Chin. Ann. of Math. 21B: 2(2000),237-248.

T H E (UT/C)-ORBIT OF E S S E N T I A L L Y N O R M A L O P E R A T O R S A N D C O M P A C T P E R T U R B A T I O N S

OF S T R O N G L Y I R R E D U C I B L E O P E R A T O R S

J I YOUQING* J I A N G CHUNLAN** W A N G ZONGYAO***

A b s t r a c t

Let 74 be a complex, separable, infinite dimensional Hilbert space, T E s (12 +/C)(T) denotes the (b/+/C)-orbit of T, i.e., (/2 +/C)(T) ---- { R - 1 T R : R is invertible and of the form unitary plus compact}. Let f~ be an analytic and simply connected Cauchy domain in C and n E N . . A ( f ~ , n) denotes the class of operators, each of which satisfies

(i) T is essentially normal; (ii) ~(T) = ~, pF(T) N ~r(T) = f~; (iii) ind ()~ -- T) ---- --n, nul (A -- T) ---- 0 (A E ~). It is proved that given T1, T2 E r and c ~ 0, there exists a compact operator K with

IIKII < ~ such that 7'1 + K E (/g +/C)(T2). This result generalizes a result ofP. S. Guinand and L. Marcoux [6'15] . Furthermore, the authors give a character of the norm closure of (/2 +/C)(T), and prove that for each T E .A(f~, n), there exists a compact (SI) perturbation of T whose norm can be arbitrarily small.

K e y w o r d s Essen t i a l ly normal , (/2 + / C ) - o r b i t , C o m p a c t p e r t u r b a t i o n , S p e c t r u m ,

S t rong ly i r r educ ib le o p e r a t o r

1991 M R S u b j e c t C l a s s i f i c a t i o n 47A55, 47B, 47C

C h i n e s e L i b r a r y C l a s s i f i c a t i o n O177.1 D o c u m e n t C o d e A

A r t i c l e I D 0252-9599(2000)02-0237-12

w I n t r o d u c t i o n

Let 74 be a complex , s epa rab l e , inf ini te d imens iona l H i lbe r t space . Le t s and /C(74)

deno t e t h e a l g e b r a of b o u n d e d l inear o p e r a t o r s and , respect ive ly , t he idea l of c o m p a c t oper -

a to r s a c t i ng on 74. W e will cal l (L /+ /C) (T) = { R - 1 T R : R E (L/+/C)(74)} the (b /+ /C) -o rb i t

of T , where

R is inver t ib le a n d of t he fo rm u n i t a r y (/2 + /C)(74) = R E s o p e r a t o r p lus c o m p a c t o p e r a t o r J "

A "~u+~ T a n d T - ~ u + ~ B i m p l y A E ( /4+]C) (T) and , respect ive ly , B E ( / 4 + ] C ) ( T ) - , t he

n o r m closure of (U + / C ) ( T ) . W h i l e "~u+~ defines a n equiva lence re la t ion , - + u + ~ does not .

A n o p e r a t o r is s t rong ly i r reduc ib le , or briefly, T E ( S I ) , i f i t does no t c o m m u t e w i th any

non t r iv i a l i d e m p o t e n t . A n o p e r a t o r is essen t ia l ly n o r m a l if [T, T*] : = T * T - T T * E 1C(74).

A n o p e r a t o r T is sa id to b e shif t - l ike if T is essen t ia l ly n o r m a l w i th ~ ( T ) = D = {z E C :

Manuscript received September 4, 1998. Revised October 10, 1999. *Department of Mathematics, Jilin University, Changchun 130023, China.

**Department of Mathematics, Jilin University, Changchun 130023, China. * * *Department of Mathematics, East China University of Science and Technology, Shanghai 200237,

China.

238 CHIN. ANN. OF MATH. Vol.21 Ser.B

Izl ~ 1} and a~(T) = OD with ind(A - T ) = - 1 and nul(A - T ) = 0 for all A E D. P. S. Guinand and L. Marcoux [6'15] proved the following

T h e o r e m G - M . Let T1, T2 be shift-like and let e be a positive number. Then there exists

a compact operator K with IIKll < e such that T1 + K ~u+~c T2.

In this paper we will strengthen the above theorem.

Let ~ be an analytic and simply connected Cauchy domain in C and n E N. Then r n) will denote the class of operators, each of which satisfies

(i) T is essentially normal; (ii) a(T) = ~, pF(T) M a(T) = ~;

(iii) ind (A - T) = - n , nul (A - T) = 0 for all A E f L

The next three results are our main results.

T h e o r e m 1.1. Given T1, T2 E .A(f~,n) and e > O, there exists a compact operator K

with IIKII < e such that T1 + K ~u+~: T2.

T h e o r e m 1.2. Let T be in .A(f~,n). Then (lg + 1C)(T)- consists of all operators A

satisfying

(i) A is essentially normal; (ii) a(A) = -~, pF(A) M a(A) = ~;

(iii) ind (A - A) = - n for all A E f L

T h e o r e m 1.3. Let T E .A(f~, n) and e > O. Then there exists a compact operator K with

[Igl[ < e such that T + g E (SI ) .

