Asymptotic variations of the Fuglede theorem

Asymptotic variations of the Fuglede theorem

Citation for published version (APA):

Eijndhoven, van, S. J. L. (1982). Asymptotic variations of the Fuglede theorem. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8218). 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 1982-18

ASYMPTOTIC VARIATIONS OF THE FUGLEDE THEOREM by

S.J.L. van Eijndhoven

ASYMPTOTIC VARIATIONS OF THE FUGLEDE THEOREM by

S.J.L. van Eijndhoven

Dept. of Mathematics & Computing Science Eindhoven University of Technology

P.O. Box 51} 5600 MB Eindhoven The Netherlands

Abstract

By Fugledets Theorem an operator B which commutes with a normal operator N also commutes with f(N) if f is a continuous function on the spectrum of N. In this paper we consider this theorem with "commuting" replaced by "almost commuting". We show that there are conditions for an operator

topology L such that f(N)B - Bf(N) is !-small as soon as NB - BN is

suf-ficiently !-small and II B II < K for some K > O.

1980 Mathematics Subject Classifications: Primary 47B47, 46C10, Secondary 47B15. Keywords and phrases: Normal operator, Fuglede Theorem, algebraic topology.

- 1

-Introduction

In the algebra S(H) of bounded operators on a Hilbert space a normal

J

*

operator N and an operator B commute if and only if Nand B commute, this is the well-known theorem of Fuglede (2). Rosenblum (5) gave an

elegant proof of this theorem, thus inspiring Moore (3) to the following

extension.

FoI' all, e:: > 0 and K > 0 theI'e exists <5 > 0 suah that II NB - BN II < <5 and

*

*

11 B II < K

impZy

II N B - BN II < e:

Rogers proved that the norm topology in Moore's theorem may be replaced by the strong and weak operator topology. We shall prove that each ope-rator topology that satisfies rather natural conditions may replace the norm topology. To prove this, we use techniques different from those of Moore and Rogers.

*

*

The main problem in all Fuglede-like theorems is how to relate N B - BN

to NB - BN. Or, equivalently, if we consider

H(') I\. :== e iAN -iAN Be , (A E <1:)

and

*

*

G(1\ ') := e -iAN Be-iAN I (A E 0:)

2

-To obtain such a relation, note first that G'(O) is represented by the Cauchy type integral

1

G' (0)

=

with integration along I~I = r in the positive sense, and then that

e i(IlN

* -

+~N)

Now, H(~) can be obtained from Ht by integration from 0 to ~ along a

straight line segment. Hence

-G' (a) = 1

21TT

-i(IlN +~N)

J

*

-I

~

=r ..:;..e

--~

2"--- { J:Ht(A)dA +

H(O)}ei(jlN*+~N)djl.

Formula (*), together with the observation that the operators e

are unitary, is the central argument in the proof of Theorem 3.

*

-i(~N +jlN)

The main difference between the approach of Moore and Rogers and our's is

in the estimation of H(~). They use the power series of the exponential

func-tion together with the identity k-1

NkB - BNk

=

L

Nj(NB _ BN)Nk- j - 1 , j=O

while we express H(~) as an integral which is then estimated in a simple way.

Results

The norm topology, the weak operator topology and the strong operato~

topo-logy are algebraic topologies on B(H) in our terminology. Here is the

defi-nition.

Definition 1. A topology T on B(H) is called algebraic if

(1.1) T is coarser than the norm topology,

(1.2) B(H) with topology T is a locally convex, topological vector space,

•

- 3

-The following lemma is an immediate consequence of the first two of these conditions. The proof depends on a simple compactness argument.

Lemma 2: Let ~ be algebraic, and Q E • a convex open neighbourhood of O.

Let T > 0 and let f: CO,T] + B(H) be norm continuous with f(t) E Q for all t E [O,T]. Then

Analogously to results of Moore and Rogers we now prove

Theorem 3. Let. be algebraic, let N E B(H) be normal and let (B ) c B(H)

*

CI.

*

be a norm bounded net with NB - B N + 0 in .-sense. Then N B - B N + 0

CI. CI. CI. CI.

in T-sense.

Proof. We may as well assume that liN II

=

1. Let Q E ~ be a convex and circled

open neighbourhood of 0, and

such that {A E B(H) IIiAIl <

!}

r

let K > 0 with II B If < K for all CI.. Fix r > 0

1 CI.

c

2

Q. Put sr

=

{A

E ~I IAI s; r} and C

=

{A E ~I

fAI

= r}. Define

r

*

-U(ll) := e-i(llN +llN) (ll E ~). Since

*

Ii U (ll) B U {ll} II s; II B II < K I we

CI. CI. have for all CI.

(i)

iAN -iAN

*

Let (A/ll) E S xC. The mapping A + U(ll)e Ae U(ll) is ~-continuous by

r r

(1.3). So there exists an open neighbourhood of 0, Q, I saYI with

A,ll iAn -iAN

*

1

U(ll)e Ae U(ll) E

4

Q , (A E Q, ) •

A,ll

For each A with II A II S; 1, let F be the mapping on Sr x C that sends

(A,ll) into U(ll) eiANAe-iANU(ll} *.AThe mappings FA are norm

~ontinuous

on

Sr x C_{r and even uniformly equicontinuous. Since Sr x}

is a finite set E := {(A.,llj,)

Ij

= l, ..• k; 1,

=

l, .•• m}

J .

that for each (A,ll) E S x C there exists (A.,ll") E E

r r J '" (IIAII S; 1). C is compact, there r in Sr x C_r such with

•

4 -Now take a 1 such that NB - B N E n Q (a ~ 0. 1) • a ·a (Aj'll.Q.> Ajll1.Q. Then F[N,B aJ(A.j ,l1t ) 1 (Aj,ll.Q.)

