A fundamental approach to the generalized eigenvalueproblem for self-adjoint operatorsCitation for published version (APA):Eijndhoven, van, S. J. L., & Graaf, de, J. (1984). A fundamental approach to the generalized eigenvalue problemfor self-adjoint operators. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol.8401). Eindhoven: Technische Hogeschool Eindhoven.

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum 84-01

January 1984

A FUNDAMENTAL APPROACH TO .THE GENERALIZED

EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

by

S.J.L. van Eijndhoven and J. de Graaf

Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven The Netherlands

A FUNDAMENTAL APPROACH TO THE GENERALIZED

EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

by

S.J.L. van Eijndhoven and J. de Graaf

Abstract

The generalized eigenvalue problem for an arbitrary self-adjoint operator

is solved in a Gelfand tripel consisting of three Hilbert spaces. The

proof is based on a measure theoretical version of the Sobolev lemma, and

the multiplicity theory for self-adjoint operators. As an application

we mention necessary and sufficient conditions such that a self-adjoint

operator in L2

(R) has (generalized) eigenfunctions which are tempered

distributions.

AMS Classifications: 46FIO, 47A70, 46E35.

Contents

Introduction

Commutative multiplicity theory

Column finite matrices

A measure theoretical Sobolev lemma with applications

The main result

Generalized eigenfunctions in the dual of a countable Hilbert space

- 2 -

Introduction

A natural problem in a theory of generalized functions is the so-called

generalized eigenvalue problem. A simplified version of this problem can

be formulated as follows. Consider the Gelfand tripel ~ c X c~. in which

X is a Hilbert space. ~ is a test space and ~ the space of generalized

functions. Let Pbe a self-adjoint operator in X. and let A be a number

in the spectrum of P with multiplicity mAo The question is whether there

exist mA (generalized) eigenfunctions in ~.

Such a problem has been studied by Gelfand and Shilov (cf.[8]) in the

framework of countable Hilbert spaces and also by the authors of the present

paper in the setting of analyticity spaces and trajectory spaces (cf.[4]).

Here we discuss a more general approach. In a separate section we show how

the theory of this paper fits into the functional analytic set-up of a

special type of countable Hilbert spaces. In fact. this set-up is a gener-

alization of the theory of tempered distributions.

The present paper is built up in such a way that each section contains

one fundamental topic and finally those separate topics culminate in the

proof of the main result in Section 4. This result can be loosely formu-

lated as follows.

Let there be given n commuting self-adjoint operators PI.PZ ••.. ,Pn in a

separable Hilbert space X. Then there exists a positive self-adjoint

. -I Hilbert-Schmidt operator R . such that the operators R P R , 2. z: I, .•• ,n,

2. . -I ·

are closable. Denote their closures by RPt

R • Then to almost all A =

O' I, ... ,An) in the joint spectrum a(PI .... ,P) with multiplicity mA there

exist mA vectors in X, say eA

I, ... ,e, such that , 1\, m

A

- 3 -

where j = I, ... ,m and i = I, ... ,n •

The proof of the above stated theorem involves three important ingredients,

discussed in three separate sections. In the first section the commutative

multiplicity theory for self-adjoint operators is discussed. By this theo-

ry the spectrum of a self-adjoint operator is split into components each

of which are of uniform multiplicity. In the second section we show that

there exists an orthonormal basis in X such that each operator of the

commuting n-set (PI, .•. ,Pn

) has a column finite matrix with respect to

this basis. Section 3 contains a general Sobolev lemma. The greater part

of this section follows from our paper [5J, but also some new results

are derived.

As already remarked the last section contains a discussion of generalized

eigenfunctions. It is worthwhile to mention one of its consequences here.

Let T be a self-adjoint operator in L2(~) . Suppose the operator

d2

2 H-(J.T HCl is closable for some Cl > 1/2 where H = - -- + x • Then T has a com-dx2

plete set of generalized eigenfunctions in HCl (L2(R) ) and is closable in

H(J. (L2

(lR) ) c S' (R) •

1. Commutative multiplicity theory

This section gives the commutative muTtiplicity theory for a finite number

of commuting self-adjoint operators. In the case of bounded operators the

proof can be found in [IJ or [9J. The unbounded case is a trivial gener-

alization.