Theorem 1.3 partially answers an interesting question posed by D. A. Herrero: Given an

essentially normal operator T with connected spectrum and e > 0, is there a K E /C(7/)

with Ilgll < e, such that T + g E (SI )?

w P r o o f of the Main T h e o r e m s

It follows from the Riemann mapping theorem that there is an analytic function r satis- fying

(i) r -- z0, where z0 is a fixed point in ~; (ii) r is injective; (iii) r = ~.

By Schwarz reflection principle r has an analytic continuation on D such that r ----

0~. Let Tr be the Woeplitz operator with symbol r Then T~ E BI ( fF) and a(T~) = -~*,

where Bn(f~) denotes the set of Cowen-Douglas operators of index n, i.e., Bn(f~) consists of operators B satisfying

(i) a(B) D f~; (ii) ran (A - B) = 7-/for all A E ~2;

(iii) nul (A - B) ---- n for all A E f~; (iv) Y { ker(A - B ) : A E ~} ---- 7-/.

(iv) can be replaced by (iv)'.

(iv)' V{ker(A0 - B ) k : k = 1, 2 , . . . } = 7 / f o r a fixed A0 e ~ (see [3]).

In order to prove Theorems 1.1 and 1.2, we need several lemmas.

L e m m a 2.1. Let All -- (ker(Tr - A)*)• Then Tel ~ is unitarily equivalent to Tr where

P r o o f . Since J~4 is a hyperinvariant subspace of Tr A~ is an invariant subspace of Tz. By Beurling theorem, J~A = TgH 2, where g is an inner function and H 2 is the Hardy space. Since T~ E BI(~*), g ---- (z -- a) / (1 -- a ' z ) , (0 < lal < 1). Thus T a can be considered as a uni tary

operator from H 2 to J~4 and T~(TgIM)TgJ = T e l for f E H 2, i.e., T~(Tr = Tr

L e m m a 2.2. LefT~ = C ~ H 2 and A E gt and l e t T = [Ab-:. T0[ E L:(n), where r ~

L - -

E = he0 | 1, e0 E ker(A - Tr and [[e0[[ = 1, a e C and a r O. Then T ~"u+lc T~.

No.2 JI, Y. Q., JIANG, C. L. et al ORBIT OF ESSENTIALLY NORMAL OPERATORS 239

P roo f . Note that eo �9 ran (Tr - ~)• By Lemma 2.1,

T r gl Tr A 4 = [ k e r ( T r • g2 T, H 2 '

where gl, g2 are rank-1 operators and we assume that g2 = f | 1, f �9 H 2. Since eo �9 ran ( T r A) • f = fleo+g3, where fl �9 C and g3 �9 ran (Tr Iff l = 0, let g3 = ( T 4 - A)x, then computation shows that

1 : ] [g~ Or [ 1 : ] _ _ [A Tel x | - x | 0 "

This i scon t rad ic tory toTr E (SI) . T h u s f l r a n d X [ )~ 0 ] X - I = [ )' 0 ] g2 Tr f~e0 | 1 Tr '

x | 1 � 9 Since f l e 0 | T 4, "~u+~c T4 ' the proof of

Lemma 2.2 is complete. By the same argument of Lemma 2.2, we can prove

L e m m a 2.3. L e t T - l = C * H 2 , g ~ r a n ( T r 1 4 9 [ )~ 0 ] g | T~ � 9 Then T "u+lc Tr

L e m m a 2.4. L e f T �9 s ' ~ @ H 2) and T = Cn Tr H~ , where F E s n) with

a(F) C f~, Cn �9 s "~, H 2) is a nonzero operator. Then for each e > O, there exists a compact operator K , JJKJl < e, such that T T K "~u+lc T4).

Proo f . We will prove the lemma by induction on n. r 0

"1

/ / where C1 ---- f | is a rank-1 operator. Choose g �9 H 2, Ilgll < e, I f n = l , T = LC1 T ~ j '

[ 0 0] T h e n j j K J j < e a n d b y L e m m a 2 . 3 ' such that f + g C r a n ( T r S e t g = g | "

T + K "~u+~c Tr Assume that the conclusion of the lemma is true when n g k - 1, and let n = k. It is

obvious that there is a unitary operator U such that [! 0 UTU* -= F,~-I C n-1 ,

Ca-1 Tr H 2

where F~-I �9 s A �9 a(F) . By the inductive assumption, we can find a compact operator g l with Jlglll < ~/2 and X1 �9 (/4 + K:)(7-/) such that 0]

XI(UTU* + K 1 ) X ~ 1 = C1 Tr "

By the assumption again, we can find a compact operator K2 such that H K2 II < (211xl IIIIX~ -1 II)

and X2 C1 Tr -t- K2 X 2 1 -- T , for some X2 �9 (/4 + 2C)(74). Thus K = U*K1U T

U*X11K2X1U satisfies the requirement of the lemma. L e m m a 2.5. Given A, B �9 s let TAB denote the Rosenblum operator given by

T A B ( X ) = A X - X B , X �9 s Let T = TABIIC(~I ). Then T* = --TBAIC'(n) and (T*)* --- TAB , where T* is the dual o f t and C1(7-/) is the set of trace class operators.

