On almost-commuting operators

On almost-commuting operators

Citation for published version (APA):

Ackermans, S. T. M., Eijndhoven, van, S. J. L., & Martens, F. J. L. (1983). On almost-commuting operators. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8302). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1983

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Department of Mathematics and Computing Science Memorandum 1983-02 March 1983 ON ALMOST-COMMUTING OPERATORS by S.T.M. Ackermans S.J.L. van Eijndhoven F.J.L. Martens

Eindhoven University of Technology

P.O. Box 513, 5600 M.B. Eindhoven

Abstract ON ALMOST-COMMUTING OPERATORS by S.T.M. Ackermans S.J.L. van Eijndhoven F.J.L. Martens

If

A

commutes with the commutator

[A,B]

then following the Kleinecke-Shirokov theorem

[A,B]

is quasi-nilpotent. Using the Fuglede theorem we shall show that for normal operators

A

the stronger conclusion [A,S]

=

0

will follow. We shall also derive asymptotic extensions of both the Fuglede theorem and of our new version of the

Kleinecke-Shi-rokov theorem in terms of operator topologies of a rather general type.

AMS Classifications 47B15; 47B47;. 47H99

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

o·

Introduction

In the Banach algebra B(H) of bounded linear operators on a Hilbert space H the following two theorems are well-known. See [IJ, [2J, [5J, [7J and [3], [8J.

(0.1) Theorem (Fuglede-PuLnam-Furuta)

*

Let N₁,N₂ € B(H) be subnormal and let B € B(H). Then

N

1

B-BN

2

=

0

im-plies

N~B

- BN;

=

O.

(0.2) Theorem (Kleinecke-Shirokov)

(0.3)

Let A,B € B(H). Then A(AB - BA) - (AB - BA)A

=

0 implies that the

spec-tral radius r(AB - BA) = 0, i.e. AB- BA is quasi-nilpotent.

We shall employ a notation [;, J* with the following meaning:_ Let

Now let A₁,A

2 be self~adjoint .operators on H. Then all B € B(H) with [A.,[A.,B]

J

=

0 satisfy the equation

J J

* *

iAAIB - iAA2

e e

=

B+iA[A.,BJ ,>"€JR.

J

*

Moreover, II e iM 1 Be -iAA2n

=

II B II, and thus we find from (0.3) that II [A.,B] II s 21>"1-1 I1BI! for all A € JR\{oL Hence [A.,BJ

=

O. SO

in-J .J< . J

*

stead of (0.2) we have proved a stronger version for self-adjoint ope-rators.

3

-(0.4) Theorem Let A

1,A2 € B(R) be self-adjoint and let

B

€ B(R). If [Aj,CAj,BJ*J* then also

[A.,B] ... O.

J

*

Now let N

1,N2 be normal operators. Then we can write Nj = Cj + iOj

where

C.,O.

are self-adjoint operators with

C.O . ... V.C.,

j

=

1,2.

J J J J J J

... 0

By (0.1) we find for each

B

EO: B(R) with

CN

j , CNj ,BJ*J*=

0

1=b.at each of the

* *

[N* CN*

BJ J . also

O.

operatora [Nj,CNj,BJ*J*, [Nj,[Nj,B]*J* and j' j' * * ~s Taking combinations we find that both [Cj,[Cj,BJ*J*

=

0

and

[O.,[O.,B]

J =

O.

Then by (0.4) [C.,BJ* ...

0

and

[OJ.,B]* ... O.

Thus the

J J

*

*

J

following theorem has been proved.

(0.5) Theorem

Let Nl,~J2 € B(R) be normal. I f CNj,CNj,BJ*J* ... 0 then [Nj,BJ* :: O. - --' .

In this formulation, similarity between the Kleinecke-Shirokov theorem for normal operators and the Fuglede theorem has become obvious. Moore, Rogers and Furuta have proved asymptotic versions of (0.1) in

terms of the uniform, the strong and the weak operator topology on B(R).

Cf.[2],[4J and [6]. In Section 2 below, we shall give an asymptotic version of (0.1) in the terms of the operator topologies on B(R) which make it a locally convex algebra and which are coarser than the uniform topology. See Section 1 below. With a slight abuse of terminology we call such topologies: algebra topologies. Next, applying this new asymp-totic Fuglede theorem, we obtain an asympasymp-totic version of (0.5).