Throughout this paper n E N will be taken fixed.

- 4 -

Let ~ denote a finite nonnegative Borel measure on mn. Then the sup-

port of ~, notation supp(~), is the complement of the largest open set

in lRn

of ~-measure zero. It can be shown that

supp(~) ~(B(x,r» > o}

where B(x, r) denotes the closed ba1l {y E lRn III x - y II ~ r,}. For the _ n

proof of this statement, see [3J, p. 153. As usual, two Borel measures

(I) (2) (I) (2) ~ and ~ are called equivalent, ~ - ~ , if for all Borel setsN

~(I)(N) = 0 iff ~(2)(N) = O. The measures JI) and ~(2) are disjoint,

~(I).l~(2), if ~(I)(supp(~(I» n supp(/2») = ~(2)(supp(~(l» n supp(~(2»)=O.

Now let (PI, ••. ,Pn) be an n-set of commuting self adjoint operators in a

separable Hilbert space X, i.e. their spectral projections mutually com-

mute. The notion of uniform multiplicity for (PI, ••• ,Pn) is defined as

follows.

1.1. Def inition

The n-set (PI, .•• ,Pn) is of uniform mUltiplicity rn if there exist m finite

nonnegative equivalent Borel measures ~(I) , ••• ,~(m) on lRn and a unitary

operator U : X + L2

(lRn , ~(I» ~ ... ~ L2

(lRn , ~(m» such that each self

adjoint operator U Pt U , t = I, .•• ,n., equals m-times mUltiplication by

the function nt

: x ~ xt

in this direct sum.

In a finite dimensional Hilbert space E each commuting n-set of self-

adjoint operators (8 1,82, ••• ,8n) has a complete set of simultaneous eigen­

vectors. An element ,\ E lRn is called an eigentuple of the n-set

(8 1, •.. ,8n) if there exists a vector e,\ E E such that

- 5 -

The set of all eigentuples of 8 1, •.• ,8n may be called the joint spectrum

of (8 1, ••• ,8n) denoted by O'(81

, ••• ,8n). In order to list all eigentuples

in a well-ordered manner one can list all eigentuples of multiplicity

one, two, etc. In fact, this is precisely the outcome of the following

theorem.

I .2. Theorem (Commutative multiplicity theorem).

The Hilbert space X can be split into a (countable) direct sum

such that the following assertions are valid

The n-set (PI, ... ,Pn) restricted to Xm, m = 00,1,2, ••• , acts invariantly in

X and has uniform multiplicity m. m

The equivalence classes <~ > of finite nonnegative Borel measures cor­m

responding to each X (Cf. Definition 1.1) are mutually disjoint, i.e. m

for all ~1 E <~1> and ~2 E <~2> we have ~I 1 ~2' etc.

In this paper we always consider the following standard splitting of X

wi th respect to (P l' ... , P n)' As in the previous theorem X .. X.., ~ XI ED X2 $ ..• •

In each equivalence class <~m> we choose a fixed measure ~ • By U we de-m m n n

note the unitary operator from Xm onto L2CR '~m) ~ ... ~ L2(lR '~m)

(m-times). Then U (P n ~X )U* equals m-times multiplication by the function m '" m m

U) Let (fA )AER denote the respective spectral resolution of the identity

corresponding to P£,o Then we define the joint spectrum as follows

- 6 -

where 0 denotes the null operator. We mention the following simple

assertion,

2. Column finite matrices

00

U m=1

supp(~ ) . m

Let L1, ••• ,Ln be n densily defined linear operators in X. Then we define

"" their joint C -domain as follows.

2. I. Definition

{ V E X I V V :N : SE:N rrdl, ... ,n}

(The set {1,2, ... ,n}:N consists of all mappings from:N into {1,2, ... ,nl-;

D(L rr (I)L rr (2) .•. Lrr (s» denotes the domain of the operator between ( ».

For the commuting n-set (P1, • .. ,Pn) the joint C""-domain is given by the

i ntersection C""(P 1) n ••• n C""(Pn

). Hence C""(P1, •.. ,Pn) is dense in X.

