Structure of Hilbert Space Operators

F\ is a linear transformation for every AeA. A Hermitian holomorphic vector bundle E over A is a holomorphic bundle such that each fibre E\ is an inner product space. Two Hermitian

Background

5

holomorphic vector bundles E and F over A are said to be equivalent if there exists an isometric holomorphic bundle map from E onto F. Let H be a separable complex Hilbert space and let n be a positive integer. Denote Gr(n,H), the Grassmann manifold, the set of all n- dimensional subspaces of H. For an open connected subset A of C fc , a map / : A—>G r (n,H) is said to be holomorphic if at each AoGA there is a neighborhood A of Ao and n holomorphic W-valued functions n

^i(z),e2(z),-"

>e„(z) such that f(z) = V ( e j( 2 )}-

If

/

:

^—>G r (n,K) is

3=1

a holomorphic map, then an n-dimensional Hermitian holomorphic vector bundle Ef over A and a map can be induced by / , i.e.,

Ef.=

{(x,z)eHxA:xef(z)}

and (j): Ef—>A,<j){x,z) = zeA. Given two holomorphic maps / and g : A—>Gr(n,'H), we have two vector bundles Ef and Eg over A. If there exists a unitary operator U on H such that g = Uf, then / and g are said to be unitarily equivalent. If there is an open subset A of A such that Ef\& is unitarily equivalent to •Eg I A! then Ef and Eg are said to be locally unitarily equivalent. Rigidity Theorem Let A be an open connected subset of Cfc and let f and g be holomorphic maps from A to Gr(n, TCj such that

V /(*) = V aw = HzeA zeA Then f and g are unitarily equivalent if and only if Ef and Eg are locally unitarily equivalent. 1.4

Some Results on Cowen-Douglas Operators

Let Q be a connected open subset of C, n is a positive integer, the set Bn(Q) of Cowen-Douglas Operators of index n is the set of operators T&JC(H) satisfying

(i)

Oca(r);

(ii) r a n ( z - T ) := {(z - T)x : x£H} = H for each z€fi; (iii) V/ fcer(* -T) = H\ (iv)

dimker(z — T) = n for each zefi.

6

Structure

of Hilbert Space

Operators

It can be proved that if Qo is a nonempty open subset of fi, then Bn(£i)cBn{yio). For an operator T£Bn(fl), the mapping z\—>ker(z — T) defines a Hermitian holomorphic vector bundle of rank n. Let (ET,TT) denote the subbundle of trivial bundle QxH given by ET := {(z,x)eflxH : xeker(z-T)

and n(z,x) = z}.

Let A'(T) be the commutant of T, i.e., A'{T) := {A&C{H) : TA = AT}, then for TeS„(fi), there is a monomorphism TT from ^4'(T) into H™,E JQ) satisfying TTX = X\ker{z_T) for XeA'(T) and zeft, or r T X ( z ) = ker(z-T) '•— X(z), where -ff^(j5;T)(^) is the set of all bounded bundle endomorphisms from E? to ETTo summarize the above and Section 1.3, we can find a holomorphic frame (ei(z), • • • ,en(z)) such that n

ker(z-T)

= \/ ek(z), ZGQ for TeB„(fi).

Fix a zoSfi, denote Hi = ker(z0 - T ) , H2 = ker(z0 - T)2Qker(z0 Hm = ker(zQ - T)mQker(zQ

- T), - T)" 1 " 1 .

We have: Theorem CD1 [Cowen, M.J. and Douglas, R. (1977)] m

(i)

E ©Wfc = V ( e

fc=i

...

(*o) : l < i < « , 0
00

r«; E®-Hk = n; k=l

(Hi) {el- (zo) : l<j
—

The following theorems will often be used in the chapters hereafter. Theorem H [Herrero, D.A. (1990)] IfTeBn(Q), then ap{T*) = 0, where T* is the adjoint ofT and o~p(T*) is the point spectrum ofT*. Theorem J W 1 [Jiang, C.L. and Wang, Z.Y. (1998)] Let Te6„(fi) and let Pz be the orthogonal projection from H onto ker(z — T) for z£fl, then (I — Pz^heriz-T)1- *s similar to T.

Background

7

Theorem J W 2 [Jiang, C.L. and Wang, Z.Y. (1998)] Let TeC(H). For given numbers > 0, there exist a positive integer n and Cowen-Douglas operators {Ai}\=1, {bj}? = i +1 such that

F-(®^)©(® s*)||<£. j=i

i=i+i

Theorem J W 3 [Jiang, C.L. and Wang, Z.Y. (1998)] Given TeBi(fl), there exist compact operators K\,K2,--- ,Kn,--- with \\Ki\\ < 7^- such that T + Ki&Bi(Q) and kerTT+K,T+K, = {0},i ^ j , where TA,B is the Rosenblum operator from C(H) to C(H) given by TA,B(X) = AX — XB forXeC{H). 1.5

Strongly Irreducible Operators

Operator T is strongly irreducible if there is no nontrivial idempotent in A'(T) ([Gilfeather, F. (1972)], [Jiang, Z.J. (1979)], [Jiang, Z.J. (1981)]). Operator T is irreducible if there is no nontrivial orthogonal projection in A'(T) ([Halmos, P.R. (1968)]). It is obvious that strongly irreducibility is invariant under similarity while irreducibility is just unitarily invariant. Denote (57) and (IR) the set of all strongly irreducible operators and irreducible operators, respectively, on H. Let K{l-L) be the ideal of compact operators on H and let •K : £(H)^A(H)

:= C{H)/1C(H)

be the canonical quotient mapping, A(H) is called the Calkin algebra. The essential spectrum of operator T is ae (T) = {AsC : A—ir(T) is not invertible in A{H)} and the Predholm domain of T is pF(T) = C\ae(T). It is well known that o-e{T) = ale{T)U
aifr(T))

are(T) :=

O-T(TT{T)).

and

8

Structure of Hilbert Space Operators

Operator T is a Fredholm operator if 0€PF(T). T is a semi-Fredholm operator if the range of T, ranT, is closed and either nulT := dimkerT or nulT* := dimkerT* is finite. In this case the index indT of T is defined by indT := nulT - nulT*. The Wolf spectrum aire (T) of T is given by alre(T)

:=

are(T)nale(T)

and PS-F{T) := C\oire(T) is the semi-Fredholm domain of T. The spectral picture A(T) of T consists of the compact set <7;re(T) and the index ind{T— A) on the bounded connected components of ps-F(T). Spectral picture theorem of strongly irreducible operators [Jiang, C.L. and Wang, Z.Y. (1996b)] Let T be in C(H) with connected spectrum cr(T). Then there exists a strongly irreducible operator L satisfying (i) A ( L ) = A ( T ) ; (ii) TeS(L)-; (Hi) If there is another strongly irreducible operator L\ with A(Li) = A(T), then L\ES(L)~, where S(L) is the similarity orbit of L, i.e., S(L) := {XTX-1 and S{L)~ is the norm closure

: Xe£(H)

is invertible}

ofS(L).

Spectral picture theorem of Cowen-Douglas operators [Jiang, C.L. and Wang, Z.Y. (1998)] Let T be in C(H) with connected a(T) and a(T)\p®_F(T). If pp{T) ^ 0, then there exists a Cowen-Douglas operator A&(S I) such that A(T) = A(A) and if there is another operator B&S(T)~, P°S-F(T)

:= {\£ps-F(T)na(T)

then BGS(A)~,

where

: ind(X - T) = 0}.

