Invariance of the analyticity domains of self-adjoint operators subjected to perturbations

Invariance of the analyticity domains of self-adjoint operators

subjected to perturbations

Citation for published version (APA):

Eijndhoven, van, S. J. L. (1982). Invariance of the analyticity domains of self-adjoint operators subjected to perturbations. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8210). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1982

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum 1982-10 May 1982

INVARIANCE OF THE ANALYTICITY DOMAINS OF SELF-ADJOINT OPERATORS SUBJECTED.TO PERTURBATIONS

by

S.J.L. van Eijndhoven

Eindhoven University of Technology

Department of Mathematics and Compo Science P.O. Box 513, 5600 MB Eindhoven

INVARIANCE OF THE ANALYTICITY DOMAINS OF SELF-ADJOINT

OPERATORS SUBJECTED TO PERTURBATIONS

by

S.J.L. van Eijndhoven

Abstract

The analyticity domain ~(A) of a self-adjoint operator A in a Hilbert space

X

is dense. If A is positive, ~(A) is equal to the test space

Sx,A

of DeGraaf's distribution theory. Here the question is investi-gated for which Hilbert spaces Y and self-adjoint, positive operators B in Y.

Especially the case that 8 is derived from A by a perturbation will have our attention.

'AMS Classifications 46FI0 47B99 47DOS

The author was supported by a grant from the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).

An

element

n

in a Hilbert space X is called an analytic vector for au operator R in X if there are constants a.b > 0 with

The linear subspace of all analytic vectors of R is called the .analyti ~

city domain of R and is denoted byCOO(R). Nelson [1] observed that a symmetric operator P in X has a dense analyticity domain if and only if

P

is essentially self-adjoint.

An

operator r in X is said to ~lytically dominate R, if

eW(r)

c

eW(R).

If .r has a dense analyticity domain and if R is symmetric, analytical dominance of r implies that R is essentially self-adjoint. Faris in [2], ch.VI,gives conditions on the commutator of rand

R

which lead to ana-lytical dominance.

In the sequel

A

will denote a positive, self-adjoint operator in X. Nelson 0] showed that a vector

n

€ X is in CW(A) if and only if there

are t > 0 and

w

€ X with

n =

e-tAw. Instead of the notation

cW(A)

we shall employ the terminology of De Graaf [3J and denote the analy-ticity domain of

A

by

SX,A.

In [3], De Graaf takes

SX,A

as the test function space in his theory of generalized functions_ It would be nice if we could classify the spaces

SX,A-

We remark that each Hilbert space X and each positive self-adjoint operator

A

in X generates a test func~ tion space

SX,A-To this end, in this paper the problem will be investigated whether there exist conditions on Hilbert spaces Y and positive self-adjoint

3

-operators

B

such that S c S

X,A -Y,B

This investigation yields results with respect to the so-called classi-fication problem, which can be translated in analytic vector terminology, of course.

As

an

immediate consequence of the theory in [3] we derive

(1) Theorem

Let G be a positive self-adjoint operator in X which generates a group t + etG, t

~

It, of continuous linear mappings on Sx,A. Then

Sx,A c SX,G • Proof

Let a,S> O. Then following [3], eaGe -SA is a bounded operator on X. Let f

~

SX,A" Then there exists t > 0 such that etAf

~

x. So we have for each a > 0

and thus f ~ SX,G"

(2) Theorem

zG Let G be an operator in X which generates a holomorphic group z ~ e

z ~ ¢, of continuous linear mappings on SX,A" Then

,

or, equivalently,

A

analytically dominates G. Proof

zG -aA

For every a > 0, the operators e e , z € C, are bounded, and the

operator valued function

zG -aA

Z.H> e e

is holamorphic. So there exists M > 0 such that II eZG. e-aA

n

_< M _,

'I I

_{z -} 1 _,

uniformly. Further n!

2'11'i

1i -aA M

hence IIG e l l s

2iT

n!

zG -aA

e e

n+l dz

z

aA

Let f €

SX,A'

Then there is a > 0 such that e f € X and

Therefore f €cf>(G) •

We note that in [4] there are given conditions on operators G in X in order that G generates a holomorphic group of continuous linear mappings

We now start with the main part of this paper and discuss the problem

o

in case 8 is obtained from

A

by a perturbation. The results are contained in two lemmas.

5

-(3) Lemma

Let

P

be an operator in

X

with D(P) ~

SX,A

which satisfies the following conditions:

(i) There exists a Hilbert space Y such that

SX,A

C Y and

A + P

is a

positive,essentially self-adjoint operator in Y. (ii) There exist d > 0 and k > 0 such that for all a > 0

Then

SX,A

c

Sy,A+P .•

Proof

Let t > 0, and let 0 < T < t be fixed. We want to estimate the operator

TA n -tA

norm of e (A + P) e • Therefore we factor as follows

t~

exp((n;J

T +

i

ttl

(l + PA-1)

e"P[ -( n:

j T +

i0

A

1 ·

0 .. A exp[ -

~

{t - T)A] •

This factorization yields the following estimate

J

--(t-T)A

:5 II Ae n , lin

~

n! (t - T)-n

n;1 (1

+

d

exp

(kT

n~j

+

k~j))

-11 n 1

s n.! (t - T) (l + d) exp(2 llk(t + T» •

Now it is obvious that for all 0', 0 < 0' < (1+d)-l(t - T) exp(-kt),

where we take 0' (A+P) e ₌₌

_1:

n==O n 0' n

=-r

(A + p) • n.