2. 2 . Theorem

"" Suppose that C (L1, ... ,Ln

) is a dense subspace of X. Then there exists

an orthonormal basis in X such that each operator L£, t = I, ••• ,n, has

a column finite matrix representation with respect to this basis.

- 7 -

Proof.

• 00

S1nce C (LI, •.. ,Ln

) is dense in X and since X is separable, there exists

an orthonormal basis (u 'k)kEN in X which is contained in Coo(L I ,··· ,Ln )·

We introduce the orthonormal basis (vk)nEm as fQl1OW~.

Set vI = u l • Then there esists an orthonormal set {v2' ••• ,v } L vI with nl

n l ~ n+2 such that the span

There exists an orthonormal set {v I""'v } L {vI' ••• ,v } with nl+ n2 n l

n2 ~ 2n+3 such that

In general, for k E m having produced {vl, ••• ,vnk

}, ~ ~ k(n+l) + I, there

exists an orthonormal set {v I""'v } L {vI' ••• ,v } with nk + nk+1 ~

~+I ~ (k+I)(n+l) + I such that

We thus obtain inductively an orthonormal basis (vk)kE:N' The basis (Vk\EN

i s complete since ~ E <VI""'V~>' Furthermore, by construction

i = I, ... ,n, the matrix «Lv.,v . )) . . lO.T is column finite. J 1 1,J E ....

for each

o

The n-set (PI, .•• ,Pn) has a dense joint ~-domain. So by Theorem 2.2, there

exists an orthonormal basis (vk)kEN such that the operators PI""'Pn have

a column finite (and hence row finite) matrix representation with respect

to this basis.

We define the positive Hilbert-Schmidt operator R by RVk = Pkvk where

(Pk)kE ~ can be any fixed i 2-sequence with positive components. Then the

- 8 -

-I I operator R PR, R , R, .. I, •.• ,n is well-defined on the span <{vk k c }ij> •

-I -I Let t .. 1, ••• , n. For the domain of R P t R we take D(R P t R ) - R(D(P R,» •

Since R is injective and self-adjoint, D(R Pt R-I ) is dense in X. Further,

-I for all f E D(R P t R ) and all k E ~ we have

Hence, the adjoint of R P R- I is densely defined. Recapitulated t

2.3. Theorem

Let TI, ••• ,Tn

be n mutually commuting self-adjoint operators in the

separable Hilbert space X. Then there exists a positive Hilbert-Schmidt

operator R -I

such that each operator RTtR , t .. I, ••• ,n, is densely de-

fined and closable.

3. A measure theoretical Sobolev lemma with applications

Our paper [5J contains a generalization of the well-known Sobolev lemma.

Here we use the results of that paper in the following concrete case: the

measure space M is the disj oint countable union of copies E,n of lRn , p

co co

Le. M = u lRn; the nonnegative Borel measure on M is given by v = ~ v p . . .. p=1 P p=l

where each v p

is a finite nonnegative Borel measure on Rn. We note that co

L2

(M,v) = ~ L2

(lRn ,v). p=1 p

On M we introduce the metric d as follows

d«x"p),(y,p» = Iix-yli n lR

Now, let B«x,p),r) denote the closed ball in M with centre (x,p) E lRn p

and radius r > 0, and let B(x,r) denote the closed ball in lRn with centre

- 9 -

x and radius r > O. Then from [6], Theorem 2.8.18, we obtain

3. I. Theorem

Let f : M + IE be integrable on bounded Borel sets. Then there exists a null

set Nf such that the limit

f(x;p) = lim v(B«x,p),r»-1 f fdv r.j.O

B«x, p), r)

exists for all (xIP) E supp(v) \ Nf • Moreover, f = f a.e. (lJ).