9

Background

Commutant theorem of strongly irreducible operators [Fang, J.S. and Jiang, C.L. (1999)] Operator T is strongly irreducible if and only if a (A) is connected for each A in A'(T). The following theorem will be used frequently in this book. Theorem CD2 [Cowen, M.J. and Douglas, R. (1977)] ator in B\(fl) is strongly irreducible.

1.6

Each oper-

Compact Perturbation of Operators

We introduce only two famous theorems on compact perturbation of operator here. Brown-Douglas-Fillmore theorem / / T\ and T2 are essentially normal operators on Ti, then a necessary and sufficient condition that T\ be unitarily equivalent to some compact perturbation of T2 is that o-e(Ti) = o-e(T2) and ind{\ — T\) ~ ind{\ — T2) for ««A^cr e (Ti). An operator T is essentially normal if T*T — TT* is compact. Voiculescu's theorem Let T£C(H) and p be a unital faithful *representation of a separable C* -subalgebra of the Calkin algebra A(H) containing the canonical images n(T) and n(I) on a separable space Ti.p. Let A •= p(w(T)) and k be a positive integer. Given e > 0, there exists KeK,(H), with \\K\\ < e, such that T -

K^T®A^^T®A^k\

where "=" means unitarily equivalent.

1.7

Similarity Orbit Theorem

Complex number A is a normal eigenvalue of T if A is an isolated point of cr(T) and the dimension of H(X,T), the range of the Riesz idempotent

10

Structure of Hilbert Space

Operators

corresponding to A, is finite. Denote the set of all normal eigenvalues of T by ao(T). The minimal index of A — T, AS/0s_ir(T), is defined by min-ind{\

— T) := min{nul(X — T), nul(X — T)*}.

Similarity orbit theorem Given T, Ae.C(H) satisfying (i) min-ind(X — T)k for all \£ps-F(A) and k>l; (iv) There is no isolated point in ae(T); then AeS(T)~. Note that this is only a sufficient condition of the similarity orbit theorem. The reader is referred to [Apostal, C , Fialkow, L.A. Herrero, D.A. and Voiculescu, D. (1984)] for the general form of the theorem. 1.8

Toeplitz Operator and Sobolev Space

Let D and C be the open unit disk and unit circle in the complex plane respectively, and let p, be the Lebesgue measure on C, normalized so that fi(C) = 1. If e„(z) = zn for ZGC and n = 0, ± 1 , ±2, • • •, then {e n , —oo < n < +00} is an orthonormal basis (ONB) for L2(C, p). Let H2 := span{en : n>0} and H°° = L°°(C,fi)nH2. If P is the orthogonal projection from L2(C,p) onto H2 and if ^eL 0 0 (C,p), then the Toeplitz operator T$ with symbol <j> is defined by T+f = P{cpf) for all feH2. If is a positive measurable function on C such that logcpGL1 (C', p) and if Q(z) = c-exp{fc ^^log)} for z€C, then Q is called an outer function, where c is a constant and \c\ — 1. The Blaschke product is a class of inner functions with the form

where k>0, c e C , with \c\ = 1 and {A.,} is a sequence of nonzero numbers 00

in D satisfying J2 (1 — l^jl) < +00.

Background

11

Factorization theorem If f&H2 and / ^ 0, then f is the product of an inner function m and an outer function Q, i.e., f = mQ. Beurling's theorem A subspace H\ of H2 is invariant under the operator Tz if and only if Hi = 4>H2 := {<j)f : f&H2} for some inner function

4>. Let Q be an analytic Cauchy domain in the complex plane and let W22(Q.) be the Sobolev space „W22 ,„,„,

f „

Tn,^

M:=[feL2(n,dm):

, x the distributional derivative of first and ] secondorderof/belongtojL2(Adm)

where dm denotes the planar Lebesque measure. For f,geWn(tt), we define (/,
then

)

W22(fl)

\a\<2

is a Hilbert space and a Banach algebra with identity under an equivalent norm. By Sobolev embedding theorem, f&W22(fl) implies that f£C(Q) and ll/llc(n)<M|l/llw»(n) for some M > 0. Thus a sequence of functions {fn}^Li converges to / in W22(Q) implies that / „ converges to / uniformly on Q. For f£W22(Q,), the multiplication operator Mf on W22(Q) is defined as follows Mfg = fg,

geW22(Q).

Let W(ti) := {Mf : feW22(Q)}, then W(fl) is a strictly cyclic operator algebra with strictly cyclic vector e(s,t) = 1. An operator algebra A on a Hilbert space 7i is said to be strictly cyclic if there exists a separating vector e such that Ae := {Ae : AGA}

= H.

Theorem J W 4 [Jiang, C.L. and Wang, Z.Y. (1996a)] (i) a(Mz) = aire(Mz) = H ; (it) A'(Mz) = W(fl); (Hi) Aa(Mz) = R(Q), where Aa(Mz) is the algebra generated by the rational function of Mz with poles outside fi and R(i}) is the closure in W22(Q) of all rational functions with poles outside fl.

Chapter 2

Jordan Standard Theorem and K0- Group 2.1

Generalized Eigenspace and Minimal Idempotents

Recall that a kxk Jordan block X 1 A

0"

Jk(X) = 0

1 A.

has the following properties (i) Jfc(A) is strongly irreducible on Ck, i.e., there is no nontrivial idempotent in A'(Jk(X))\ (ii) nul(X- Jfc(A)) = l,ker(X-Jk(X)y g ker(XJk{X))i+l,l<jk; (iii) A'(Jk(X)) consists of all kxk lower triangular matrices B 0 where di£C,i

B = o-k

• • • a-2

= 1,2,-

, fc\

a\_

It follows from (iii) directly that (iv) A'(Jk(X))/radA'(Jk(X))~C. For j4eM n (C) and XGCT(A), ker(X — A) is called the eigenspace of A related to A. If there is a positive integer m,m < n, such that ker(X - A)m = ker(X -

A)m+1,

then ker(X — A)m is called the generalized eigenspace of A related to A. By 13

Structure

14

of Hilbert Space

Operators

elementary matrix theory, we have the following proposition. Proposition 2.1.1 Let Ae.Mn(C) and \£a(A) with nul(X - A) = 1 and let M be the generalized eigenspace of A related to X, then A\tf~Jk{X) for some k,k < n. A nonzero idempotent PGA'(T) is minimal, if for each idempotent Q&A'(T), ranQcranP implies Q = 0. Proposition 2.1.2 Let A&Mn(C) and let P be a minimal idempotent in A'(A), then there exists a number AsC, such that ^4|ranP~«^fc(A), where k = dimranP. Proof Since PEA'(T), ranP and ran(I — P) are invariant subspaces of A. {Claim 1} B :— A\ranp has a unique eigenvalue XB- Otherwise, it follows from the theory of linear algebra that P is not minimal. {Claim 2} B is strongly irreducible and UUI{XB — B) = 1. Since P is minimal, BG(SI). Using the basic theory of linear algebra and the fact that P is minimal again, we have HUI{XB — B) = 1. Thus the proposition is a conclusion of Proposition 2.1.1. Let {Afc}£=1 be all of the eigenvalues of A e M n ( C ) , the multiplicity is included, by Proposition 2.1.2 and the theory of matrix, we have the next proposition. Proposition 2.1.3 {PxiliLi such that (i)

Let A€Mn{C),

then there exist minimal idempotents

E Pxk = / c - and PXk Px3 = 0 for k ± j ; fc=i

(ii)

A\ ranP\.

