point g.) Proof U(G) is a unitary representation (U(g)* = U(g-‘)) of G in %?(U(g)* = U(g)+) thus U(G) is also unitary in B(Z) and the representatives U(G),,are unitary in B(H,.) for every r E I. Our hypothesis of continuity of U(G) on HF implies the strong continuity of U(G),, on Hr for any r E I and the strong continuity of U(G) on Z’. So we may apply the usual SNAG theorem in &’ and in every Hr and obtain the decomposition (2) in the strong sense in % and the corresponding decomposition of U(G),, in the strong topology of Hr. The E(g) so obtained commute with all elements of B(Z) commuting with U(G), i.e. E@) E U(G)” c W’ = %2(since %?is a von Neumann algebra). Thus the spectral family {E(g)) consists in orthogonal projections of our NHS. Since each representative U(g),, is decomposed in the strong sense on H,, (2) is valid on HI in the inductive limit topology sense and once again if f, h E Ht are such that (U(g)f I h) is defined, one has: = (Ub9rr.L ushi),

= {B(g)d(E(ML

u,;h;)r = i g(g) d
n

4.2. EXAMPLE. Consider the following class of NHS: HI = lim indLZ(R, r(p)-‘&) strictly positive functions on R. (If one where I = {r(p)} is a family of measurable demands that r(p) be continuous, HI is then the NHS considered in [13]. If r(p)’ and Y@)-” are locally integrable, Ht = L:,,(R) and HI# then equals L:(R). If r(p)” and r(p)-’ are locally square integrable, Ht = L&(R) and HI# equals L:(R).) In HI consider the unitary representation of G = R given by: (U(x)f) (p) = e’““f(p)

for every f E Ht.

It is the extension to Ht of the usual representation of the same form on L2(R, dp). Obviously, that representation restricted to every H, is unitary and strongly continuous SO we may apply the preceeding theorem and obtain the decomposition

( W.fl h) = i eirXd(E(p) f j h) --m

for any f, h E Ht such that it is defined. The E(p) SO obtained are the multiplication operators by the characteristic functions x(- 03 ,p]. Thus that multiplication operator usually defined on L2(_R)extends now to the whole NHS and belongs to the class V.

SPECTRAL PROPERTIES IN A CLASS OF OPERATORS

369

4.3. EXAMPLE. The Nelson construcfion [5] [6]. To each continuous unitary representation of a Lie group G, in a Hilbert space H,, we can canonically associate an NHS, namely a scale of Hilbert spaces constructed as follows. Let U(G),, be a continuous unitary representation of G in Ho. The Lie algebra of G is represented by self-adjoint operators in Ho. Let X1 . . . X4 be a basis of the rep d

resentation

of the Lie algebra and d = l+ c Xf, the Nelson operator. d is essentially i=l

self-adjoint on a dense domain 2~ c H,, (e.g. the Girding domain). Consider its selfadjoint closure 0 and construct the scale of Hilbert spaces associated to x1/‘: For n > 0, let H, be the domain of p/Z in H,,. H, is a Hilbert space with the scalar product (f, h)” zz @‘@f, jf’“‘Zh)o. For n < 0, denote by H, the completion of H,, w.r. to the same scalar product. We obtain a scale of Hilbert spaces and lJ H. = H-, is a nested Hilbert space. In that scale the representation U(C&, of G can be extended into operators defined everywhere in H_, . It can be shown that they leave each H, invariant and are bounded in each H,,. Now for an Abeliun group, U(G) commutes with d; it follows that U(G),, is unitary in H,(Vn). Indeed: IlU(glfll, = 11~~2U(g)fllo = IIU(g)~~2fllo = il@“fll~ = Ilfll.. So we have finally U(g) E V for every g E G and we can apply our SNAG theorem to obtain a spectral family {E@)} c V. Thus in the case of an Abelian Lie group, the Nelson construction gives us always a NHS such that the considered representation belongs to %‘. It has also been shown that with such a construction the elements of the Lie algebra are everywhere defined operators on H_ m. In fact they belong to the class of the “good operators” [14], i.e. operators A with the following domain property: for every m, there exists an n such that the representative A,, from H. into H,,, exists and, conversely, for each n there exists an m such that A,, exists. 5. SNAG theorem for other classes of representations