(For convenience we note that f = (fl,f2, ••• ) and f(x,p) =

over, lim V(B«x,p),r»-1 f fdv = lim v (B(x,r»-I r.j.O B«x,p) ,r) r.j.O p

f (x). More-p .

f f dv .) B(x,r) p p

Let X denote a separable Hilbert space and R a positive Hilbert-Schmidt

operator on X. Let (vk)kE~ be the orthonormal basis of eigenvectors of

R with eigenvalues Pk > 0, k E 1N Further, let U denote a unitary operator

00 n from X onto p~1 L2 (B , Vp), The series

follows that L"" P~ IUvkl2 E L1(M,v). K=I 00

Following Theorem 3.1 there exists a v-null set N = U N (disjoint union), p"l p

i.e. each set N is a v -null set, with the following properties. p p

There exist functions !P k.p E (Uvk\. k E ~, P E :N. such that

3.2. i)

H)

!Pk,p(x)

2 l!Pk,p(x)I

lim v (B(x ,r»-I qO p

-I lim v (B(x, r» NO p

f CUvk) p dvp'

B(x,r)

3.2. iii) v V PE:N XESUpp(\I) \N P P

00

- 10 -

These conditions onU N lead to the proof of the following result. P p=l

See [5J, Lemma 2.

3.2. Theorem co

Let P E :N and let x E supp(\I ) \ N • Set e(p) = P P x

L Pk ~k,p(x) vkand k=l

e(p)(r) = \I (B(x,r»-l I Pk

( f (Uvk

) d\l )vk

' r > O. x P k=l B(x,r) P p

Then we have

e(p) e(p)(r) are members of X , x ' x

lim ~e(p) - e(p)(r)" = o. o x x X r+

Let Q£, t = I, ... ,n, denote the multiplication operator, formally defined

by

We recall that nt denotes the function nt(x) = xt ' X E 1Rn. It is clear that 00

Q£ is a self~adjoint operator in ® L2(~n, \I ). So p=1 P

the operator

is self-adjoint in X. The follQwing lemma says that the . vectors

-I candidate eigenvectors of the operator R P t R •

3.3. Lenuna

Let t = 1, ••• ,n. Then the linear span <{e~p) (r) I r> 0, p E :N, X.E sUPP(\lp)\Np }>

is contained in D(RP R- 1). Further we have for all x E supp(\I)\ N t . P P

II -

Proof

Let p E N and let x E SUpp(V ) \ N • Then for each r > 0 the series P P

00 00

1. ( f (Uvk)p dVp)vk = L (f n (t~~p(~ r) (Uvk)p)dvp)vk represents the k=1 B(x.r) k"l lR •

elements U*~~~~.r) in X where ~~~~.r) denotes the characteristic function

of the ball B(x.r) as an element of L2

(lRn .vp), Hence U*~(p) E D(P o )

B(x. r) ~

and e~p)(r) RU*~~~~.r) E R D(PR,) = D(RPR,R-I

) for all R, = I ..... n .

Next we prove that lim (RPoR-I)e(P)(r ) -x e(p)(r)=O. Then by Theorem 3.2 r+O '- x R, x

the proof is complete. To this end. observe that for all r > 0

= kII Pk(Vp(B(x.r»-1 f (YR, - XR,)~k'P(Y)dVp(Y»)Vk' B(x.r)

We estimate as follows

II (R P R-I)e (p) (r) - x e (p) (r) 112 ~ JI. x R, x

~ (Vp(B(x.r»-1 f lyR, - xR,12 dVp(Y»)'

B(x.r)

The first factor in the last expression tends to zero as r + O. Because

"" of assumption 3.2.iii) the second factor is bounded by L P~I~k (x) 12 + I

k= I .p for sufficiently small r > O. o

- 12 -

4. The main result

In the introduction we have given a non-rigorous formulation of the main

theorem of the present paper. The results of the previous sections culmi-

nate in the proof of this theorem.

Again, let (P1, ... ,Pn) denote an n-set of commuting self-adjoint operators

in X. Following Section 1 there exists a standard splitting

and disjoint finite nonnegative Borel measures 1100

,11 1,112"" on lRn

such m

n that each X is unitarily equivalent to the direct sum ~ L2(~ ,11m), m j~1

Furthermore, the n-set (P1, •.. ,Pn) acts invariantly in each Hilbert summand

Xm and is unitarily equivalent to the n-set (QI, ... ,Qn)' Here Q~ restricted m

n to ~ L2(lR, 11 ) is the operator of m-times multiplication by the func-

• 1 m J= tion llJl.'