~Jmfc(Afc), where rrik =

dimranP\k;

n

(Hi) A = Y, +A\ranP^ • 2.2

Similarity Invariant of Matrix

Jordan standard theorem Let A£Mn(C), sum of finitely many Jordan blocks, i.e., i

A~Q)Jmk(Xk). k=i

then A is similar to a direct

Jordan Standard

Theorem and

Ko-Group

15

By Propositions 2.1.2, 2.1.3 and Jordan standard theorem, the similarity of two nxn matrices A and B depends completely on their eigenvalues and generalized eigenspaces. A quantity (or quantities) or property (or properties) V is similarity invariant if operator A has V and A~B imply B has V. For a subset 11 of C(H), similarity invariant (or invariants) V is completely similarity invariant if AGTZ, then A~B if and only if BE.% and A and B have same V. One of our aims is to find or determine complete similarity invariants of operators. But if H is an infinite dimensional separable Hilbert space, it is very difficult to obtain complete similarity invariant of operators in £(7i). As a matter of fact, some operators in £(Ji) even have no eigenvalues. To provide a new idea for finding of the similarity invariants of Hilbert space operators, we interpret the Jordan standard theorem in the view point of i^o-theory. The propositions below are from [Blanckdar, B. (1986)] and [Aupetit, B. (1991)]. Proposition 2.2.1 K0(Mn(C))^Z and V(M n (C))^N, where N {0,1, 2, • • • } andZ = {0, ± 1 , ±2, • • • }.

=

Proposition 2.2.2 Let A be a unital Banach algebra and let P be an idempotent in A and R&adA. If P + R is still an idempotent in A, then there exists an invertible element XGA such that X{P + R)X_1 = P. We know that A'{Jk(X))/radA'{Jk{X))=C for each Jordan block Jfc(A). If P is an idempotent in A'(Jk(ty), it follows from the structure of A'(Jk(X)) that P = ICk + R, where R is a lower triangular idempotent. Thus P~A'{Jk(\))Ic>< by Proposition 2.2.2. From Proposition 2.2.1 and the definition of .RVgroup, we have the following proposition. Proposition 2.2.3 Lemma 2.2.4

\J{A'{Jk{X)))^N

Let A\, A2€(SI)
and

K0(A'{Jk(X)))=Z.

satisfying i = 1,2,

then at least one of the following is true (i) AX~A2; (ii) If X,Ye£{H) with AXX = XA2 and YAx =• A2Y, then XYeradA'(Ai) and YXGradA'(A2). Proof If A\ is not similar to A2 and AXX = XA2, YA\ = A2Y. Thus AXXY

= XA2Y

= XYAX

Structure of Hilbert Space Operators

16

and XYtA'iAi). If XY&radA'iAx), then XY = A + R for some AeC,A ^ 0 and R&radA'{Ai), since ^'(A^/racM'(;4i)5=!C. Therefore XY is invertible. Similarly, YX is invertible. This implies that X and Y" are both invertible, i.e., Ai~A2. The contradiction indicates that XYGradA'{A\). Similarly, YX&adA'{A2). i

Lemma 2.2.5

Let A£(Mn(C)),

then A'{A)/radA'(A)^

*£

Mki(C),

i=l

where k\ + k2 H Proof

\-ki—n.

By Proposition 2.1.3 and Proposition 2.1.2, we have m

A~^®Jfci(Ai). i=l

For simplicity we prove the lemma for m = 2 and consider the following two cases. {Case 1} k\ = k2 and Ai = A2, therefore Jfc^Ai) = Jk2{^2)- A simple computation indicates that 0"

Oil

A'(A) = <

a2

An A12 A21 A22

Oil

012 Oil.

Oik

from which we have

radA'(A) = <

Ru -R12

0 a 0

0

Ri

>.

R21 R22

/?••• a 0 Thus A'(A)/radA'(A)^M2{C). {Case 2} fci ^ k2 or Ai ^ A2. If k\ ^ k2, without loss of generality we assume that k\ > k2. By Lemma 2.2.4, if XGkerTj ( hJ (A } and YekerTj .. w .. ., then XY&adA'(JkM),YX&adA'{Jk2(\2)).

Jordan Standard Theorem and KQ-Group

17

Set J =

Xn X±2 X21 X22

XneradA'(JkM), X22eradA'(Jk2(X2)),

X12X21&adA'(Jkl^i)) X21X12£radA'(Jk2(\2))

It is easily seen that J is a two sided ideal of A'(A).

}•

We claim that

Xu - A X\2 X2\ X22 — A is invertible for each A 7^ 0 and Xu X2i

Xi2 &J. X22

In fact, observing that 1 0 -(Xn-A)-1!

Xu — A XX2 X2\ X22 — A

X\\ — A X\2 0 (X22-A)-X12(X11-A)-1X2i. where Xl2(Xlx

- \yxX2X&adA\J^{\2)). (X22 — A) - X\2{Xii

is invertible. Therefore

Thus — X)~

X2\

X\\ — A X12 is invertible. X21 X22 — A

The claim above indicates that CF{X) — {0} for each X&J.

Thus

J = radA'(A) and A'(A)/radA'(A)^C®C. This complete the proof. 1

Theorem

2.2.6

Let

AeMn(C)

and

i4~E®4W(ni).

i/ien V(-^'(^))=N ( Z ) and K0(A'(A))^Z«\ where Jki{Xi)(ni) orthogonal direct sum of rii copies of Jki{Xi).

denotes the

18

Structure of Hilbert Space

Proof

By Lemma 2.2.5, A'(A)/radA'(A)S*

Operators

£ ©M„,(C). By Proposition i=l

2.2.2 and Proposition 2.2.3 we have

i

\ / { E ®^(C)}SN("

\/(A'(A))= \j{A'{A)/radA'(A)}^

2=1

and K0(A'(A))=iZ^. Theorem 2.2.6 is another form of Jordan standard theorem. The following result gives the complete similarity invariant for matrices in terms of ifo-grc-up. m

Theorem 2.2.7

Let A,BeMn{C)

and A = £ © 4 ? i=i

and Aki is not similar to Akj for i ^ j . Then A~B exists an isomorphism h such that

with

Akie{SI)

if and only if there

h(K0(A'(A®B)))^ZW and h[IA,(A@B)} = 2niei + 2n2e2 + ••• + 2nkek, where 0 ^ n i £ N , i = 1, 2, • • • , k, {e,}^ =1 are the generators of the semigroup N ' m ' of Z^"1' and IA'(A®B) is the identity of A'(A®B). m

Proof

If A~B,

,„

then A®B~ £ ®A£ni).

.

The "necessary" part follows

i=l

from Theorem 2.2.6. Proof of "Sufficient" part.

Since

BGM„(C),

B ~ £ ®Bk

>

where

j=l

Bkj£(SI), and

Bkj + Bkj, if j rf, f.

It follows from

h[lA'(A
h(K0(A'(A®B)))^Z^ + 2nkek

that I = m. By Lemma 2.2.4, for each i?fc. there exists an Ak. such that Bkj~Aki and m^ = n s . This implies that A~B. 2.3

Remark

Theorem 2.2.6 and Theorem 2.2.7 are different forms of Jordan standard theorem and can be obtained in different ways. Lemma 2.2.4 is due to [Cao, Y., Fang, J.S. and Jiang, C.L.(2002)].

Chapter 3

Approximate Jordan Theorem of Operators

In the operator theory of finite dimensional space, or in matrix theory, Jordan canonical theorem is one of the core contents. The Jordan theorem gives the complete similarity invariant of matrices. But in the infinite dimensional Hilbert space case, it is very difficult to find complete similarity invariant for operators. We can only give an approximate Jordan theorem, or obtain complete similarity invariant for some special class of operators. In this chapter, we give some different kinds of approximate Jordan theorems considering strongly irreducible operators as the replacement of Jordan blocks in matrix theory.