E

'4

Q for every E E as soon as

[N,B ] _a

=

NB _a - B N). Let (A,l1) _a E S _r x C r and a 2: a 1 • Since

we can find (A.,ll.) E E with J N . K < -4r So 1 1 1 + F[N B J(A.,llt) E

'4

Q +

'4

Q

= 2

Q. , a J a ~ a 1 (with II [N,B JII< 2K, a

Since a "does not depend on the choice of (A,ll) E S xC; we have shown

r r

(ii) U(l1)e· i AN (NB - B N)e -iAN U(l1)

*

E -1 Q

a a 2

for every (A,l1) E S x C and a ~ a 1•

r r

Let H denote the function H defined in the introduction, in which B is a

replaced by B " Then relation (*) gives a

Applying (i),(ii) and lemma 2 we obtain

-( i i i )

l

_{llU(l1)H ' (A)U(l1) *dA} + U(l1)B U(l1) * E r Q

o

a a

for a ~ a

1 and 11 E Cr "

Finally, applying (iii) and, again, lemma 2 we conclude that

1 2

5

-In the next theorem N*is replaced by feN) with f E C(cr(N», i.e. f is

a continuous complex function on the spectrum cr(N) of N. Theorem 3 is

a special case of Theorem 4; the former is essential as a preparation

to the proof of the latter, though.,

Theorem 4. Let T,N and (B ) satisfy the conditions of Theorem 3, and

CI.

let f E C(cr(N». Then f(N)B - B feN) + 0 in T-sense.

CI. CI.

Proof. Let Q E T be a convex and circled neighbourhood of O. We divide

the proof into two steps.

Step one. Assume first that f is a polynomial p, i-j

c .. A A J.J

say_ For each i , j E ::N with i + j ::> m, we have

and

Consider the mappings

and '1': A +

I

i+j::>m . j-l *j-k-l *k c. . NJ.

(I

N A N ) J.J k=O

6

-Since, is algebraic, ~ and ~ are ,-continuous. Hence there exist convex

and circled open neighbourhoods of 0 I Q ¢ € " QljI € " say, with

According to Theorem 3 there is an open neighbourhood Q'jI of 0 such that

such that

* * ...

N B - B N _{~ Q'jI} whenever NBa - BaN € Q'jI. Now take a_l

a a NB - B N € Qqr

n

Q~ for all a :?: a l • Then a a

*

*

=

'¥(N B - B N ) + ~(NB - B N) € Q a a a a as soon as a 2 ale

Step two. Let f € C(cr(N», and let K > 0 be a bound for IIB

a II. Fix p >~O such that

{A € B(H) IIiAIl < p} c

Since cr(N) is compact, there is a polynomial p(A,X) with

sup If(A) - p(A,A)

I

< p. Sinceflf(N) - p(N,N*) II < p and IIBall < K AEcr(N)

(i) (f(n) - p(N,N »Ba

*

€

3

1 Q and Ba(f(N) - p(N,N

*

» €

3

1 Q.

According to the first part of the proof there exists a

1 such that

(ii) p(N,N )B

*

'I< 1 a - Ba p(N,N ) E

3

Q, Combining (i) and (ii), and taking a :?: 0.

1 our proof is complete.

Remark 1. In Theorem 3 and Theorem 4 we may take the (ultra-)weak or

(ultra-)strong operator topology of B(H). In each case it is necessary

to require the net to be bounded. This may be shown by a construction

.

'

7

-of Bastians and Harrison (cf. the pro-of -of the last part -of Theorem 3

in [1 J) •

Remark 2. Let N

1,N2 E B(H) be normal. Analogously to relation (*) in

part I, we have

*

*

1 NlB - BN2 = _21Ti with H (A) iAN 1 -iAN₂ = e Be t (A E 0:),

and further more

Hence we may prove the following Putnam-like version of Theorem 4.

Theorem 5. Let, be algebraic, let N

1,N2 E B(H) be normal, let f € C(cr(N

1) u cr(N2»,and let (Ba) c B(H) be a normbounded net with

N1 Ba - BaN2 + 0 in T-sense. Then f(N

1)Ba - Baf(N2) + 0 in ,-sense. References

1. J.J. Bastians, K.J. Harrison, Subnormal weighted shifts and

asymp-totic properties of normal operators. Proc. Am. Math. Soc., 42 (1974),

475-479.

2. B. Fuglede, A commutativity theorem for normal eperators. Proc. Nat.

Acad. U.S.A., 36 (1950), 35-40.

3. R. Moore, An asymptotic Fuglede theorem. Proc. Am. Math. Soc., 50

8

-4. D.O. Rogers, On FUglede's theorem and operator topologies,

Proc. Amer. Math. Soc. 75 (1979), 32-36.

5. M~ Rosenblum, On a theorem of Fug1ede and Putnam. J. London Math. Soc., 33 (1958),376-377.