In this section, we generalize the SNAG theorem to other classes of representations by operators (unitary or not) in a NHS. We start from a paper of R. Hirschfeld [7] in which the SNAG theorem is proved for equicontinuous representations in projective limits of Hilbert spaces. Applying this result to Hy which is dense in every H,, we shall see whether it can be extended to some of the H,‘s or even to H,, and what kind of representations of G and of spectral families can be obtained in that way. 5.1. Let G be a locally compact Abelian group and U a representation of G by operators in the NHS HI. We do not require anymore that the U(g) leave each HP invariant. We only need that the product U(gl) U(g2) be defined for all g, , g,, belonging to G and that it be equal to U(gIg2), (Those products are always defined if for every g E G, U(g)

FR. DEBACKER-MATHOT

370

is a good operator (Section 4.3). But this is a sufficient condition, not necessary.) Moreover, the representation can be unitary or not. We consider now the restriction of U(G) to the projective limit H?+and we demand: 1) U(G)HT c H,#. 2) U(G)l@ is a continuous representation of G in HF, i.e. the map: (g,f) m+U(g)f is continuous from Gx HP into HP2 (the latter being considered with the projective limit topology defined by the set of norms {II . /lrlr E I}). Moreover, for applying the result of Hirschfeld we need one additional hypothesis: 3) lJ(G)[HF is equicontinuous, i.e. for any q E I, there exist p E Z and a constant c > 0 such that for any f E HF and every g E G: ll.%)fll,

< cllfll,

(EQW.

Remark. In virtue of 2 this last condition is implied by 2) if G is compact. With those hypotheses there exists a uniquely defined spectral family of operators {E@)lg E 61, continuous on HF, such that:

This is the result of Hirschfeld [7]. The technical part of his proof consists in a construction of new Hilbert spaces (H,,) such that Hy is also the projective limit of those H4, and such that the representation on HT is the projective limit of unitary representations u(G),,,, in each H4,. In those spaces, the usual SNAG theorem holds and so we have spectral families (EC&,,,) in each Hq,. The spectral family {E@)) on H$+is then the projective limit E(g) = limproj E(g),,,, . The new spaces H4, are constructed as follows: Being an Abelian group, G possesses at least one invariant mean M (i.e. a linear normalized positive functional defined on the algebra L”(G), translation and inversion invariant under the group action). For any f, h E HF, the scalar product (U(g)f, U(g)h), in any H4 belongs to L”(G) by condition (EQUI), thus we may define new Hermitian, non-negative, forms ( * , * ),, on Hy by: (_f?A),, = M[(U(g)L

%+),I.

Dividing by the kernel of those forms if necessary, we obtain new scalar products and new Hilbert spaces H4, by completion. U(g)jHT can be extended onto each H4, to unitary operators U(g)4,4s. The topology defined on HF by the new norms (11. II,,} is equivalent to the previous one (given by {II * [I,} because of (EQUI) and the fact that M is inversion invariant [7]. That means for each q E I there exists p such that HP, E H4 and HP c H4,. The em2 This is equivalent

to the following

two conditions

[15], [16]:

a) For any f e i!ZF, the map g _+ U(g)f is continuous from G into H#. b) For every compact subset K E G, the set { U(g)l~? }g d K} is equicontinuous.