1. Auxiliary results

The uniform, the strong and the weak operator topologies are algebra topologies on B(H). A formal definition of an algebra topology can be given as follows.

(1.1) Definition

(I.I.1)

(1.1.2)

(1.1.3)

A topology L on B(H) is called an algebra topology, if

L is coarser than the uniform operator topology.

B(H) with topology L is a locally convex topological vector space.

The mapping X + EXC from B(H) into B(H) is T-continuous for each pair

B,C

E B(H).

Before asymptotic extensions of (0.1) and (0.5) are proved, we give the following lemmas.

(I.2) Lemma

Let T be an algebra topology on B(H), and let V € T be a convex open neighbourhood of

O.

Let T > 0, and let the operator valued mapping F: [O,T] + B(H) be continuous with respect to the usual topology on the real interval [O,T] and the uniform topology on" B(H). Let a: [O,T] + [O,~)

be continuous. If F(t) € a(t)V for all t € [O,T], then

T T

f

F(t)dt

a

€

(f

a(t)dt)V.

a

The proof of the previous lemma depends on a simple compactness argu-ment and is omitted.

5

-(1.3) Lemma

Let T be an algebra topology on B(H). Let B,C be operator valued map-pings from ~n into B(H) which are continuous with respect to the uni-form operator topology on B(H) and the Euclidean topology on ~n. Let

D c ~n be compact, let K > 0 and let W € T be a neighbourhood of

O.

Then there exists a T-open neighbourhood

V

of 0 such that B(s)XC(s) €

€ W for all sED and for all X € V wi th II X II

s

K.

Proof: We may assume that W is balanced and convex. Let s € D be fixed. Then there exists

n

_s € T with 0 E

n

_s such that

B(s)XC (s) E

i

W

for all X €

n •

Since T is coarser than the uniform operator topology s

there exists r > 0 such that the ball in B(H) with radius r and centr~ .

o

is contained in W. The mappings FA: D -+- B(E), /lAIl s I are introduced by

FA(s)

=

B(s)AC(s) , s € D.

These FA are continuous with respect to the uniform operator topology and they are even equicontinuous.

Since D is compact, there is a finite set E cD, E

=

{sl,sZ, ••• ,sN} with the property: for each s ED there is an s. E E such that for all

J A € B(H) with IIAII

s

1 we have

lr IIFA(s) - FA(s.)II~< - - .

J ZK

N

Now take

V

=

n n •

Then B(s.)XC(s.) €

!W

for all X

j=l Sj J J

=

1, ••• ,N. Let s € D and let X E V with II X II S K. Then

€ V and all j

=

·X

can find s. E

E

with J and hence Thus we obtain r

2" •

B(s)XC(s) - (B(s)XC(s) - B(s.)XC(s.»+ B(s.)XC(s.) E

w.

J J J l

o

Rosenblum's elegant prooof of the Fuglede theorem depends on the simple iAN* ilN

fact that for a normal operator N E B(H) the operators e e ' , A € ~, are unitary and hence have norm one. We prove a similar result for subn01::maLoperators 0

(1.4) Lemma

iAN*

iXN

Let N E B(H) be subnormal. Then II e ell:::; 1 for all A E ~.

Proof: Since

N

is subnormal, there are

N

₁₂

,N

₂₂

E B(H) such that

N

.-

o

-iIN iAN*

is normal as an element of B(H~). So the operators e e

,A

E ~,

are unitary and for suitable P(A) E B(H), A E ~

) peA)

I

eiAN22

J

,

7 -iAN*

o

e ... PC-A) Thus we find (II i(AN*

+:\N)(x\

II _

I

> ... sup. \ e ) ~) -II xU 2+11 yll 2"'1 y

(

i(AN*+XN)(x)

\

~ sup II e 0 IIH(QI) = II xII-I .

o

2. AsYmptotic theorems

The main problem in the proof of (0.1) is how to relate

NIB - BNZ

and

*

*

NIB - BNZ"

Or, equivalently, if we consider

and

how te relate

H'(O)

and

G'(O).