Proo f . Recall that C1(7-/) is isometrically isomorphic to the dual/C(7/)* of/C(7-/). This isomorphism is defined by C1(7/) S K ++ Cg �9 1C(7/)*, where CK(X) ---- t r ( K X ) , X �9 lC(7-l).

240 CHIN. ANN. OF MATH. Vol.21 Ser.B

Thus

r = C K( r (X) ) = t r [ K ( A X - X B ) ] = t r ( K A X ) - t r ( K X B )

= t r ( K A X ) - t r ( B K X ) = t r ( - -~-BA(K)X) = q~---rvA(K)(X).

Therefore T* = - -TBAIc 1(7./).

Since s is isometrically isomorphic to the dual C1(74) * of C1(7-/), by the similar

arguments we can prove tha t (r*)* : tAB.

L e m m a 2.6. Given T �9 A(f~, n) and e > O, then there exists a compact operator K with

Ilgl[ < e such that T + g ~u+~: T ( '0 .

P r o o f . Note tha t Tr admits a lower tr iangular mat r ix representat ion with respect to the

O N B {ei~~176 of H 2. Set f l4k = V { e ij~ : j = o, 1 , . - - , k} and denote by Pk the orthogonal

projection onto ~ fl4k. By Brown-Douglas-Fillmore theorem UTU* = T (n) + K , where U k : l

is a uni tary operator and K is compact . Set K1 = P,~KP,~ - K , and m will be determined

later. By Lemma 2.1, we can find a uni tary operator U1 such tha t

F 0 . . . 0 ]

c1 T~ U I ( U T U * + K 1 ) U ; = . . 0 ,

" .

cn o T~ where F �9 s Ck is a finite rank operator (k = 1 , . . . ,n) . Fix m so tha t IlKlll < e/8 and a ( F ) C f~ / s . Thus we can find an operator C �9 s " '~) such tha t [IC[[ < e/4

and a ( F ' ) C ~ , where F ' = F + C. Therefore there exists a compact operator /(2 with

IIK=ll < e/4 such tha t

U I ( U T U * + K 1 ) U ; + K2 =

0 00] C1 Tr =AI.

o T,

I F t 0 ] t-1 _-- T~b. By Lemma 2.4, there is an X~ E (/4 + / C ) (C '~m ~ H 2) such tha t X~ C1 Tr X1

Thus we can find X 1 �9 (/~ -~- ](~)(C nm (~ (H2)( '0) such tha t

[ T~ o ~ o ] X1A1X11 = C~. Tr = A2,

LC._I 0 T~

where C~ (k = 1 ,2 , - - . , n - 1) is a finite rank operator . Since .A'(T~) does not contain any

compact operators, ker'rT~T~]lC(H~ ) : {0}. By Lemma 2.5 [ranTT~Tr : CI (H2) . Thus for each k (1 < k < n - 1) we can find compact operators Dk and Ek satisfying

T c E k - EkTg, = - C ~ - Ok and IIDkll < e/(llXlll[[Xlll[8k). Set

I ~ ~ i] [i 0 0 K3 D1 0 . . . E1 1 . . . 0 ---- . , X 2 : . .

o~ "~

D -1 0 . . . . t 0 . . . 1

No.2 JI, Y. Q., JIANG, C. L. et al ORBIT OF ESSENTIALLY NORMAL OPERATORS 241

Then K3 is compact, ILK311 < /(811XxlllIx lll) and X2 e (/4 +/C)((H2)(")) . Calculation

indicates that X2 (A2 + K 3 ) X ; 1 : TOO. Thus, Z (T + g ) x -1 = T ~ , where g = U* g l U +

U*Ui*K2UxU+X~IK3X1 is a compact operator with [IKll < e, x -- X 2 X i U i U is invertible and of the form unitary plus compact.

Lemma 2.6 implies tha t .A([%, n) C (/4 +/C)(T(n)) - .

L e m m a 2.7. L e t A , B 6 s Assume that7-/ : V { k e r ( A - B ) k : A �9 F , k > 1}

for a certain subset F of the point spectrum ap(B) of B , and ap(A) N F = 0 ; then TAB is injective.

P r o o f . Let p be a monic polynomial with zeros in F and let x �9 7-/be any vector such that p ( S ) x : 0; then A X = X B implies p ( A ) X x = X p ( S ) x = O.

Since p(A) is injective, we infer X x = O. It readily follows that ker X D V { ker(A - B ) k :

A �9 F, k > 1} = 7/. Hence, X = 0. L e m m a 2.8. [1,The~ 4.15] Suppose that T �9 s is essentially normal and a(T) =

a~(T) Uao(T). Assume, moreover, that C(a(T) ) = R a t ( a ( T ) ) - . Then T -+u+~: N, where g is a normal operator such that a ( N ) = a(T) , ao(T) = ao (g ) , and nul ( A - N ) = dim74(A; T)

for all A �9 a0 (N), Rat ( a ( T ) ) - is the uniform closure of rational functions with poles outside a(T) .