We remark that the operator norms are taken with respect to B(X), the algebra of bounded operators on X.

Let f E: SX,A. Then there exist t > 0 and T, 0 < T. < t, such that

tA

e f E: X. Now f can be written as

f == e -0' (A+P) -TA (TA a (A+P) e e e e-tA) e tA f.

S• -TA{ TA 0' (A+P) -tA) tA f S Y h 1 f S

1nce e e e e e E: x,A C , t e resu t € Y,A+P

is obtained, completing the proof.

o

A corollary of Lenma (3)colltains conditions on P such that A analytical-ly domiuates the perturbed operator

A

+ P.

(4) Corollary

Let P be an operator in X with D(P) ~ SX,A for which there are constants d > 0 and k > 0 such that

~ 7

-then ~,

Ace

Ol(A + p). ( c: X) •

Proof

As in the proof of Lemma (3) the following estimate can be derived

Bence for f € SX,A t

where a = (1 + d) t -1 exp(fct), and t is chosen sUfficiently small. 0

Remark: In Lemma (3) and Corollary (4) we could as well demand

....

As a preparation to the following lemma the function space B + (ll) is introduced

For W € B+(JR) , the operator log ~(A) is self-adjoint, and it has a

(5)

II [10g1jl(A)]n e -tA 1\

for all n ~ E and all t > O. To this end, let t > 0, n ~ E and take

T, 0 < T < t. Then there exists MT > 1 such that

-TX

1jI(x) e < MT ' x ~ 0 • Hence

(10g1jl(x»n _. S (log M + XT)n • _T From the latter inequality we derive

sup [(logt/{(x»n e -tx] s sup [(log MT + XT)n e-tx]

x~O x~o

So the following estimate holds true

-TX where 0 < T < t and M

=

sup (1jI(x) e ) .

T _x->0

(6) Lemma

Let P be a well-defined operator in X with D(P) ~ SX,A which satisfies

(i) There exists a Hilbert space Y with SX,A C Y such,that A + P is

a positive, essentially self-adjoint operator inY, and Sy,A+P C X.

9

-for all (X e lB.. (Here we take the operator norm. with respect to X.)

Then \,A+P c Sx,A • Proof

-t(A+P) With the aid of Duhamel1_{s iteration principle the operator e ,}

t > 0, can be expressed by a series

where -tA - e

,

P (t) n t' t

-(-I)nJ

J

n-1 -tA( trAp ':"'tlA) .. - .

~.(·

....

~Ap ~t~A)dt

d

• • •• ,. e e e • •• e E!' J • •• to.

o

0

0. - 1,2, •••

We shall prove that for every t> 0 the series

i

lIe

hA

P (t) II

n=O n

converges. To this end we factor the integrand as follows

'. e-tA ( ttA e P e -tt A) ••• e (triAp -:tne A )

=

t - .... -A

=

e 2 0. - t A (t.-n-le)A -(t.-n-jt)A II (e

"21.i

10gl/1(A»(e J 20. (1 0 gl/1(A»-lpe, J 20. ).

Straightforward computation yields t ,--A lie'? t}A -tlk t _n Ao - t A 11 (e Pe ) ••• (e' Pe )lIs

Since (by 5) for all T, 0 < T < t ,

and since

i-

j _n-j n (t.- t)A ' -(t.- t)A

n

lie J n (logtP(A»-IP_e .1 n i l S j-O , n

kl

t.-n-j t

I

S 'IT d e J n j-O , S dn _e 2ukt _, we. derive

If we take 0 < T < d-l e-2kt , t en t e serles satlS Y h h • . f'

11

-and bence

~t

(A+P)

II < ~ e ,1lCI •

Now let

6

~ Y, and let t > O. Then for 0 < \ < t

-t(A+P)

1 -t~A

( ILA ,

-~(A+P» -(t-~){A+P)l

e o . . e e e e 0

and

-(t-T) (A+P)t X

e n ~ ,

by assumption. This yields tbe result

-t(A+P)

1 S

e U €

x.A.

o

A combination of Lemma (3) and Lemma (6) yields the following theorem.

(7) Theorem

Let

P

be a well-defined operator in

X,

D(P) ~

s.x,A'

which satisfies the following conditions

(i) There exists a Hilbert space Y such that

SX,A

C Y,

A

+

P

is a

posi-tive, essentially self-adjoint operator in Y and

Sy,A+P

eX • ....

(ii) There exists ~ € B+(lR) , d > 0 and k > 0 such that

for all P ~ lR.

Then

12

-An application of the theory of this paper can be found in [5] where it is proved· that

for all a,S> -1. Here we take

d2 2 1 1

A . - - + x - (2y+ 1) - - ,.

y dx2 x dx

where y > -1.

References

[1] Nelson, E, Analytic vectors,

Ann.

Math •• 70 (1959), 572-615. [2] Faris, W.G. Self-adjoint operators. Lecture Notes in Math., 433

(1975), Springer.

(3] Gruf, J. De, A theory of generalized functions based on holomorphic semigroups, Tn-report, 79-WSK-02, Eindhoven, University of Technology, 1979.

[4] Eijnhoven, S.J.L., Van, Ph.D.thesis, to appear in 1983.

[5] Eijndhoven, S.J.L., Van, On Hankel invariant distribution spaces,

EUT-report, 82-WSK-Ol, Eindhoven, University of Technology, 1982.