3.1

Sum of Strongly Irreducible Operators

[Radjaval, H. and Rosenthal, P. (1973)] proved that every operator in C(Ti.) is a sum of two irreducible operators. In this section, we will prove the following result. Theorem 3.1.1 Every bounded linear operator on H. is a sum of two strongly irreducible operators. In order to prove the theorem, we need the following lemmas. Lemma 3.1.2 Assume that Te£(H) with indT = —1 and min-indT = 0. If B\ and B2 are two left inverses ofT and eo is a unit vector in (ranT)-1, then there exists an f£H such that B\ — B2 + /<8>eo. Proof Set f =(B1B2)e0 and A = B1-B2/<8>e0. Since indT = - 1 and since min-indT = 0, ranT is closed and dim^anT)1= 1. Thus for each XGH, there is a number a€C and x\€H such that x = ae^ + Tx\. 19

20

Structure

of Hilbert Space

Operators

Since BiT = B2T = I, Ax = a{Bi - B2)e0 + B{Txi - B2Txx - a = af + X1-X1af- < Txi,e0 > f = 0.

eo -

(f®e0)Tx!

Thus, A = 0 and B\ = B2 + f®e0. Lemma be a unit ThenTB Proof and y€7i

3.1.3 Let TGC(H) with indT = — 1 and min-indT = 0. Let eo vector in (ranT)1- and B be a left inverse of T with Beg = 0. = 1 -e0®e0. Set A = TB + e0(g>eo — / . Since for each XGH there exist a e C such that x = aeo + Ty,

Ax = aTBeo + TBTy + a(e 0 ®eo)e 0 4- (e 0 ®e 0 )Tj/ — ae0 — Ty = 0 + Ty + ae0 + 0-ae0-Ty = 0. Thus A = 0 and TB = I - e0e0. Proposition 3.1.4 BG(SI),

Given T, eo and B as that in Lemma 3.1.3.

If

thenTe(SI).

P r o o f If there is a nontrivial idempotent PGA'(T), kerT* is an invariant subspace of P*. Since indT = —1 and min-indT = 0, dimkerT* = 1. Therefore, P*eo = Aeo and A = 0 or A = 1. Assume that A = 0 (otherwise consider I — P), then P =

0 0

Q IranP'

kerP ranP*

and

T=

T i T12

0 T2

kerP ranP*.

Set

x=

IkerP

Q

0

IranP'

kerP ranP* ' p =

X-'-PX

0

0

0

IranP*

kerP ranP*

and f = X-1TX =

T: 0 0 T2

kerP ranP*.

Note that P* = P , PeA'(f),indf = - 1 and min-indf = 0. Thus Ti and T2 are injective with closed ranges and indT\ + indT2 = indT = — 1. Assume that e 0 = ei + e2, where eiGkerP and e2€ranP*. Since P*eo = 0 and e 0 ^ 0,ei ^ 0 and Q*ei = - e 2 . Since Tj*ei + T^ex + T2*e2 = 0, Ti*ei = 0. Thus indTi = - 1 and indT2 = 0. This implies that T2 is

Approximate

21

Jordan Theorem of Operators

invertible and T\ is left invertible. Suppose that B2 is the inverse of T 2 and B\ is a left inverse of Ti satisfying kerB\ = (ranTi)1-. Thus Biei = 0. Set Bt 0 0 B2

B =

kerP ranP* '

then B is a left inverse of f and BP = PB. Set B = X ^ R X " , then B is also a left inverse of T. It follows from T*ei = T,*ei = 0 that jrhi is a unit vector in (ranT)1-. By Lemma 3.1.2, there exists a vector / such that J5 = B + /®ijU|. From-BX^eo = X ^ B e o = 0,Xe o = e 0 + Q e 2 + e 2 and BXeo = 0, we have: 0 = 6(ei + Qe2 + c 2 ) = 5 i e i + 5 i Q e 2 + 5 2 e 2 + < e2, ^

> / + < Qe 2 , p ^ > /

= 5! Qe2 + £ 2e2 + | | e 1 | | / - g | r / . If e 2 ^ 0, it follows from BiQe2±B2e2 Thus /:

ei 2

|e2|| -||e1||2

and B2e2 7^ 0 that BiQe2+B2e2

BxQe2 +

ei

Ml 2 -INI 2

^ 0.

B2e2

and D

1

" 1

"r

B

BiQe20ei

l l « . „ l | 2 _ n „ , 112

I|e2|r-||ei|| j?2£2®ei

IMP-IK

n

<J

52

kerP ranP*

Moreover, herP - $$•

+ Qe2®el - < 2 | a ^ e i ® e i

0

TB =

(3.1.1) e?<»ei e2<»

i-ranP*

By Lemma 3.1.3, TB = I — eo®eo = I — ei®ei — e\

e\ - e2
Therefore,

TB

-e 2 ®ei *

kerP ranP*

(3.1.2)

Structure

22

of Hilbert Space

Operators

Compare (3.1.1) with (3.1.2) we have

INI 2 - INI 2 = 1. Since || e i || 2 + ||e 2 || 2 = ||e 0 || 2 = 1, ||e 2 || = 0 and e 2 = 0. Thus, Pe0 = Pex = 0. Therefore, T{PB - BP) = P(TB) -

(TB)P

= P(I - e 0 ®e 0 ) - (/ - e 0 <8e 0 )P = - ( P e 0 ) ® e 0 + e 0 ®P*e 0 = 0.

But T is injective, thus PB - BP = 0, i.e., PB = BP. This contradicts B&(SI). Therefore TG(SI) and the proof of the proposition is now complete. Lemma 3.1.5 Let A, S&C(H), \\A\\ < \ and let S be the froward unilateral shift. Then for each A G C with |A| < | , S + A — A is a Fredholm operator and ind(S + A - A) = - 1 , min-ind(S + A - X) = 0. Proof

Note that for each x£H, \\(S + A-

X)x\\>\\Sx\\ - \\Ax\\ - | A | H > ( 1 - piDllzll.

Thus S + A — A is bounded below, ran(S + A — A) is closed and dimker(S + A - A) = 0,min-ind(S + A-

X) = 0.

On the other hand, it follows from | | ( ^ - A)5*|| < 1 that I + (A - X)S* is invertible. Therefore, ind{S + A - A) = ind[S + {A-

X)S*S]

= ind[(I + (A - X)S*)S] = ind[I +{A-

X)S*} + indS

= 0 + (-l) = - l .

Approximate

23

Jordan Theorem of Operators

Thus S + A — A is a Fredholm operator. Lemma 3.1.6

Assume that Te£(H),

X0eC,ind(T

- A0) = - 1 and

min-ind(T — Ao) = 0. Then T~T\ran(T_\0). Proof Define X : H —> ran(T - A0) by Xx = (T - A0)x for x€H. Since ran(T — Ao) is closed and T — Ao is injective, X is invertible. Note that T\ran{T-x0)Xx

= T(T - X0)x = (T-

\0)Tx

=

XTx,

thus T\ran(T-\o)X

=

XT,

i.e., T\ran(T-\0)~T.

Lemma 3.1.7 Given matrix representation

AGC(H)

A =

that admits the following lower triangular

"Ao a2\ 0 a32 0

0

eo ei e2

(3.1.3)

with respect to the ONB {efc}£L0. 7/||A|| < | , then (S + A)*e#i(fi), where S is the unilateral shift. Proof Since ||A|| < \ and |A0| < \, ind(S + A - A) = - 1 and min-ind(S + A - A) = 0 for |A| < - . Thus (S + A - A)* is surjective and nul(S + A - A)* = 1. In particular, ker(S + A-

A0)* = {ae0 : aGC}.