To obtain such a relation we first represent

G'(O)

as a Cauchy type integral by

G' (0) ... 21ri

r

GClli dll

(2.1)

with integration along I~I

=

r in the positive sense. Then

Now H(~) can be obtained from

H'

by integration from 0 to ~ along a straight line segment. This leads to the following relation.

~

G' (0) = 2'IT

r

~

-2e

-i~Nie -i~Nl{f

H' (A)dA + H(O)

}ei~N2ei~N2d~

1~1=r

0

. N* '-N

Formula (2. 1) and the observation that the operators e -l.~ .• 1 e -l.~ 1 and

i~N2

illN;

e e ~ II e ¢,.have norm not exceeding 1, establish the central argument in the following theorem.

(2.2) Theorem

Let '[ be an algebra topology on B(H), and W be a 't-open neighbourhood

*

of

O.

Let NI and

N2

be subnormal operators and let

K

>

O.

Then there

.

0

N*B - 8N* f 11

eXl.sts a 't-open neighbourhood

V

of such that 1 2 ~

W

or a

B ~ B (H) wi th II 8" ::; K and N 1 8 - 8N 2 ~ V.

Proof: We again may assume that W is balanced and convex and also that

II N 1 II and _{"N2 "} do not exceed one. We take r > 0 such that

I

K -illN* -i~Nl

{X ~ B(H) IIXII S;

-r}

C !W, and we put U

1 (ll) ::: e Ie and

U () _{2 ~}

₌

_e

i~N2

_e illN; _{, ~ ~ ¢. Then by Lemma} ( _1.4, ) II U ( ) 8U ( ) _{1 II 2 ~} II _{::; K 1.f} •

11811 ::; K, and hence

(i)

for all II ~ ¢ and all B ~ B(H) with II 8 II ::; K"

9

--iAN2

2

(A,~) ~ e U2(~)' (A,~) e ~ • From this lemma we obtain aT-open neighbourhood V of 0 such that

(ii)

for all

(A,~)

e

{(A,~)

E

~21

IAI

,(~) ~

r} and for all B E B(R) with

liB II ::; K and [N.,B] e: V.

J

*

Finally, we shall ins'ert formula (2.1) to obtain the desired result. So let B E B(R) with IIBII ~ K and [N.,B] e: V. Then we use (i), (ii)

J

*

and Lemma (1.2) in order to get

(iii) _e:

₂

r _W.

~

Applying Lemma (1.2) again we can conclude that

*

Z1Tr 1

[N. , BJ e:

2iT

2 ·

rW = W.

J

*

r

In this section we also want to obtain an asymptotic version of (0.5). To this end we first prove an asymptotic version of (0.4) and then

o

apply Theorem (2.2). For self-adjoint Al ,A

Z e: B(R) the operator AIS -SAZ

can be expressed by

u t

(2.3) i(AIS - SAZ)

=

f

₈

)0)

=

u-1(f

S(U) - £S(O» - u-1J

I

fg(s)dsdt

o

0

( isAI -isA2 .

where fS s)

=

e Se , s E lR. Formula (2.3) leads to an asymptot~c

version of (0.4), which is proved by similar arguments ,as Theorem (2.2).

(2.4) Theorem

of

O.

Let A1,A

Z E B(H) be self-adjoint and let K > O. Then there exists aT-open neighbouthood of

0

such that [A.,8] E W for all 8 E B(H) with

J

*

both II 8 II S K and [A., [A. ,8J] E V •

J J

* *

Proof: Once more we may assume that

W

1S balanced and convex, and also that II A 1 II and II A_Z II S 1. For each 8 E B (H) we define the mapping f8 as

. isA_{1 -isAZ}

l.n (Z.3) by f

8(s) == e 8e , s E ]R. Since T is coarser than the uniform operator topology, there exists r > 0 such that the ball

{C E B(H) IIICII S r} is contained in W. We take u

=-.

4K _r Since

-1 -I

II u (f

8(u) - f8(0» II S ZKu it follows that

(i)

Applying Lemma (1.3) to the mappings s + e isAt and s + e-isA2 we obtain

a 1."-open neighbourhood V such that for all C € V with II C II S 4K and ~ll .