An operator is almost normal if it has the form of normal + compact.

L e m m a 2.9. Suppose T �9 s is almost normal, a (T) is a perfect set with Lebesgue measure O, N is normal with a ( N ) = a(T) ; then for each e > O, there exists X �9 (/4+/(:)(7-/) such that

(i) X T X -1 - N �9 ]C(74); (ii) I I X T X -1 - Nil < e.

P r o o f . Since m ( a ( T ) ) = O, it follows from [8] tha t R a t ( a ( T ) ) - = C(a(T) ) . By Lemma 2.8, for each 5 > 0 there is an X1 �9 (/4 + ]C)(74) such that I I X I T X ~ 1 - N u < 5. Since T is almost normal, X I T X 1 1 is also almost normal. Therefore there are normal operator M and compact operator K such that X 1 T X ~ -1 = M + K. Since IIM + K - Nil < 5,

II [ (U + K)*, ( U + K)] I[ < 411NIl 5 + 252. I. D. Berg and U. R. Davidson [2] asserts tha t there exists a positive valued continuous function f on [0, c~), f (0) -- 0, such that for each almost normal operator Q, there is a compact operator K(Q) with [IK(Q)II <- f(l[Q*Q - QQ* [11/2) and Q + K (Q) is normal. According to this theorem, we can find a 5 and a compact operator

K1 such that

(i) I]Klll < e/4; (ii) X 1 T X 1 1 + K 1 = M + K + K 1 is normal;

(iii) a ( M + g + K1) C a(T)~/4. Since T is almost normal and a(T) is a perfect set with m (a (T ) ) = O, a(T) = a~(T). Thus

a ( M ) = a(T) t3ao(M). Since M + K + K 1 is normal, a ( M + K + K 1 ) = a ( T ) U a o ( M + g + K 1 ) .

Thus we can find a compact operator K2, IIK2H < e/2, such that a ( M + g + g l + K2) = a(T) = a ( N ) and M + K + K1 + K2 is normal. It follows from Voiculescu theorem that there is a uni tary operator U and a compact operator Ks with ILK311 < e/2 such that

U ( X I T X 1 1 q- K1 --b K2)U* -- U ( M + g + g l q- K2)U* = N + K3.

Thus U X I T X I I U * - N �9 ]C_.(74), [ [ U X I T X l l U * - N [ [ < e, and X :-- U X l satisfies the requirements of the lemma.

Let T �9 s A is an approximate normal eigenvalue of T if for each e > 0, there is a unit vector er such that [[(A - T)e~[[ < e and [[(A - T)*e~[I < e. If A �9 al,.~(T) C al~(T), by Apostol-Foia~-Voiculescu Theorem [9], there exist a compact operator K and an infinite

242 C H I N . A N N . O F M A T H . Vol.21 S e r . B

dimensional subspace 7{, such that T ---- 7{~ + K. Let {en},,~__, be an O N B of 7{,.

Since e~ -~ 0 weakly, Ilge.[I -~ 0 (n -~ oo), i.e. II(A - T)enJ[ -+ 0. If T is a hyponormal operator, then [[(A - T)en[[ 2 - [[(A - T)*en[[ 2 -+ 0 (n -+ oo). Thus A is an approximate normal eigenvalue. Thus we get the following proposition.

Proposition 2.1. Assume that T is essentially normal. Then al~(T) is contained in the set of approximate normal eigenvalues of T.

Using Propostion 2.1, we get

T h e o r e m 2.1. Given T, N �9 s such that hire(T) is contained in the set of approxi- mate normal eigenvalues ofT, g is a normal operator with a(N) = he(N) = F, r c ol~(T) and given e > O, there exists a compact operator K with IIKll < e such that

A ~ Proof . Let { n}n=l be a sequence of complex numbers such that {)h~}- = r and card {m : A,,~ = A,~} = c~ for each n = 1, 2 , . - . . By the definition of approximate normal eigenvalue, there exists a unit vector el such that I1(,~1 -T)eill < ~/16 and I1()'1 -T)*elll < e/16. Thus, we have

IIP~1(~1 - T)P~ll l < e/16, IIe~l(~l - T ) e ~ t II = IIP~t(A: - T)*PexI[ < e/16,

and IIP~ (A1 - T)P~ x [[ < e/16, where P~I and P ~ denote the orthogonal projections onto

Vie1} and, respectively, [Vie1}] • Under the decomposition 7{ = Vie1} ~ [V{el}] • T

admi t s the represen ta t i~ A1+tnT21 T12]T1 J" Set KI= [ tilT21 T12] " then K1 �9 K : ( 7 { ) ' 0 '

[A01 0 ] Clearly, Ilglll < e/8 and T - g l = T 1 "

we can ge t / (2 �9 K:(n), [[K21I < e/2 4 such that

T - K I - K 2 = A2

0

alre(T1) ---- hire(T). Repeat the argument,

~] V{e~} V{e~}

•

By induction, we can find an orthonormal sequence el, e2,- �9 �9 , en, ' " �9 in 7{ and a sequence g l , g 2 , " " , g n , . . , in/(:(7{) such that []K,~H < e/2 n+2 (n = 1, 2 , . . . ) and