Therefore ran(S + A - A0) = \J{ek : fc>l} := Hx. Set T = (S + A)\Hl. Lemma 3.1.6, S + A~T. Thus r a n ( T - A ) * = Hi

By

24

Structure

of Hilbert Space

Operators

and nul{T - A)* = 1 for all AGfi := {A : |A| < i } . Clearly, T* admits a strict upper triangular matrix representation with respect to {ek}^Li- Thus \J{kerT*k,k>l}

= Hi.

This implies that T*eBi(ft). Thus (S + 4)*eSi(fi). Lemma 3.1.8 Given matrix representation

that admits the following upper triangular

BGC(H)

0

eo

&22

ei e2

0 0

B = 0

with respect the ONB {ek}f=0. If \\B\\ < 1, then S + BG(SI), the unilateral shift. Proof Since \\BS*\\ < 1, (I + BS*) is invertible. Set A = S*(l +

where S is

BS*)-\

Thus A - A = S*(I + BS*)-1^ - \{I + BS*)S) for |A| < \\I + BS*\\-\ Note that (/ — X(I + BS*)S) is invertible. Therefore, A — A is surjective and nul(A — A) = 1. Since A admits a strict upper triangular matrix representation with respect to the ONB {efc}£L0, \J{kerAk,k>l} = H. 1 Denote D, = { A G C : |A| < (1 + HSU)- }, then AeB^Q.) and AG(SI). Because of A{S + B) = A(I + BS*)S = I, A is a left inverse of S + B and Ae0 = A(e0 + BS*e0) = A(I + BS*)e0 = S*(I + BS*)-1(I

+ BS*)e0

= S*e0 = 0. It is easy to see that eo£[ran(S + B)}1-. By Proposition 3.1.4, S +

Be(SI).

Approximate

Lemma 3.1.9 representation

25

Jordan Theorem of Operators

Let BEC(?{)

admit the following upper triangular matrix

b13

0 bu

B

&22 ^23

•••

eo

*

ei

0

with respect the ONB {ek}kL0- If\\B\\ < a S C satisfying

then there exist

JQ,

(ii) \\B\\ < I and B admits the following matrix 0 0

0

•••

and

representation

eo

^22 &23 ' • '

B =

BGC(H)

ei e2 ;

&33

0

(Hi) S + B + ae0®e0~S + B and S + B + ae0®eo£(SI), where S is the unilateral shift. Proof Denote H 0 = {Ae0 : AeC}, Hi = V{e™ : n > l } . Then S =

0 0 eie0 Si

Ho Hi

and

0e0®/ 0 B±

B

Ho Hi

under the decomposition of the space H = Ho ©Hi. Note that Si is unitarily equivalent to S, i.e., Si=S. Set g = Si(IHl

+ SiB*i)~lf

J2(-±)nSi(SiB*i)nf.

= 71=0

Then 5-Lei and

n*n< f > r 111/1 = ! ! %

10

i=0

Since / = B*e0, ||/||<||fl*|| < ^ . Therefore || 5 || < §. Let a = - < ei,g >, n | a | < | | / | | < £,i.e. a satisfies (i). Set B =

"0 0 Ho 0 Bi + e i O g Hi

26

Structure of Hilberi Space

Operators

i.e., B = B - e 0 ® / + eiC Then \B\\ = \\Bx+e&g\\<\\B\\

+

\\g\\
and "0 0 B =

0 •••" eo

&22 ^23 • ' •

ei

.0 i.e., B satisfies (ii). Set X =

0 J Wl

i.e., X = I + eo®g. It is obvious that X is invertible and X Note that X-1(S

x

= I — eoi

+ B + ae0®e0)X

eoQs>e0 — e0<x> 0 IWl

ae 0 ®e 0

e0®/

eoQjeo e 0 ®g 0 IWl

a e 0 ® e 0 - (e 0 ®9)(e 1 <8>e 0 ) e 0 ® / - (eo®g)(Si + Bx) ei<8»eo 5i+Bi

0

a e 0 ® e 0 - < ei,g > (e 0 ®e 0 ) e 0 ® / - e 0 ®(5i + B\)*g ei®e 0 Si + B\

eo®e 0 e 0 ® g 0 /Wl .

0 0 ei<8>e0 S i + £?i

/Wl J

eo®eo e 0 ®g

L 0

/ Hl J

0 0 ei®e 0 ei®g + Si + Bx = S + B, i.e., S + B + a e 0 ® e 0 ~ 5 + B. By Lemma 3.1.8, S + Be(SI). S +B +

ae0®e0£(SI).

Therefore,

Approximate

Jordan Theorem of Operators

27

The following lemma is due to [Jiang, C.L. and Wu, P.Y. (1998)]. Lemma 3.1.10 For given TeC(H), there exist ONB { e f c } ^ 0 and A,B£C(H) satisfying (i) T = A + B; (ii) A admits a strict lower triangular matrix representation with respect to {efc}£L0; (Hi) B admits an upper triangular matrix representation with respect to {ek}%0. We are now in a position to prove Theorem 3.1.1. Proof of Theorem 3.1.1 By Lemma 3.1.9, we can find ONB {efc}£L0 and A\, B\ &£(H) such that T = A\ -f B\, A\ is strict lower triangular and B\ is upper triangular with respect to {efc}^ :1 . Denote b =< Bieo,eo >, then b is the upper left entry of the upper triangular matrix of Bi. Set Ai = A\ + 6eo®e0, Bi = B\ - be0®e0 and r = B3 rB2. 10(||A 2 || + ||B 2 || + 1)

Then || — -S3 j| < 375 a n d —B3 admits the following matrix representation 0 * *

eo ei e2

-B3 0

with respect to {efc}£L0. By Lemma 3.1.9, there is an aGC with |a| < § such that S — B3 + aeo®eo£(SI), where S is the unilateral shift. Denote Ti

1

(S - Bi + ae0

then TiG(57). Let A3 = rAi + aeo®eo and AQ = rb + a, then ||^4.i||l2|| + ]«| < and A3 admits the matrix representation (3.1.3) with respect to ONB {e/c}£°=oSince \b\<\\As

28

Structure of Hilbert Space

|Ao|<|r-6| + H < i By Lemma 3.1.7, S + A3G{SI).

+

Operators

i
Define T2 = ±(S +A3), then

T2G(SI)

and

Ti+T2 = ±(B3 - S - ae0®e0 + S + A3) = ^{rB2 - aeo<8>e0 + rA2 + ae 0 ®e 0 ) = B2 + A2 = B\— 6e0®e0 + A\ + 6eo®e0 = A1 + B1 = T.

Theorem 3.1.1 indicates that every bounded linear operator acting on an infinite dimensional, separable Hilbert space can be expressed as a sum of two strongly irreducible operators. In the proof, we used the strongly irreducibility of Cowen-Douglas operators of index 1 frequently. As a matter of fact, in finite dimensional Hilbert space we have the following theorem. T h e o r e m 3.1.11 Every operator on a finite dimensional Hilbert space is a sum of two strongly irreducible operators. Proof Let T e £ ( C n ) and A = T - (£)trT, where trT is the trace of T, i.e., the sum of diagonal elements in the matrix representation with respect to some ONB. If rankA
'Oa 0 ••• 0" 0 aO • • • 0 A^ 0

0 0 0

Approximate

29

Jordan Theorem of Operators

Let b = ( ^ ) t r T , c ^ 0 and c ^ a, then b a — c 0 ... b a — c ...