Following Section 2, there exists an orthonormal basis (vk)kE~ such that

each operator in (P1, ... ,Pn) has a column finite matrix representation

with respect to (vk)kE ~ • Let R denote the positive Hilbert-Schmidt opera­

tor defined by RVk = Pkvk where (Pk)kE~ is a fixed sequence in Jl. 2 with

-I Pk > O. Then the operators R P JI. R are closable in X for each JI. = 1, ••• , n.

-I We denote the respective closures by R P JI. R •

Following Section 3 there exist Ilull-sets N ., j &:: I, .••• ,m, with respect m,J

to 11m' m = 00,1,2, ••• such that the limit

(m) (jIk • (x)

,J lim 11 (B(x,r»-I

m nO J

B(x,r)

- 13 -

(m) ( (m) (m) (m)) m n (Uvk ) = (Uvk ) I , (Uvk ) 2 "",(Uvk)m E ~ L2(~ '~m) •

j=1

In addition, for all m = 00,1,2, ••• and I ~ j < m+l, the series

(m) e . =

X,]

and

f B(x,r)

with r > 0, X E supp(jJ ) \ N ., converges in X. m m,]

By Theorem 3.2 and Lemma 3.3 it follows

4.1. Lemma

Let m = 00,1,2, ••• and let j E ~ with I ~ j < m+l. Then for all

X E 8 upp (~ ) \ N • m m,]

lim (lIe(m~(r) - e(m~ II) = ° r+O X,] X,]

For each R. I, ••. ,n and all r > 0, e(m~(r) E D(RP n R- I ), and X,] '"

Since the operator R P R-I

1.S closable, we obtain from the previous R.

lemma

4.2. Corollary

o

Le t m = co, I , 2, • •• and 1 e t ] E "N, ~ ] < m+l . Then for all X E supp(jJ )\N . m m,]

and for all £ = I, ... ,n

- 14 -

(m) is in the domain of R P R- I e . x,] R.

R P R- I (m) e . x e(m~ R. x,] X.J

m

We observe that for each m = "".1.2.. • • the set N = UN. is a m j=1 m,]

~ -null set. Now we are in a position to formulate the main theorem. m

4.3. Theorem

Let (PI.P2 ••••• Pn) be an n-set of commuting self-adjoint operators. Then

there exists a positive Hilbert-Schmidt operator such that the operators

R P R- I are closable. R.

be the standard splitting of X. and ~""'~1'~2""

be the corresponding multiplicity measures. Let m = 00.1.2 ••••• Then there

is a ~ -null m

ther e exist

(m) e .

X.J

Proof

set N with the following property: for all x € supp(~ )\N m m m

m independent vectors e(m~ € X. I ~ j < m+l. satisfying x,]

(m) = Xn e ., t = I, ... , n.

'" X.J

Th e proof of this theorem ~s a compilation of the results given in the be-

gi nning of this section.

Remark

o

o

Le t P be a self-adjoint operator .in X and let R be a positive Hilbert-Schmidt

-I -I operator such that R P R is closable. The spectrum of R P R can be larger

t han the spectrum of P. An interesting example is the following. In ~2(~)

d . -TH I 2 d 2 take P i-d ,then a (P) = lR. FurthertakeR=e with H = -(x --+ I) and

x . 2 dx2

T > O. Then RPR- I = i cosh T d~ +i x sinh 't. Each A € a: is an eigenvalue

of this operator. I ts eigenvector is x» exp ( - ~A x --21 (tanh Thlh which is an cos :r

L2(lR)-function. This continuous set of eigenvectors is closely related to

the so-called coherent states.

- 15 -

4.4. Corollary

Let R-1(X) denote the completion of X with respect to the inner-product

(u,v)_l a (Ru,Rv)X' Employ the notation of the previous theorem

Each operator P~ is closable in R-1(X)

co

R-le(m~ = I ~(m~(x) Vk

E R-1(X) is a simultaneous generalized eigen­X,J k=l k,J

vector of the n-set (PI, ••• ,Pn) with eigentuple x = (xI' ••• ,xn) where

P~ denotes the R-1(X)-closure of P~, ~ = I, ••• ,n.