T - n = l = ~--'~ Kn m 1 ,2 , . . . n=l 0 T,~+I [ v l e l ' ' ' " ' e m i l •

@

Set C1 = ~ K n ' N 1 = ~Anen@e '~ ; t h e n T - C 1 = [ n=l TooO ] w i t h r e s p e c t t ~

the decomposition n ----- V i e , : 1 < n < o o } ~ [V{en}] • and Cl � 9 Hc1H < e/4. Applying Voiculescu's non-commutative Weyl-von Neumann Theorem to N1, we can find compact operator C~, HC~[[ < e/8, such that N1 -4- C~ -~ N ~ N1. Therefore, 0]

T - C, - C2 ~ - N1 0

0 Too

where C2 is compact and HC2H < e/S. Finally, we can find a compact operator C3 with

No.2 Jl, Y. Q., JIANG, C. L. et al ORBIT OF ESSENTIALLY NORMAL OPERATORS 243

r N O1 IIC~ll < d4, s u c h t h a t T - C l - C 2 - C 3 ~ _ ] n T i . Set K = C 1 + ( 7 2 + C 3 . T h e n K

L ~

satisfies all requirements of the theorem. L e m m a 2.10. Let T E Jr(D, n), e > O. Then there exist an operator T' 6 M(D, n) and a

compact operator K such that IIKI[ < e, T' admits a lower triangular matrix representation with respect to some O N B of n and T' + K "*u+tc T.

P r o o f . By Apostol 's triangular representation theorem T = Tl n~(T) and

7-/0 is a hyperinvariant subspace of T, (2.1)

Tz admits a lower triangular matrix representation, (2.2)

ps-F(T) C p(To). (2.3)

By Brown-Douglas-Fillmore theorem T ~_ T ~ + K, K is compact, and r ( T ) is a unitary element in the Ca]kin algebra ~4(n) = s i.e., 7r(T*)Tr(T) - 7r(1) = 0. Thus

[ ~ ( T ; T o ) - ~(1) ~ ( T S B )

*r(B*To) 7r(B*B + Tt*!/})- ~r(1)] = [00 : ] "

Since D C p(To) C p(Tr(To)), ~r(To) is invertible and 7r(To) -1 = 7r(To)*. Thus 7r(To) is a unitary element in A(no (T ) ) . Since To is invertible, by Brown-Douglas-Fillmore theorem To -- U0 + Ko, where Uo is unitary and Ko is compact." By (2.3) a(Uo + Ko) COD. Since

7r(T) is a unitary element in ,4(n) , TT* - l E IC(n), i.e., [ToT~ + BB* - TtT[BTt*- 1 ]

is compact. Since 7r(T0) is a unitary element, ToT s - 1 is compact, thus BB* and B are compact. Since Tt* E Bn(D) and D rl a(T~) = ~ , by Proposition 2.1 kerrT$T? = {0} and therefore ker~'TzTo = {0}. By Lemma 2.5, ran'CToT~ is dense i n / c ( n l , n 0 ) . Thus for 6 > 0, there exist compact operators E, G E s no (T) ) such that T o G - GTI = B + E and

[JEll < 6.

[~ 1 G ] n ~ [~ o E ] n ~ T h e n X l ( ( T o e T t ) + K 1 ) X { 1 Set XI = h i ( T ) and K1 = h i ( T ) "

= T, where X1 6 (b /+ /C) (n ) , g l is compact and Ilgll[ < 6/2. Since a(To) C OD, Lemma 2.9 indicates that there are compact operator K2 and X2 �9 (L/+ K:)(n) such that I[K211 < 6 and X2((No 63 Tt) + K2)X~ -1 = To 63 Tt, where No is a diagonal normal operator with a(No) = a(To). By Theorem 2.1 we can find a compact operator Ks and a unitary operator U such that IIK3II < 6 and V((No 63 Tl) + K3)U* = Tt, i.e., U*(Tt - UK3U*)U = No 63 Tl. Thus let 6 = e/(4[lX2[llIX21[[), g = UK2U* -~2 r.z U . 2 - - u ~3 + U X 2 1 K 1 X 2 U *, T' = Tl satisfy the requirement of the lemma.

L e m m a 2.11. Let T �9 ,4(D,n) , 0 < e < 1/10 and IIT*T - I I I < e. Then there exists a

W T W - T~ zs compact and IIW*TW T('011 unitary operator W such that * (n) . - - ~ z II < ~ "

P r o o f . Consider the polar decomposition T = UIT I of T, where U is a partial isometry, r a n U = ranT. By the Wold decomposition theorem (see [17]), U ~ T (n) 63 V, where V is a unitary operator. By the assumption IIIYl 2 - 111 < ~, I z2 - 11 < E for an z �9 a(ITI). Since e < 1/10, z > 1/2 and e > Iz 2 - 11 = I* + l l l z - 11 > 3lz - 11/2. Thus ~(ITI) c { z �9 R : I~ - 11 < 2 , / 3 } . Since ITI is se i f -adjo int , I I I T I - 111 < 2E/3 ~ d t h e r e f o r e

liT - UII = IIUITI - UII = IIU(ITI - 1)11 < IIUIIIIITI - 111 < 2 e / 3 . N o t e t h a t T ~ T~ ('~) + K , where K is compact. Thus [TI 2 - 1 = T ' T - 1 _.~ ~r~v(n) ~")a-w~* ~.,ffv(n)~ + K ) - 1, and IT[ 2 = 1 +K~ a n d ITI = 1 + K2 for s o m e K ~ , K2 �9 X : ( n ) .