6c 0 ••• 0 6 c ••• 0 T=

o o

+ b c b

0

b a—c b

0

Since Ti~T2~J n (6)€(SI), T is the sum of two (57) operators. If rank A > 1, by the matrix theory, we have 0 a 12 • • • a l n _ i a-21 0 a 2n fln-n &nl

0 a

ai„

a„_i„

n 2 • • • O-nn-l

0 ai2 aJ3 • • • 0 a 23 • • •

0

aln a2„

0 021

0 0

+ 0 a„_i„ 0

a-n-n

O-n-12

•

O-nl

On2

•

•

0 ^ n n -- i 0

where a^ ^ 0(i ^ j) [Choi, M.D., Lausee, C. and Radjavi, H. (1981)]. It is easily seen that T is the sum of two (SI) operators.

3.2

Approximate Jordan Decomposition Theorem

If cr(T) is disconnected for T£L(7i), by Riesz decomposition theorem T£(SI), and by upper semi-continuity of spectrum, there exists an e > 0 such that T + Ag(SI) for all AeC(H) with \\A\\ < e. If a(T) is connected, we have the following theorem stated in section 1.5. Spectral picture theorem of (SI) operators Given T€JC(H) with connected spectrum <J(T), there exists a (SI) operator L satisfying (i) A(L) = A(T); (ii) TeS(L)-; (Hi) If there is another Lj6(57) such that A(Li) = A(T), then LIG5(L)".

30

Structure of Hilbert Space Operators

By the above spectral picture theorem, we get the following theorem. First approximate Jordan decomposition theorem [Jiang, C.L. and W a n g , Z. Y. (1996b)] Let T&C(H), then there exists A&C(H) such that A is the direct sum of at most infinitely many (SI) operators and satisfying (i)

TGS(A)-;

(ii) IfB€£(H)

andBeS(T)-,

then

BGS(A)~.

The interested reader is referred to [Jiang, C.L. and Wang, Z.Y. (1998)] for the proof of the theorem. In general, we call this theorem the unique decomposition theorem with respect to similarity orbit. In 1988, D.A. Herrero raised the following conjecture in personal communication: Every operator with connected spectrum is a small compact perturbation of (SI) operator. Or precisely, given T€C(H) with connected a(T) and given an e > 0, there exists a compact operator K with \^K\\ < e such that T + Ke(SI). Following theorem confirms Herrero's conjecture. T h e o r e m 3.2.1 Let T&C(H) (or T&Bn(9)) with connected a(T). given e > 0, there is a compact operator K, \\K\\ < e, such that T + Ke(SI)

(orT +

Then

KeBn(n)n(SI)).

We will prove the theorem only in the Bn(Cl) operator case. Reader is referred to [Ji, Y.Q. and Jiang, C.L. (2002)], for the general case, which is a little more complex. Before we begin our proof, we need the following lemmas. Lemma 3.2.2[Fialkow, L.A. (1981)] ing are equivalent: (i) ar(A)nat(B) = 0; (ii) TA,B is surjective; (Hi) ranTA,B contains IC(H).

Let A, B^C(H),

then the follow-

Lemma 3.2.3 [Herrero, D.A. and Jiang, C.L. (1990)] Let A,BeC(H). Assume that H = \J{ker(X - B)k : Aer,fc>l} for a certain subset T of the point spectrum o~p(B) of B, and ap(A)r\T — 0, then T A,B is infective. Let 0 be a non-empty open connected subset of C such that (Q)° = Q, where (Q)° denotes the interior of the closure Q of Q. Given zo&Q, there exists a probability measure /z supported by T := d£l, the boundary of Q, satisfying f(zo) = fr fd^i for every function / analytic on fi [Herrero,

Approximate

Jordan Theorem of Operators

31

D.A. (1990)]. Let M(T) = "multiplication by A" on L 2 (T,/i). The subspace H2(T) of L2(T,fj,), spanned by the functions analytic on Q, is obviously invariant under M(T), i.e.,

M(r) =

# 2 (r) M+(T) Z 2 2 o M_(r) L ( i » e # ( r y

Lemma 3.2.4[Herrero, D . A . (1990)] Let M{T),M+{T) and M _ ( r ) 6e as above, then (i) M(T) is normal and both M+(F) and M-(T) are essentially normal; (ii) a{M{T)) = o-e{M{T))=
a ( M + ( r ) ) = a ( M _ ( r ) ) = fi; *nd(A - M+(T)) = -nul(X - M+(T)) = -nul{\ - M+(T))* = - 1 for all \€il; (Hi) If Q is simply connected, then \\Z\\<m^ ', where m denotes the planar Lebesgue measure. It is obvious that M + ( r ) * € # i ( £ r ) and M-(T)eBi(Q),

where Q* := {A :

Aeft}. An operator is quasitriangular if ind(\ — T)>0 for all A€/9 s _ir(T). The Weyle spectrum aw{T) of TeC{H) is defined by aw{T) := D M ? 1 + # ) •' K&K(H)}. It is well known that aw(T) := {AGC : A - T is not a Fredholm operator of index 0}. L e m m a 3.2.5[Herrero, D . A . (1990)] Suppose that T£C(H) is quasitriangular and a(T) = aw(T). Let T = {A„}^L1Ccr(T) satisfy that each clopen subset of o~(T) intersects the closure ofT. Given e > 0, there exist a compact operator K, \\K\\ < e, and an ONB { e ^ } ) ^ such that *

~ai

T+K

02

ei e2

.0 where the set {a-i}^ll = T and card{j : aj = ai} = oo for each i. Lemma 3.2.6[Herrero, D.A. (1990)] Given T£C{H), nonempty subset rco"; r e (T) and e > 0, there exists a compact operator K, \\K\\ < e, such that

T +K=

0 AJ H2'

32

Structure of Hilbert Space

Operators

where N is a diagonal operator of uniformly infinite multiplicity with o-(N)=T,a(A)

= a(T),alre(A)

=

alre(T)

and ind{A - A) = ind(T - A) for each A£p s _ir(T). Lemma 3.2.7

Suppose that j, _

T i T12

~ [O T2

n2

satisfies the following conditions:

(i) (ii)

T2eBn(n)n(SI);

nul{T\ — Afc) =.n, k = 1, 2, • • • and \J{ker(Xk - Ti) : k = 1,2, •••} = Hi; (in) Bk := Pfcer(Ti-Afc)'?i2|fce7-(T-Afc) is injective, where Pfcer(Ti-Afc)* *« t/ie orthogonal projection from H onto ker(Ti — Afc)*. Then Te£ n (ft)n(SJ). Proof {Claim} fcer(T - Afc) = ker{Tx - Xk). Assume that Ti - Afc T12 0 T2 - Afc for xGHi and y£H2, then (Ti - Afc)x + T i 2 y = 0 and (T2 - Afc)y = 0. Since Pfcer(T 1 -A t )*(Tl - Afc) = 0,

Pfcer(Ti-Afc)*Ti22/ = 0. Since Pfcer(Ti-Afc)*Ti2 is injective, y = 0. This implies that ker(T - Xk) = ker(T\ - Xk)

Approximate

Jordan Theorem of Operators

33

and \/{&er(r-Afe):*>l}=Wi. Since cr 0 (Ti)nfi = {^fc}fcLi> there exists connected open subset fiiCnnp(Ti). Since T 2 eS„(fi),\f{ker{T 2 -\) : Aefti} = W2. Furthermore, \/{ker(T

- A) : Xe{Xk}^=1UQi}

= Hi®H2.

