liT- Ur U*II IIN-N II+IIKII - Almost commuting matrices and a quantitative version of the Br

for appropriate choice of K. Hence T is the limit of operators T, unitarily equivalent to T~N,. The operator T, is essentially normal with " n i c e " spectrum X,, and ind(T,-2)=0 for 2 ~X,. Thus it suffices to prove the theorem for " n i c e " spectra. As we have remarked, we can assume a nice spectrum has one component.

Let T be essentially normal with nice essential spectrum X, and let t=z~T. Let h be a homeomorphism of X onto a rectangle with parallel rectangular holes R. (For example, if X has p holes, take

R = {(x, y): 0 ~<x ~<p, [y[ ~< 1 } \ Id {(x, y): [ x - ( j - 1/2) I < I/4, }Yl < 1/2}.)

j=l

Then s=h(t) is essentially normal with spectrum R and i n d ( s - 2 ) = 0 for 2 ~R. Let S be any operator such that st(S)=s and IIIm SIl=llImsll 9

Apply Lemma 5.1 to S ( n - 1) times to obtain a compact perturbation of S unitarily equivalent to E~=1~S j where each Sj is essentially normal with spectrum equal to a rectangle with a rectangular hole. By Corollary 4.7, each Sj is normal plus compact.

Thus, there is a normal operator M such that sr(M)=s and o(M)=ee(M)=R. Then h-t(M) is normal and st(h-l(M))=h-I(s)=t=stT. Thus T is a normal plus compact

operator. []

THE BROWN--DOUGLAS--FILLMORE THEOREM 149 To prove the BDF theorem, it is necessary to examine how the preceding proof works on individual summands of T. This has to be done with operators, not in the Calkin algebra. So the homeomorphism h will be taken to be analytic in a neighborhood of the spectrum so that the Riesz functional calculus is available.

5.4.

Proof of Theorem

1.1 (BDF). Let T be essentially normal with

o~(T)=X

and ind(T-2)=0 for all 2 ~X. By the previous theorem, T is quasidiagonal. Hence there is a (small) compact perturbation K so that

o o

T' = T - K = E @ T ~ N

n = l

where Tn act on finite dimensional spaces ~n, and N is normal with

o(N)=oe(N)=X.

Let X, be a nice region containing X as in the previous proof. Then we can further take X~ so that the interior of X, is conformaUy equivalent to the interior of a rectangle R with n parallel slits. (As conceivable worst

X , \ X

has a small hole which maps to the infinite component of R complement.) L e t f b e the conformal map. Choose a region S such

thatf(X)cSc~qcR

and so that ~r is a rectangle with p parallel rectangles removed.

Let

Ye=f-l(s).

Let N, be a normal operator such that

o(N,)=ae(N,)= Y~.

The spectrum of T~ is contained in Y, except for finitely many n; otherwise we would have a cluster point of eigenvalues for orthogonal eigenvectors, hence a point of essential spectrum outside X. Extend f arbitrarily to a neighborhood of Y, U U ~ l o(T~) so t h a t f t a k e s o(T') into S. Let

T~=E~i~Tn~)N,,

and let

S,=f(T~).

The operator S, is given by the Riesz functional calculus, and one readily obtains

oo oo

s,-- sneM.

n = l n = l

The operator

M=f(N~)

is normal with

o(M)=oe(M)=~q.

Since S, is essentially normal,

en=ll[SL sjIP

tends to zero as n tends to infinity.

The region ,r is the union o f p regions

Aj, 1 <.j<.p,

each of which is a rectangle with a rectangular hole. The regions Aj and

Aj+ 1

intersect in a rectangle Rj, l<~.j<~p-I. Let

~>0 be the minimal width of the Rj's. Let the projection o f A s onto the x-coordinate be [2s, gj]. Apply the Cutting lemma 5.2 to each summand S~ in turn ( p - 1 ) times. That is, one obtains normal matrices M~,j. with spectrum contained in

Rj, l<.j<~p-I

so that

S~Y.~-~Mn,j

can be approximated by an operator of the form

E~f~S,.j

with

AjI<~ReS~,j<.pjI.

Notice that the error incurred in making this perturbation is the

150 I. D. BERG AND K. R. DAVIDSON

maximum of the ( p - l ) errors since these changes are on orthogonal pieces Remark 3.3). Thus, if one writes

M,,-Ej=1 t~M.,j,

_ p-i we have

S.O)M.-~j=I ~S.,j <~

67max{e.,

e2./6}.

(see

For n sufficiently large, this error is 67en. (Incidentally, the proof of Lemma 5.2 shows that Mn acts on a space of dimension no larger than that on which

Sn

acts.)

By the extended Weyl-von Neumann theorem, M has a small compact perturbation unitarily equivalent to E~> 1

~M~t~M.

S t -~ ~ O)(S.O)M.)O)M-Ko

n = l

where K 0 is compact and

IIg011 t0=ll[S, *, S ]ll

Let

K 1 = Z ~ ) S.t~M.- ~)S.j ~0.

n = l j = l ' /

This is compact, and IIKlll~<67max{e0, e~/6}. Let

K=Ko+K I.

Then

[Igll-< 68

max{e 0, e~/6}

and

j = l \ n ~ > l /

N o w Mm~;=I(~)M j is normal with spectrum Aj. Let

S,~.TS,,~1@S,,,j@M s.

