A fundamental approach to the generalized eigenvalue problem for self-adjoint operators

problem for self-adjoint operators

Citation for published version (APA):

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

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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

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 L

2(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

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 m_A (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 ••..

,P_n 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

RP

t R • Then to almost all A = O'I, ... ,A_n) in the joint spectrum a(PI .... ,P) with multiplicity m

A there exist m_A vectors in X, say e

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

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, .•. ,P

n) 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

H

Cl

(L

2

(R) )

and is closable in

H(J.(L

2(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.

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),

i f

~(I)(supp(~(I»

n supp(/2») =

~(2)(supp(~(l»

n

supp(~(2»)=O.

Now let (PI, ••. ,P_n) 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, ••• ,P

n) is defined as follows.

1.1. Definition

The n-set (PI, .•• ,P

n) is of uniform mUltiplicity rn if there exist m finite nonnegative equivalent Borel measures

~(I)

, •••

,~(m)

on lRn and a unitary operator U : X + L

2(lR

n_,

_{~(I» ~}

_...

_~

_L

2(lRn ,

~(m»

such that each self adjoint operator U P

t U , t = I, .•• ,n., equals m-times mUltiplication by the function n

t : 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

(8₁, •.. ,8

The set of all eigentuples of 8

1, •.• ,8n may be called the joint spectrum of (8₁, ••• ,8_n) denoted by O'(8

1, ••• ,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, ... ,P

n) 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 cor-m

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

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

assertion,

2. Column finite matrices 00

U

m=1

supp(~ ) . m

Let

L1, ••• ,L

_n 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 _SE:N V _{rrdl, ... ,n}} :N :

(The set {1,2, ... ,n}:N consists of all mappings from:N into {1,2, ... ,nl-; D(L_rr(I)L_rr(2) .•. L_rr(s» denotes the domain of the operator between (

».

For the commuting n-set (P1, •.. ,P

n) the joint C""-domain is given by the intersection

C""(P

1

)

n ••• n

C""(P

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

2.2. Theorem

""

Suppose that

C (L1, ... ,L

n) 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.

Proof.

• 00

S1nce

C (LI, •.. ,L

n) 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,··· ,L_n)· We introduce the orthonormal basis (vk)nEm as fQl1OW~.

Set vI

=

u_l• Then there esists an orthonormal set {v₂' ••• ,v } L vI with n

l

n

l ~ n+2 such that the span

There exists an orthonormal set {v _n I""'v } L {vI' ••• ,v } with

l+ n2 nl

n

2 ~ 2n+3 such that

In general, for k E

m

having produced {vl, ••• ,v

nk}, ~ ~ k(n+l) + I, there exists an orthonormal set {v I""'v } L {vI' ••• ,v } with

n_k+ n_k+₁ ~

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

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

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

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

for each

o

The n-set (PI, .•• ,P

n) has a dense joint ~-domain. So by Theorem 2.2, there

exists an orthonormal basis (vk)kEN such that the operators PI""'P_n have a column finite (and hence row finite) matrix representation with respect to this basis.

We define the positive Hilbert-Schmidt operator R by RV

k = Pkvk where (Pk)kE~ can be any fixed i

-I

I

operator R PR, R , R, .. I, •.• ,n is well-defined on the span <{v

k 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 P

t

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, ••• ,T

n be n mutually commuting self-adjoint operators in the

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

operator R such that each operator RTtR -I , 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 L 2(M,v) = ~ L2(lR n , 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

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 N_f 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) \ N

f• Moreover, f

=

f a.e. (lJ). (For convenience we note that f

=

(f

l ,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 P

k >

0,

k E

1N

Further, let

U

denote a unitary operator

00 n

from X onto p~1 L2 (B , V_p), 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 (Uv_k\ . 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

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 _k=l

L

Pk ~k,p(x) vkand

e(p)(r) = \I (B(x,r»-l

I

P

k(

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 n_t denotes the function nt(x)

=

x

t' 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)\N_p}> is contained in D(RP R-1). Further we have for all x E supp(\I)\ N

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)v_k =

L (f n

(t

~

~p(~

r) (Uvk)p)dvp)v_k 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 L

2(lR n

.v

p), Hence

U*~(p)

_{B(x. r)} E D(P_~ o)

and

e~p)(r) RU*~~~~.r)

E RD(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

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, ... ,P

n) 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 11

00,111,112" " on lR

n such

m

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

m),

m j~1

Furthermore, the n-set

(P1, •.. ,P

n) acts invariantly in each Hilbert summand Xm and is unitarily equivalent to the n-set (QI, ... ,Qn)' Here Q~ restricted

m

n to ~ L

2(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, ... ,P

n) has a column finite matrix representation with respect to (vk)kE~ • Let R denote the positive Hilbert-Schmidt opera-tor defined by RVk

=

Pkv_k where (Pk)kE~ is a fixed sequence in Jl._{2 with}

-I

P

k > 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

(m) ( (m) (m) (m)) m n

(Uv_k)

=

(Uv_k) I , (Uv_k) 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,]

(m) is in the domain of R P R-I e . x,] R. R P R- I _e (m) _. x e(m~ R. x,] X.J m

We observe that for each m

=

"".1.2.. • • the set N

=

U N . is a m j=1 m,]

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

m

4.3. Theorem

Let

(P

I

.P

2 •••••

P

n) 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 there 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

The proof of this theorem ~s a compilation of the results given in the be-ginning of this section.

Remark

o

o

Let 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

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

d . -TH I 2 d2

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~

+ixsinh 't. Each A €

a:

is an eigenvalue

of this operator. Its 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.

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)

V

k E R-1(X) is a simultaneous generalized

eigen-X,J k=l k,J

vector of the n-set (PI, ••• ,P

n) 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 P

t in R-1(X) by

Further,

P~

F =

R-I(RP~R-l)RF,

F E

Dom(P~).

I t is clear that

P

_t 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)RF

s -+ 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""'P

n lie at the 'periphery' of the Hilbert space X.

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, ••• ,P

n) 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 a l , ••• , 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, ••. ,P

n have closed -I

extensions in

R

(X). Also, it follows that to almost each eigentuple x = (xl'···,xn) E: o(P1, •.• ,P

n) with multiplicity mx there are mx simultaneous generalized eigenvectors E

(m~

X,J

-I

supp(lJ ) . m and that a(P1, ... ,P

n) \ ( 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 L

2([-I,IJ) has generalized eigenfunctions which are hyperfunctions on [-I, 1 J, i f 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}!

References

[IJ Brown, A., A version of multiplicity theory, in 'Topics in operator theory', Math Surveys, nr.13, AMS., 1974.

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