Thus Te£ n (ft). Now we will prove that TG(SI). Assume that PGA'(T) is an idempotent. Since ker(T — Xk) = ker(T\ — Xk), k = 1,2, • • • , Hi is an invariant subspace of P, i.e., P =

PiPn 0 P2

Hi H2

Since T2&{SI), P 2 = SI-^2,S = 0 or 1. Without loss of generality, we can assume that 5 = 0, i.e., P2 = 0. Thus ranPcHi. Note that PS-A'(T) and T€Bn(fl). If P ^ 0, then T'|ranp6^ r n (0) for some m, l < m < n . Since T| ranP = Ti\ • a n P i Tii =

Ti\ranp£Bm(Q),

It is easy to see that ranP is an invariant subspace of T\. Therefore, Ti =

0 T^

ranP HiQranP'

By the condition (ii) of the lemma, Ti — A is invertible for AGfi\{Afc}^=1. Thus we can find an invertible XGC(H),

x=

^ 1 1 -X"l2 _X2i X22.

ranP HiQranP

such that Xn X\2 'Tn - A £ 0 2"22 ~ X 2 i X22_

' ^.

7o" 0/

Simple computations indicate that Xu(Tu — A) = I Thus Tn — A is invertible. This contradicts Tn&Bm(n). Therefore Px = P2 = 0 and TG(SI).

34

Structure of Hilbert Space

Operators

Lemma 3.2.8[Ji, Y.Q., Jiang, C.L. and Wang, Z.Y. (1996)] Suppose that B is essentially normal and B(=Bn(Cl). Then for given e > 0, there exists a compact operator K with \\K\\ < e, such that T + K£Bn(Cl)C\(SI). Lemma 3.2.9[Herrero, D.A. (1987)] Suppose that R&C(7i) and satisfies the following conditions: (i) o~(R) and o~w(R) are connected and contain a connected open set Cl; (ii) ind{\ - R)>0 for all \€ps-F(R); (Hi) PS-F(R)DQ

and ind(X — R) = n for all AGH.

Then given e > 0, there exists a compact operator K, \\K\\ < e, such that R-KeBn(fl). Lemma 3.2.10[Herrero, D.A. (1984)] LetT€C(H) be quasitriangular operator and \£ae(T). If o~w{T) is connected, then given e > 0 and natural number m, there is a compact operator K, \\K\\ < e, such that nulT^ — km, k>\, and \J{kerT^ : k>l} = H, where Te = T + K - A. Lemma 3.2.11 Let T&C{H) be quasitriangular operator and let Cl be a connected component of ps_F(T) such that crw{T)L)Cl is connected. Given e > 0 and natural number n, there exist a sequence {^k}k^=i of complex numbers in Cl and a compact operator K with ||K|| < e, such that {Afc}2L1C
*

A2

T + K^

cn C"

0

where pflF{T) := {\£ps-F{T) : ind{\ - T) = 0}. Proof Note that dClCo-ire(T). We choose a dense subset {/ifc}^ of dCl. Then each clopen subset of o-w{T) intersects {pk}%K'=i- By Lemma 3.2.5, there exists a compact C\, ||Ci|| < §, such that

\J{ker{T + d - pky : j>l, k>l} = H and dim \J{ker(T

+ Cx - pk)j : j > l } = oo

for all k>l. Without loss of generality, we can assume that ap{T* + Cl) = 0. Since ClCP^F(T)

= p%(T

+ C1),

Approximate

35

Jordan Theorem of Operators

nncr(T + C 1 ) = 0 . Denote

M1=\J{ker(T

+

C1-fi1)j:j>l},

then M. is an invariant subspace of T + Cx and dimM.\ = oo. Let

iwr + dju,,, then /UiGcre(Ti) and cr(Ti) = aw{Ti) is connected. If dim\f

{ker(T + Ci - fnY : i = 1,2,-•• , j > l } e A 1 i < oo,

then fcer(T+ for all j>l.

d-Hi)'CM

Denote

Mk = yikeriT+d-fii)'

: l
{ker{T + Cx-fnY

where k>l,j>l. Without loss of generality, we can assume that dimMk

:

l
= oo. Thus

T+ d 0 Since \J{ker(Tk — Hk)j '• j>l} = Mk, crw(Tk) = cr(Tfc) is connected and Cp(Tk*)c{Jik}- Thus Tk is a triangular, /ifeGl. By Lemma 3.2.10, there exists a compact operator Kk, \\Kk\\ < fz, such that Mfc

*" (J-k

Tk + Kk

cn C"

cn>

fc = l , 2 , - .

0 Since {/Ujtj^iCcftl,

we can choose pairwise distinct numbers {A (fc)

.

j

fc>l, j > l } in ft such that \fXk — A^KJ| < -^

for all j and fc. Thus there

36

Structure

of Hilbert Space

Operators

is a compact operator Kk with \\Kk\\ < pr such that iW

C"

c

\(2)

Ak = Tk + Kk + Kk*

i< 3 >

C"i ^ = 1>2, •••

Furthermore, there is a compact operator C2,11C21 i < § satisfying •

'J4I

Mi M2 M3'

T + Cx + Co = 0

where C 2 = ^ ©(-Kk + ^fc)- Set if = C\ + C 2 , then K is compact and fc=i i ( * >J : J>1,/s>l} as {^k}kLi- ^ i s n o t ; difficult to see H^ll < e. Rearrange {A^that K and {Afcj^j satisfy all the requirements of the lemma.

Now we are in a position to prove Theorem 3.2.1. Proof of Theorem 3.2.1 For T&B n (fi), assume that dQ.Caire(T) (otherwise, replace Q with the component of PS-F(Q) containing fi). Denote $ = (Q)° and T = d$. Then rccr, r e (T). By Lemma 3.2.6 and Lemma 3.2.9, we can find a compact operator K\, \\Ki\\ < | such that T +

K^

Tx * 0 iV W2:

where Ti€# n (fi), N is a diagonal operator of uniformly infinite multiplicity and CT(JV) = T. Let M(T) be given as in Lemma 3.2.4, then M(T)^ = n

0

M(T) is normal and

0 M + (T) M{T)

(n)

0 It is clear that a(M(T)W)

0

Z

fc=i fc=i

= T.

0 M_(r) fc=i

Approximate

Jordan Theorem of Operators

37

By Voiculescu theorem [Voiculescu, D. (1976)] there exists a compact operator Fx such that \\Fi\\ < § and N + F^MIT)^. Thus Ti T + Ki+

K2^

*

*

o 0M+(r)

Jz

k=l

0

k=l

0 M_(T)

0

fc=i

where K2 is compact, K2 = 0©i<2 and H-ft^H < f • Set

o ©M+(r) fc=l

It is obvious that a(Bi) = CT(T), Q C p ^ B i ) and ind(Bx - A) > 0 for all A in cr(Si)npir(-Bi)\f2. Therefore B\ is quasitriangular, fi is a connected component of pp{B\) and Q,r)aw(Bi) = cr(T) is connected. By Lemma 3.2.11 we can find a sequence {Afc}^ °f pairwise distinct complex numbers in Q and a compact operator E, \\E\\ < JQ such that {Afe}^=1Ccr0(yl1) and C"

Ai

c

Ai = B + E^

Summarizing the arguments above, we can find a compact operator if3, ||#31| < y| such that A\ T + Kt + K2 + K3*

Denote B2

=

0 M_(r).

*

n3

n

0

©M_(T)

fc=i

Note that B 2 is essentially normal and

k=i

B2€Bn{$)cBn(£l)

[Cowen, M.J. and Douglas, R. (1977)]. By Lemma 3.2.9,

we can find a compact operator F2, \\F21| < ^ such that A2 = B2 +

F2eBn(n)n(Sl).

Thus there is a compact operator K4, \\K^\\ < ^ such that T + Kx + K2 + K3 + X 4 = [ ^ U

A 12

.