Then

S,-K=-E~=~@S~,j

and each S,,j is an essentially normal, quasidiagonal operator with

2fl<.-ReS~,j<.lt/L

Thus

oe(S,j)

is contained in the strip {zEC:;tj--.<Rez-.</zj) intersect

oe(S),

which is Aj. Because S,,j contains M s as a summand,

tr~(S,~.j)=Aj.

Repeat this procedure with the operator S,,j and a conformal map onto an annulus.

Then by the proof of Theorem 4.4, one obtains a compact perturbation of S,,j which is normal with spectrum

Aj and,

as above, this compact perturbation is achieved by attaching to an, j a normal summand Mn, j with

lira

II(S,,,st~M,,,s)-N,,,jII = O.

n-..~ oo

T H E B R O W N - - D O U G L A S - - F I L L M O R E T H E O R E M

There is a compact operator Cj of norm at most e0 so that

S,,F Ci = Z ~(S.,j~M..j)~Mi"

n>~l

t ~

Define Cj-E.~I~(S.,j(~M~,j-N~,j). This is compact, and

s ,FcFc;=

n ~ l

which is normal.

Recombining

151

all the terms, one obtains normal operators Mn-Ej=l(~Mn, j P and

N.-r.J=~N,,,j P with spectrum contained in S so that lim ]]Sn~)M.-N.I ] = O.

n---~oo

Consequently, as f is conformal from a neighborhood of Y~ onto a neighborhood of lim Ilz,,@f-l(M.)-f-'(N.)ll = O.

n.-.> ~

The operators f-l(M,,) and f-~(Nn) are normal with spectrum in Y,. This (finally) recaptures the proof of Theorem 5.3 with the important additional information that the compact perturbation is obtained by adding on to each summand T~ a normal summand

A.=f-l(Mn) so that T ~ A ~ is close to a normal B. for n large. (Furthermore, A. acts on a space of dimension no larger than the domain of Tn.)

Choose a sequence ek decreasing to zero. For each k, obtain normal matrices A., k and B., k with spectra in Y*k so that

lim IlTn~m,,,k-B,,,kll=O.

n----~ o0

Choose an increasing sequence of integers Nk such that for all n>~Nk ]lZn| <-- %

Choose normal matrices A~ and B. with spectra in X so that

Ila..k-a, ll<e k

and

IIn..k-n,ll<ek

for Nk<.n<Nk+ v

So then

IIT,@A,--n,ll<--ek

for Nk<~n<Nk+ r For l<~n<Nl, choose Bn arbitrarily and An to

152 I. D. BERG A N D K. R. D A V I D S O N

be vacuous. No good norm estimate holds for these terms. Finally, T'= E ~ O)Tn@N is a (small) compact perturbation of an operator unitarily equivalent to

T" = E,>>_~@(T~ ff)A~)@N.

Let

K=En>~l@(Tnt~A-Bn)t~)O.

This is compact since lim~_.| and so

T"-K=

E~>,~B~N

is normal with spectrum X. []

To get a quantitative version of BDF, it now suffices to examine the way the compact perturbation is obtained in the previous proof. It is the fact that each summand Tn in the block diagonal form of T is perturbed

individually

by adding on a normal summand that makes this possible.

Definition

5.5. Let X be a compact subset of C, and let Nor(X) be the set of normal matrices with spectrum contained in X. For t~>0, let

Xt

denote {2 E C: dist(L

X)<.t}.

Let

~ ( X ) denote the set of matrices T such that (i)

lilT*, T]lll/2~<t

and (ii)

I1(T-21)-%--<

dist(2,

Xt)-t

for all ~. ~Xt. Define a function

fx:

[0, oo)--,[0, ~) by

fx(t)

= sup inf

dist(T@N,

Nor).

TE ~,(X) NE Nor(X)

LEMMA 5.6 (Absorption principle),

lim_~o+fx(t)=O.

Proof.

Choose T~ in 6Pv~ so that

dist(T~N, Nor)>89

Let

T=En~1~T~N.

This is essentially normal and quasidiagonal. Furthermore,

ae(T)

equals X since if dist(2, X ) >

1/k,

k - I

S~ = E + 0 + X ( T ~ - M ) - ' + ( N - M ) - '

n = l n>~k

is bounded by k2+k and is an inverse modulo ~ of

T-2I.

From the proof of 5.4, one obtains normal summands Nn of N so that dist(Tn~Nn, Nor) tends to zero. Hence

limn_~fx(1/n)=O.

Since

fx

is monotone, lim/__,0fx(t)=0. []

5.7.

Proof of Theorem

1.2. Let T be an essentially normal operator with ae(T)=X and

ind(T-M)=O

for 2 ~X. By Theorem 5.3, T is quasidiagonal. Given any 6>0, there is a compact K0 with [[K0ll<8 so that

THE BROWN--DOUGLAS--FILLMORE THEOREM 153 where N is a diagonal normal operator with o(N)=o~(N)=X, each T~ is finite rank and T~ belongs to 5e, for all n>~l. From the proof of 5.4, and Lemma 5.6, one obtains finite dimensional summands N~ of N and normal matrices M~ so that

IIZ~Nn-M~ll<fx~e)

and lim~_.~

IILeN -M II=O.

Thus one obtains a compact operator K so that T - K is normal, and

Dans le document Almost commuting matrices and a quantitative version of the Brown-Douglas-Fillmore theorem (Page 28-33)