244 CHIN. ANN. OF MATH. Vol.21 Ser.B

Since V is unitary and U -~ T ('~) (B V, by Theorem 2.1 there is a unitary operator W such that

I I W * V W - T~n)ll < ~/3, (2.4)

K3 := W * U W - T ('~) is compact. (2.5)

Thus W * T W - T (n) = W * ( T - U ) W + W * U W - T (n) = W*[U(IT[ - 1)]W + K3 --

W * ( U K 2 ) W + K3. It is compact, and s

[ [ W * T W - T(n)[[ < [[W*(T - U)W[[ + [[g3[[ _ [ [T - UI[ + -~ < e.

P r o p o s i t i o n 2.2. Let T e ~4(D, n) admit a lower triangular matr ix representation

T - - [ X F 0S]741742' where, F i s a d i a g o n a l o p e r a t o r o n a f i n i t e d i m e n s i o n a l H i l b e r t s p a c e 7 4 ~ ,

S C ~4(D,n) and [[S*S - 1[[ < 5 < 1/10. Then there exists Q 6 (U + K ) ( T ) - satisfying

dist (Q, /4(T(~))) < 45 and Q - W * ( T ( n ) ) W �9 ]C(74), for some unitary operator W .

Proof . Case 1. Assume that a (F ) C {A : [A[ < 1/(1 + 35)}. We will proceed by induction on the dimension m of 741 to prove that there exists Q �9 (/4 + ]C)(T) such that dist (Q,/4(T('~))) < 5 and Q - W * T ( n ) W is compact for some unitary W.

[ A OS] V{e}==741 w h e r e x e 7 4 2 a n d x ~ t O ( x = O i s c o n t r a _ W h e n m = 1, T = x | 742 '

dictory to T �9 A ( D , n)). If x �9 ran (S - A), computation shows that T ~ A ~ S. This is also contradictory to T �9 ~4(D, n). Thus x ~t ran (S - A). For z | e �9 s

[ 1 : ] [ ~ 0 ] [ ~ : ] [ ~ 0] T Nu+g =

z | x | S - z | y | S '

where y = x + ()~ - S ) z . Since A - S is a Fredholm operator, we always can choose z so that y �9 ker()~ - S)* and y ~ 0. For c~ �9 C ((x ~t 0) and w �9 742, we have

[ - o~ 0 "~U+/C Lw| e 01] [~ 01] [y:e OS 1 [0 1] [-col| 01]--[v~e 0S] ' T

where v -= ~y + ()~ - S)w.

Set Q1 = v | e . We win choose adequate a and w so that IIQ~Q1 - 111 < & Note

that

O~Q1 : [ ')~12 + ( e | 1 7 4 e | . (S 'v ) | e S*S

We intend to choose a and w so that

IAI 2+llvll 2 - 1 = 0 or IIvlI=X/1-1AI 2, (2.6)

s * . = 0 (2 .7)

and a r O.

Since S* is a Fredholm operator and nu lS -- O, S * S is invertible. Set S * S = A. Thus

S* v = S*[ay + C A - S)w] = a S * y + (AS* - A )w = a S * y + A ( A A - 1 S * - 1)w.

Since A > 0, a ( A ) C (1-5 , 1§ Therefore IIA-111 < 1 / (1 -5 ) . It follows from IIS*II < 1+5 that

IIA-1S*II < IlA-1llllS*ll < (1 -1-5) / (1 -5) < 1-1-35.

No.2 JI, Y. Q., JIANG, C. L. et al ORBIT OF ESSENTIALLY NORMAL OPERATORS 245

By assumption, II`kA-1S*II < 1. Thus ( `kA-1S * - 1) and (,kS* - A) axe invertible. Choose w = (-a`k*)(AS* - A ) - l y , we have

S* v -= aS* y + (`kS* - A)( -a )C) ( `kS* - A ) - l y = a`k* y - aA* y = O.

Since

Ilvll = l i l y + (`k - S ) ~ l l - - [ l l~y l l 2 + II(`k - s ) ( - ~ ` k * ) ( ` k s * - A)-~Yl l2] 1/2

- - I~1 [IlYll 2 + II`k*(`k - S)(`kS* - A ) - ~ y l l 2] 1/2

and since `k and y are f ixed, we can choose c~ ~ 0 so t ha t Ilvll = (1 -I`kl2)ln. Thus

[10 0 ] and ] I Q T Q I - I I I < 5 . S i n c e Q l � 9 for the chosen v, Q~Q1 -- S*S

Q1 �9 ,4 (D,n) . By Lemma 2.11 dist (Q1,/4(T(~))) < 5 and QI - W * T ( ~ ) W is compact for some unitary operator W.