Proof.

We only prove the closability of Pt. We define the domain Dom(Pt ) of the

operator Pt in R-1(X) by

Further, P~ F = R-I(RP~R-l)RF, F E Dom(P~). It is clear that Pt extends

-I I in R (X). We prove that P~ is a closed operator in R- (X). To this end

-I let (F) .... , be a sequence in R (X) with S SE JL~

lim F = F E R-I(X) and lim P F = G E R-1(X) • s s

s~ s~

Then RFs -+ RF and (R P~ R-I)RFs -+ RG as s -+ 00. Hence F E Dom(P~) and

G = P~F, because R P~ R- 1 is closed in X.

Remark

Since for the eigenvalues of R any positive t 2-sequence can be taken it

is clear that the improper eigenvectors of the operators PI""'Pn lie at

the 'periphery' of the Hilbert space X.

o

- 16 -

5. Generalized eigenfunctions in the dual of a countable Hilbert space

An application of the previous sections is in the field of generalized

eigenfunctions and countable Hilbert spaces.

As in Section 4, (P1, ••• ,Pn) is an n-set of commuting self-adjoint opera­

tors in X, and R denotes a positive Hilbert-Schmidt operator with the

-I property that the operators R Pi R , i al, ••• , n, are closable in X. The

countable Hilbert space ~X R is defined by ,

~X,R =

00

n s=1

where in each RS (X) the norm is defined by II QJ lis = II R-s !pIIX' QJ E: RS (X) •

With its natural topology, the space ~X,R is a nuclear Frechet space.

The strong dual of ~X,R can be represented by the inductive limit

IjIX,R = U SE:JN

-s s Here R (X) denotes the completion of X with respect to the norm IIR wll,

w ( X. On IjIX,R the inductive limit topology is imposed. In [2] a set of

seminorms has been produced which generates a locally convex topology equi-

valent to the inductive limit topology. The Gelfand tripel ~ ReX c IjI R X, X,

places the theory of tempered

-I work. (Take X = L2 (:IR) and R

distributions in

d2 2 =--2 + x .)

dx

a functional analytic frame-

From Corollary (4.4) it follows that the operators P1, ••. ,Pn

have closed

-I extensions in R (X). Also, it follows that to almost each eigentuple x =

(xl'···,xn) E: o(P1, •.• ,Pn) with multiplicity mx there are mx simultaneous

generalized eigenvectors E (m~ X,J

-I j = I, ... ,mx inR (X) C IjIx,R' i.e.

- 17 -

supp(lJ ) .m

and that a(P1, ... ,Pn) \ ( U m=1

(Cf. Section I).

supp (lJ » ·has lJ -measure zero, m = "",1,2, ••• m m

Remark

The generalized eigenvectors E(m~ as constructed in this paper can be X,J

embedded in a trajectory space [7J. There they constitute a Dirac basis.

For these concepts and for a rigorous foundation of the genuine Dirac-

formalism, see [4J and [3J.

Remark

A self-adjoint operator P in L2([-I,IJ) has generalized eigenfunctions

which are hyperfunctions on [-I, 1 J, if e -tL P e tL is densely defined and

closable for each t > O. Here L denotes the positive square root of the Legendre

{ d 2 d}! operator, i.e. L = - dx(I-x )dx ' Cf. [lOJ.

- 18 -

References

[IJ Brown, A., A version of multiplicity theory, in 'Topics in operator

theory', Math Surveys, nr.13, AMS., 1974.

[2] Eijndhoven, S.J.L. van, A theory of generalized functions based on

one parameter groups of unbounded self-adjoint operators, TH­

Report 81-WSK-03, Eindhoven University of Technology, 1981.

[3] Eijndhoven, S.J.L. van, Analyticity spaces, trajectory spaces and

linear mappings between them, Ph.D. Thesis, Eindhoven Universi­

ty of Technology, 1983.

[4] Eijndhoven, S.J.L. van, and J. de Graaf, A mathematical interpretation

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