SPECTRAL PROPERTIES IN A CLASS OF OPERATORS

371

beddings are bounded and have dense range. In fact, if one considers (EQUI) with g = e one has also HP E H4. Let us go back to our representation U(G). The condition (EQUI) says that for every q E I, there exists p E I (independent of g) such that the representatives U(g),, exist for every g E G. For that p, we have in the corresponding HP,, the spectral family E(0),,,, (6 = Bore1 set of 6) which determines a projection-valued measure in HP,. Now for that p there exists r E I such that H, c HP, c H4 (apply (EQUI) again). So combining E(8),,,, with the bounded embeddings of Hr into HP. and of HP* into H, we can define a spectral family {E(O),) of b ounded operators from Hr into H4. Since we have also Hr E H,, c Hg, we can summarize the preceeding discussion as follows: For every q E I there exists an r E I such that U(g),, exists for every g E G and satisfies the relation

This relation is the continuation to the (q, r)-representatives of the corresponding relation (3) on HI#. As such, it holds also as a relation between operators in the NHS, in the weak sense w.r. to the partial inner product. The fact that {E(O),,,,} are orthogonal projection in H,, implies the usual properties of a spectral measure for the operators in the NHS defined by the representatives {E(8),) so that finally we have proved the following: 5.2. THEOREM. Let G be a locally compact Abelian group, 6 its character group and U(G) a representation of G by operators of an NHS HI such that U(G) restricted to H,# is a continuous representation of G in H,#. Then there exists a unique family {E(O)/ 0 = Borel sets of G> of operators acting in HI, equicontinuous on H,# and such that: a) E(G) = lHr. b> E(U 0")= cm9

63,...0,a countabIe sequence of disjoint Bore1 sets of G).

c) Thenmap 0 -: =
= ~B(g)d.

FR. DEBACKER-MATHOT

312

So we obtain automatically the SNAG theorem for the adjoint with the spectral family {E(e)*}. S’mce the initial representation

representation U(G)* has representatives

U(G),, for every 4 E 1, the adjoint representation possesses representatives (V(G)*)-,4 for every 4 E I (see the definition of the adjoint (1.2)). That means that U(G)* is everywhere defined on the NHS. The same is true of the E(0)*. As for the E(8), they satisfy E* = (E*)2 but are not projections

on sub-NHS

either.

We have thus proved:

5.4. THEOREM. Let G be a locally compact Abelian group, 6 its character group and V(G) a representation of G by operators acting in an NHS HI such that the adjoint representation V(G)* is equicontintious on HF (this implies that V(G) is defined everywhere in HI). Then there exists a unique family (F(O)] 8 = Borel sets of e} of operators in HI, everywhere defined, satisfying the properties a) to d) of Theorem 5.2 and such that for every g E G: V(g) = i g(g) dFW c; in the weak sense w.r. to the partial

inner product

of HI.

n

In that theorem and in the preceeding one, the representations we have considered need not leave each H, invariant. In particular, they need not possess representatives in HO (the central Hilbert space of the NHS) that means they can be representations of G by unbounded operators in HO. The spectral decomposition we got by our SNAG theorems, is thus valid for unbounded representations in a Hilbert space considered as operators in a suitable NHS. In fact, for the first theorem we have seen that the representation operators are always defined on a dense domain HP E H,, (by the condition EQUI). Considered as operators in HO they can be unbounded but they are defined on a dense domain. For Theorem 5.4 the representation operators are bounded from HO into some H4 bigger than HO. Their domain as operators in HO can be very small or even zero. They can be very singular objects. 5.5. Consider now a unitary representation U(G) in the NHS HI. (Unitarity by itself is a very weak requirement in a NHS. In particular the existence of (fl h) does not imply the existence of (Uf j Uh) even if U is defined everywhere [12].) We shall see that in this case, the SNAG theorem is just the extension of the same theorem in Ho. First we remark that a unitary representation in the NHS (U(g)* = U(g-‘)) which is equicontinuous on Hz# is automatically unitary in the central Hilbert space Ho ((U(g),,&,, = U(g-l),,,). Indeed, by the construction described above (5.1) U(&oP exists and is unitary on HO, which is the completion of Hy w.r. to the new scalar product (.> . >,,,. The latter, however, coincides on HP with the original ( * , .h:

U-3h)w =

M[(U(g)f,

= MKf SO Hoe coincides

with H,,and

Wg)h),]

= MKUtg)f

IUkW)l

Ih)l =
V(G),.,,

is simply the set of

(0, 0)-representatives of U(G).