To complete the induction step, we now assume that the result is true when F is an (m - 1) x (m - 1) diagonal matrix.

Let T = [xF O ] s a t i s f y t h e c o n d i t i o n o f t h e l e m m a , w h e r e F i s a n m x m m a t r i x . Then

T - ~ 0 `k X[ T1 X1 x2

where F ' is an (m - 1) x (m - 1) diagonal matrix, X I = X1 x2

T1 is precisely of the form handled when m = 1, and so we can find R �9 (/4 + K:)(7-/~) such

that if R - 1 T i R := Q1, then Q1 �9 A ( D , n ) , HQ~Q1 - 1][ < 5, I[R-XTIR - T(~)II < 5 and

R - 1 T I R - T (n) is compact, where T1 is acting on 7-/~. Then

F' 0 0 T*~U-t-K: [; R 0 - 1 ] [X~ T 1 ] [; OR] : [R-~11tX~ R-1T1 R] :-~-T'. Thus T ' �9 (/4 +/C) (T) and therefore T' �9 A ( D , n ) and satisfies the same condition as T does. By the inductive assumption, we can find Q �9 (/4 + K:)(T') -- (/4 q- K:)(T) such

that dist (Q, u(T(n))) < 5 and Q - W * T ( n ) W �9 lC(n) for some unitary operator W. This

completes the proof of Case 1. Note that our distance estimate is actually 8 as opposed to 45 in this case.

Case 2. a ( F ) N {`k: I`kl -> 1/(1 + 35)]. r O.

By an appropriate choice of basis, we can assume that the eigenvalues { ` k l , " " " , ` k i n ] . of F are listed in nonincreasing order of absolute value. Thus

`kl

" ' . 0

T ~ _

3Yi �9 �9 �9 Xr

0 `kr+l

" o

`km

Xr-.t-1 �9 �9 �9 X m ~ -

= Lic ToJ'H~

2 4 6 C H I N . A N N . O F M A T H . V o l . 2 1 S e r . B

where IA, I �9 [1/(1 + 3~),1) if and only if 1 < i < r, G = (zl," " ,x~) �9 s

T o =

I A~+I 0 ] " . . 0

X r + 1 . . . X m S

Set Irk = n~ E (/4+1C)(7i). Then Y k T Y [ I = -~ Ai ~ T o

L~-G T0 -

(k --~ c~), i.e., T --ru+lc Ai ~ T 0 . Moreover, it is not difficult to check that To satisfies

all the conditions of Case 1. Because of this, we can conclude that Ai ~ T 0 "~u+tc

Ai ~ Q 1 , where q l �9 (12 +/(:)(To), IKQ1 - TOO I < (i and Q1 - ~(n) is compact. Let 2: ~ Z

Q = Ai (~Q1, thus Q �9 ( / 4 + K : ) ( T ) - . Since iiA, I I i < 0 1 / ( 1 + 3 5 ) - 1 i < 36

(1 < i < r), we may easi lyf ind A~- �9 DD such that iAi-A~i < 35 (1 < i <_ r). Since (-)

T ~ is essentially normal and OD C a~(T~ ), by the arguments of Proposition 2.1 we have

T~ (n) = W [ [ • A '~mT(")] W* ~- K, where g is a compact operator with tlgl] < 5, W is a L\/= 1 ] J

unitary operator. Thus

i = l i = 1 T

/----1

Moreover, Q - W*T(z")W is compact, since Q1 - Tz ('0 is compact. P r o p o s i t i o n 2.3. Suppose that T �9 .A(D, n). Then given ~, 0 < $ < 1/10, we can

find a compact operator K with i i K i l < $ such that T (~) + K ~u+~c T . In particular,

T ('~) �9 (/4 + E ) ( T ) - , and so (/4 + ]C)(T('~)) - = (/4 + ~ ) ( T ) - . P roo f . By Lemma 2.10, T --~u+~: T', where T ' �9 ,4(D, n) and is a lower triangular. Thus

T'* �9 Bn(D). From the definition of Bn(D) operators we may assume that the diagonal entries {)~k}~=l of T ~ are pairwise distinct and form a dense subset of D. Thus, for each k > 1, we have

T' " Gk 0 ---- * Ak 0 :---ZkT~"

E l l . - . Zlk T~ Since Tt*T ' - 1 =: K �9 we obtain

L T Zk Tt*1 ] /qk with respect to these decompositions of the spaces. Since K is compact, we can choose N large enough so that if k > N, ]]K4~[[ < ~/4. Let S = T~. Then S E ~4(D,n) and

]]S*S - lil < 6/4. By Lernma 2.11, d(S,/4(T(~"))) < $/4. Since the eigenvalues of GN are

No.2 JI, Y. Q., JIANG, G. L. et al ORBIT OF ESSENTIALLY NORMAL OPERATORS 247

all distinct, we can find an invertible matr ix R such that F = R - 1 G N R is diagonal. Let

X = ZNR. Then

~ 0] [0 01]:[ 0] o By Proposition 2.2 we can find an operator Q e (/d + K:)(To)- = (/d + / C ) ( T ) - and a

uni tary operator W such that

- W T) W is compact, (i) Q * (=)

(ii) IIQ-W*T(n)wII <6. Thus, setting K = WQW* "P('~) -z , we have IIKII < 6 and q , ( n ) - r ~ - ~ z . . . . u+pc T. Therefore

T ('*) E (H+IC)(T)-. By Lemma 2.6, (H+K:)(T~('~)) - = (H + / C ) ( T ) - .