S E [O,u]

(H)

e~e isAte -isA2 E -·1 W •

u

Let 8 E: B(H) with II 8 II S K and let [A. ,[A. ,8J J E: V. Then by (H) we

J J

* *

.

find

(Hi)

!

fS(s) €

~W

, S E [O,uJ.

u

Applying Lemma (1.2) twice we get from (ii)

(iv) u-I ( ( fii(s)ds dt •

Iw •

o

0

,

.

,11

-We have employed the Fuglede theorem (0.1) to extend our result (0.4) for self-adjoint operators to (0.5) for normal operators. Instead of the asymptotic version (2.4) of (0.4) we shall prove an asymptotic Kleinecke-Shirokov theorem for normal operators. For this we need the asymptotic Fuglede theorem (2.2).

(2.5) Theorem Let Nt ,N

2 ~ B(ll) be normal and let "C be an algebra topology on B(ll). Let

W

be a T-open neighbourhood of 0 and let K > O. Then there exists a "C-open neighbourhood V of 0 such that [N. ,B] ~ W for all B € B(ll)

J

*

with liB II !> K and [N.,[N.,B]] e: V.

J J

* *

Proof: We may again assume that

W

is balanced and convex. We write

N.

=

C.

+ iV. with self-adjoint

C.

and

V.

satisfying C.V.

=

V.C., j

=

1,2.

J J J J J J J J J - -- .

Applying (2.4) twice we can find a "C-open neighbourhood

U

of 0 in

B(H),

which may be taken balanced and convex, having the property that for

IIBII!> K, [c.,eC.,B]] e: U and [V.,[V.,B]] e: U it follows that

J J

* *

J J

* *

[C.,B] € ~W and [V.,B] € !W, hence [N.,B] €

w.

J

*

J

*

J

*

So it only remains to prove that there exists a "C-openneighbourhood of 0, V, say, such that liB II !> K and [N.,[N.,B]] e: V imply

J J

*

*

[C.,[C.,B]] e: U and [V.,[V.,B]] €

u.

To this end we use (2.2).

J J

*

*

J J

* *

*

*

There exists a T-open neighbourhood U

1 of 0 such that [N.,[N.,B]] _{J J}

_{* *}

e: U for all B e: B(ll) with IIBIJ!> K and [N.,[N\B] ]

J J

*

*

e: Ut

" Moreover, there

*

exists a "C-open neighbourhood U

2 of

0

such that [N.,[N.,B]*] _{J J}

_*

=

*

[N.,[N.,B] ] J J

*

*

Hence, if II B II e: U

1 n U for all B e: B(ll) with IIBII !> K and [N.,[N.,B] ] _{J J}

_{* *}

e:U2"

!> K and [N.,[N.,B]] € U n U

2, then the operators

all belong to U. Since U is balanced and convex this implies [C.,[C.,SJ J E U and [V.,CD .. ,E] J E U.

J J

**

J J

**

o

Acknowledgement

The authors are grateful to Dr. P. v.d. Steen for helpful remarks during the research for this paper.

References

[IJ Fuglede, B., A commutativity problem for normal operators, Proc. Nat. Acad. USA, 36 (1950), p. 35-40.

[2J Furuta, T., Normality can be relaxed in the asymptotic Fuglede-Putnam theorem, Proc. American Math. Soc., 79 (1980), p. 593-596.

[3J Kleinecke, D.C., On operator commutators, Proc. American Math. Soc., 8.( 1957), p. 535-536.

[4J Moore, R., An asymptotic Fuglede theorem, Proc. American Math. Soc., 50 '(1975), p. 138-142.

[5J Putnam, C., On normal operators on Hilbert space, American J. Math., 73 (1951), p. 357-362.

[6J Rogers, D., On Fuglede's theorem and operator topologies, Proc. American Math. Soc., 75 (1979), p. 32-36.

[7J Ros.enblum, M., On a theorem of Fuglede and Putnam, J. London Math. Soc" 33 (1958), p. 376-377.

[8J Shirokov, F.V., Proof of a conjecture of Kaplansky, Uspekni Mat •. Nauk., 11 (1956), p. 167-168.