SPECTRAL PROPERTIES IN A CLASS OF OPERATORS

313

Now a unitary representation U(G) lies in the intersection of the two classes of representations considered before. Putting Theorem 5.2 and Theorem !5.4 together, we can prove : 5.6. THEOREM. Let G be a locally compact Abelian group, G its character group and U(G) a representation of G by unitary operators of an NHS HI which is equicontinuous on HF. Then there exists in HO a unique spectral family of orthogonal projections {E(0)OOIO Bore1 sets of G> which can be extended in a spectral family (E(e)> of operators in H,, zerywhere defined, satisfying E(8) = E(e)* = E(8)* and such that for every g E G:

U(g) = 1 g(g)dE(g) c*

(4

in the weak sense w.r. to the partial inner product of Ht. Proof The existence of the spectral family {E(O)} of operators in HI such that (4) holds, follows from Theorem 5.2. As U(G) is unitary, the representatives U(G)OO and &(O),, exist (because HO = Ho,) and they are related by the usual SNAG theorem. The E(B),, are usual Hilbert spaces projections in H,,, thus E(Q-,, = E(B)&, = (E(O),,)&-, = (E(8)*)Oo; therefore E(e) = E(O)* = E(B)* as operators in Ht. Finally, the E(8) are defined everywhere on Ht by Theorem 5.4. n Remarks. 1. Also in this case, the E(8) are not projections on sub-NHS. They do not leave each Hr invariant except Ho. In fact they are projections on vector subspaces of HI which are not NHS by themselves. 2. In the three SNAG theorems of this section, we have obtained an operator-valued spectral measure. The only case we had really a (nested-) projection-valued spectral measure was for representations in the class W (Theorem 4.1). 3. The SNAG theorem in %’is of course a particular case of Theorem 5.6 but as we did in 4.1 it can be proved in a totally independent way which does not necessitate the construction of new Hilbert spaces as described in 5.1. 5.7. Let G be a locally compact group, H a normal, closed, Abelian subgroup and Z? its character group. The action of G on & is given by: g: h^-

fi,: h,(h) E h^(ghg-l),

g E G, h E H, h’ E 8.

Consider now a representation U of G by operators in an NHS Ht which is a continuous representation of G in HF (with the projective limit topology, which implies that the operators U(g)* are defined everywhere) and such that U(H)jn~ is equicontinuous. By the SNAG Theorem 5.2 there exists a unique operator-valued spectral measure E defined on I? such that: U(h) = i h(h) dE(I;), 2 (weak sense in HI).

Vh E H

374

FR. DEBACKER-MATHOT

Compute

now for any g E G: U(g-‘hg)

(weak

= 1 h^(g-%g)dE(hj

(5)

= s I;,- i (/z)dE(^h) = s h^(h)dE(j;,)

sense) and (U(K1)

U(h) WgYl Q = (WA) %)J-I

%?‘)“k)

= s &+Wr)

for any k E HI, any j”E HF. Comparing Stieltjes transform:
Ug)fl

LW1)*k)

(5) and (6) we have by the uniqueness

L(e)W9fI

k) = (W-9

of Fourier-

k)

for any fg HF, k E HI, 0 = Bore1 set of fi, 8, = {h^,lh^~ 0}. Since that equality holds for any k in H,, fe HF we have: E(8,) = U(g-l)E(B) U(g) as operators in the NHS HI. That means: E is an imprimitivit~~ system with basis fi for the representation U [17]. 5.8. EXAMPLES. Take fir G the n-dimensional HI = lim indL2(R”, r(pP2d”p) w b ere r(p) and positive

functions

on 5”. In that

NHS

Euclidean group En = R”lxl SO(n) and r@)-’ are arbitrary continuous, strictly

consider

the following

representation:

(p) - = e’?X_f(R-lp) - . For the translation part, we have exactly the n-dimensional generalization so that U(?, 1) leaves each L2(r@)) invariant and: (U(Z) R)f)

of Example

4.2,

U(X) 1) = s e’!: LIE(p) where for a Bore1 set 0 of R”, E(8) is the multiplication operator by the characteristic function x(e). NOW, the rotation part does not leave each L2(r(p)) invariant. U(g, R) is defined everywhere

in HI and

all the Hilbert

spaces

maps

LZ(r(p_)) unitarily

L2(rCp)) are mixed.

LZ(r,(p))

onto

The imprimitivity

where property

r@)

= r(Kp>. So

gives here:

u(g-1) x(e) u(g) = x(0,) where

g = @, R) and 8, = {Rp+~l p E 01.

This equality

is valid

everywhere& u(g-9,,R

In the same NHS we may rotations in a given 2-plane we get:

HI. In terms

x(e)PR,R u(g)l.,,

of representatives

we have:

= x(03,, .

apply Theorem 5.6 to each subgroup SO(2) consisting of R”. So if R = R, E SO(2) is a rotation by an angle

U(0, R$ = +f eimqE(m) , m----m

of 9,

SPECTRAL

PROPERTIES

IN A CLASS OF OPERATORS

375

in the weak sense w.r. to the partial inner product of HI, which yields for the operator E(m) :

(E(mlf)@I = f

2

e- im9f((R; ‘E)

for every f E HI. In this last example we can see explicitly what are the new Hilbert spaces constructed in 5.1. We defined the new Hermitian forms as follows :

Since the group considered is compact, we may interchange the two integrals and so we obtain as a new space a weighted L2-space again with a weight function r’ given by the SO(2) average of the original weight r: 2n

1 1 _& rl(P)Z = o 2~ r(R@’ s -

’

Acknowledgments

I would like to thank Dr J. P. Antoine for his help and his encouragements out the realization of this work and also for improving the manuscript.

through-

REFERENCES

PI F. Debacker-Mathot:

121A. Grossmann:

Comm.

Comm. Math.

Math. Phys.

Phys.

42 (1975),

2 (1966),

183.

1.

[31 J. P. Antoine, A. Grossmann: J. Funct. Anal. 23 (1976), 369; 23 (1976), 379. 141 J. P. Antoine, A. Grossmann: preprint Marseille 75/P. 711. [51 E. Nelson: J. Funct. Anal. 11 (1972), 211. Physics, A. 0. Barut, ed., D. Reidel, Dordrecht 1973. bl B. Nagel: in: Studies in Mathematical [71 R. Hirschfeld: Math. Ann. 194 (1971), 180. Gauthier-Villars, Paris 1969. @I J. Dixmier: Les C*-algsbres et /curs reprbentutions, York 1966. [91 T. Kato: Perturbation theory for Iinear operators, Springer Verlag, Berlin-Heidelberg-New York 1970. DOI S. Sakai: C*-algebras and W*-algebras, Springer Verlag, Berlin-Heidelberg-New Opkrateurs duns les espaces d produit interne partiel, ThBse, Louvain 1975. [ill F. Debacker-Mathot: WI A. Grossmann: Comm. Math. Phys. 4 (1967), 190. Comm. Math. Phys. 4 (1967), 203. H31 A. Grossmann: J. R. Fontaine: in Proceedings of the 3’* International Co/1141 J. P. Antoine, F. Debacker-Mathot, loquium on Group Theoretical Methods in Physics, Marseille, vol. 1 (1974), 138. r151 F. Bruhat: Bull. Sot. Math. France 84 (1956), 97. iI61 G. Warner: Harmonic analysis on semi-simple Lie groups Z, Springer Verlag, Berlin-HeidelbergNew York 1972. ofgroupsandquantum mechanics, Benjamin, New York 1968. 1171 G. Mackey : Inducedrepresentations