Now we are in a position to prove Theorems 1.1, 1.2 and 1.3.

P r o o f o f T h e o r e m 1.1. By Lemma 2.6, there are a compact operator Kx with IIKI II <

e/2 and an operator Y E (/g +/( : ) (7/) such that T1 + K1 = YT(~n)Y -1. Let r be the

analytic function in Lemmas 2.2-2.3. Then r is analytic in D and r = r is analytic in

~ . We can find 61, 62 > 0 such that r D D ~ . Set A = r E A(D,n) , B = T(~ n). Let ml = max{Ir : Izl = 1 -I- 62} and m 2 : m~x{ll(z - B ) - I I I : I~l = 1 + 62}. By Proposition 2.2, there are a compact operator K2 and an operator X E (H +/C)(7/) such

that X(B + K2)X -1 = A, and IIK~II < m l q l / [ ( , ~ + 1)m~1,62}, w h e r e m E N satisfies

[mlm2(1 + 6~)l/m < ~/(211YIIIIY-111). Th~ rmpnes that a(B + K1) = D . Therefore

r + K1)X -1) = Xr + K1)X -1 = r = ~b o r = T2.

Set

Thus

K3 = r + K2) - T ('0 = ~b(B + K2) - r

= 21ril / . ]=1+ , , ~b(z) [(z - B - / (2) -I - (z - B) -1] dz

1 ~ d~(z)(z B) -1 Z [ (z-- B)- lK2]ndz. 21ri i=l+z2 n=l

27rra1(l + ~2)m2 oo

IIKall < ~ II( z - B ) - l l P l l K 2 l P - 2 ~

n-~-I

= 1 mlm2(1 + 62) < .

< mlm2(1 + 62 (m + 1) n m 21lYIIIIg-lll

Since K2 is compact, K3 is compact. Thus T('~)+K3 "~uwcT2, i.e., there is a Z E (H-t4C)(7/)

such that T~'~)+K2=ZT2Z -1. Thus

T1 +K1 + Y K 3 Y -1 = Y ( T ('0 + K 3 ) Y -1 = Y Z T 2 Z -1,

and K = K1 + Y K 3 Y -1 satisfies the requirements of the theorem.

P r o o f o f T h e o r e m 1.2. If A satisfies (i), (ii) and (iii), then for each ~ > 0 by [9,

Theorem 3.48], there exists a compact operator K , IIKII < ~, such that A + K E r By Theorem 1.1, A + K E ( / / + / C ) ( T ) - . Thus A E ( L / + / C ) ( T ) - . If A E (/4 + K~)(T)-, then X,~TX~ 1 --7 A (n --+ eo) for a sequence of invertible operators {X,~} C (/4 + ~)(7/) . Since lr(X~)7f(T)7r(X~ -1) -+ ~r(A) (n -+ eo) and since each ~r(X~) is a uni tary element,

248 CHIN. ANN. OF MATH. Vol.21 Ser.B

ar = a~(A) : Of~ and nul (A - T) = nul (A - A) for all A e [a(T) U a ( A ) ] \ a e ( T ) . Thus A

is essentially normal , 12 C a ( A ) and for A e f~, ind (A - A) = - n . Conversely, assume tha t

A �9 a ( A ) N p ( A ) bu t A ~ ~ . Since A �9 p ( X n T X ~ I ) , i n d ( A - A) = 0 -- n u l ( A - A). Thus

A �9 p (A) , a contradict ion. Thus a ( A ) = ~2.

P r o o f o f T h e o r e m 1.3. By T he o re m 1.1, it suffices to prove t h a t there exists a compac t

opera tor K , Ilgll < e, such tha t T (~) + K �9 ( S I ) , or (T~)(~) + g �9 ( S I ) , where r is an

analyt ic homeomorph i sm from D onto ft. Since T~ �9 Bl(f~), by [12, L e m m a 2.3], we can

find compac t opera tors K 1 , K 2 , . " , K n such t h a t [[Kill < e/2, Ai : : T~ + g i �9 BI(~'~) (i : 1 , 2 , - . . ,n ) and kerTA, Aj : {0} (i ~ j ) . Since a t ( A 1 ) n a l ( A i ) ~ O (i : 2 , 3 , . . - , n ) ,

by [9, Theorem 3.53] there are compac t opera tors C2, C 3 , ' - . , Co such t h a t Ci ~ ranTA1n,

and [[Ci[I < e/2 i (i : 2 , 3 , . - - , n ) .

Set

I K1 C2 . . . Cn

K2 K ----- " .

0 Kn

T h e n g is compac t and [[gl[ < e. I t is easily seen t h a t (T~)(~) + g �9 ( S I ) .

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