A2

38

Structure

of Hilbert Space

Operators

According to Lemma 3.2.7, it is sufficient for us to find a compact operator E12 such that ||£ 1 2 || < fi and Pfcer(A!-Afc)*(^12 + •£;i2)Uer(A1-Afc) is injective. Since {Xk}k^L1Cap(A1)nPF(A2) and nul(A\ — Xk) — n,

X\

cn

G\2

C"

A2

"Gi * " 1-0

0 G2

Woo

Jno where Xi G\2 G13 • • X2 G23 • •

Gi

C" C"

A3 '•

and G 2 = A » - Since A2eBn(Q),ker(A2

- Xk) = ker(Gx - Xk) and

Ax Ex E2 G = 0 Gi * L 0 0 G2

©C"

fc=l

00

©C"

fc=l

We need only to find a compact operator i*3, H-F3JI < <^, such that Pker(A-\k)*(El

+ ii3)Uer(G1-AJ,)

Approximate

39

Jordan Theorem of Operators

is injective. Since \k£0o(Ai)

for k>l,A\

has the following expression:

"Ho

Ax = A3 C23 C13

C"

A2 C12

c cn

Ai

Thus Afc £ a(Aoo) for all &>1. Under the above decomposition,

*

*

*

£1 = E31 E32 E33 • • •

:

*

E21 E22 E23 • • •

:

*

En

:

*

E12 E13 • • •

where E^ is an operator from one n-dimensional space to another for i, j>l. It is obvious that we can choose F n with \\Fn\\ < | such that En + Fu is invertible. Inductively, we can choose Fjj with \\Fjj\\ < ^ such that Ej+ij+i + WjX~1Vj + Fj+i:j+i is invertible, where Wi = (Gj,j+i> Gj-i,j+i, • • • . Gitj+i, Ej+iti, • • • ,

-\? cj,j-i Xj-i Xj — Xj

' ••

c

j,i

" • Cj-1,1

Ai

Ej,i

Ej+ij),

Aw + Au

Ej-iti

En + Fn A2 Xj

40

Structure

of Hilbert Space

Operators

and

Ej-i,j+i

Vj =

Fi,j+i

Set

F2 =

0

F33

'•

0

:

F22

Fn

then F2 is compact and j|JF211 < -§ils From the construction we can see PkeriAt-x^'iEi + Fi)\ker(A2-\k) injective for all k>l. Therefore the proof of the theorem is now complete. Definition 3.2.12 A sequence {Pj : l<j
n-

Definition 3.2.13 Let TeC(H) and V = {Pj : l<j
Approximate

Jordan Theorem of Operators

41

the orthogonal sum of at most countably many (57) operators. Definition 3.2.14 Let Vx = {Pj • l<j
ranQv(j))

such that 5UP{||Xj||,||X-1||,l
= TranQ^u)Xj,

l<j
In Chapter 4, we will show that V\ and Vi are similar about T if and only if there exists an invertible operator X&A'(T) such that {XPjX-1

: l<j
= {Qj :

l<j
Definition 3.2.15 Suppose that T has strongly irreducible decomposition. T is said to have a unique strongly irreducible decomposition up to similarity if any two of the (SI) decomposition of T are similar. The Jordan canonical theorem in finite dimensional space means essentially that each nxn matrix has a unique (SI) decomposition up to similarity. For operators in £(H), it is very difficult to obtain a "Jordan canonical theorem". In Chapter 5 of this monograph we will prove the following theorem. Theorem FJ Every Cow en-Douglas operator has a unique (SI) decomposition up to similarity. Using Theorem FJ, we obtain the following two approximate Jordan canonical theorems. Theorem 3.2.16 Given T&C(H) and e > 0, there exists Ae£(H) such that A has a unique (SI) decomposition up to similarity and \\T — A\\ < e. Theorem 3.2.17 Let TGJC-(H) be quasitriangular. Assume that cr(T) consists of finitely many components and each component intersects pp(T).

42

Structure

of Hilbert Space

Operators

Given e > 0 there exists a compact operator K with \\K\\ < e such that T + K has a unique strongly irreducible decomposition up to similarity. Theorem 3.2.16 can be proved by using of Theorem JW2 in Chapter 1 and Theorem FJ, and Theorem 3.2.17 can be proved by Theorem 3.2.1 and Lemma 3.2.4. 3.3

Open Problems

1. Given TGC(H) and e > 0, does there exist a compact operator K with \\K\\ < e such that T + K has unique (SI) decomposition up to similarity? 2. Is every operator in C(H) the sum of two Cowen-Douglas operators of index 1? 3. Given TGC(H) and e > 0, does there exist an integer p, l

CP(H) = {KeK{H)

:^A£ <

oo,\ n £a p (K*K)±}

n=l

and

3.4

Remark

The concept of unique strongly irreducible decomposition up to similarity appeared first in [Jiang, C.L. and Wang, Z.Y. (1998)]. Theorem 3.1.1 is given by [Yue, H. (2002)]. Before that, [Jiang, C.L. and Wu, P.Y. (1998)] proved that each operator in £(H) is the sum of three (SI) operators, and each triangular or compact operator is the sum of two (SI) operators. Theorem 3.1.11 is given by [Jiang, C.L. and Wu, P.Y. (1998)]. Theorem 3.2.1 is due to [Ji, Y.Q. and Jiang, C.L. (2002)]. Theorem 3.2.16 and Theorem 3.2.17 are both proved by [Jiang, C.L.(l) ].

Chapter 4

Unitary Invariant and Similarity Invariant of Operators In this chapter % always denotes a complex, separable infinite dimensional Hilbert space. One of the basic problems in operator theory is to determine when two operators A and B in C(H) are unitarily equivalent or similar. In infinite dimensional Hilbert space, this problem has no general solution. What we can do is to find the answer for some special classes of operators. A quantity (quantities) or a property (properties) P is unitary (or similarity) invariant (invariants) if A has P and A=B(A~B) implies that B has P. For example, reducibility and strong reducibility are unitary invariants while strong reducibility is only the similarity invariant. For a subset R of C(H), unitary (similarity) invariant (invariants) P is completely unitary (or similarity) invariant (or invariants) if AGR, then A=B (or A~B) if and only if BGR and A and B have same P. From this point of view, one of the basic problem in operator theory mentioned above is to determine the completely unitary or similarity invariants. We have seen in Chapter 2 that eigenvalues and generalized eigenspaces are completely similarity invariants of nxn matrices. [Conway, J.B. (1990)] showed that two *-cyclic normal (or subnormal) operators A and B in £(H) are similar if and only if the scalar-valued spectral measures induced by them a equivalent, while they are unitary equivalent if and only if they are similar. Here, an operator A is normal if A* A — AA*, and A is subnormal if there exist a normal operator JV and an invariant subspace M. of N such that N\M = A. For two injective unilateral weighted shifts, the boundedness of the ratios of the products of their weights is the completely similarity invariant [Shields, A.L. (1974)]. There have already been a lot of results on the similarity invariants of operators, especially that of non-adjoint operators, which can be found in, for example, [Herrero, D.A. (1987)], [Herrero, D.A. (1990)], [Conway, J.B. (1990)].

43

44

Structure of Hilbert Space

Operators

In this chapter, we will discuss further the unitary invariants and similarity invariants of non-self-adjoint operators.

4.1

Unitary Invariants of Operators

We begin with a famous theorem. Schur theorem Each nxn matrix X is unitary equivalent to the orthogonal direct sum of irreducible matrices. For T££(H), let W*(T) denote the von-Neumann algebra generated by T. By von-Neumann double commutant theorem we can easily prove the equivalence of the following conditions. (i) T is irreducible; (ii) A'(W*(T))=I; (hi) W*(T) = C(H). The following proposition tells us that the Schur Theorem can not be generalized to C{Ji). Proposition 4.1.1 Let Ne£(H) be a self-adjoint operator with crp(N) = 0, then N is not orthogonal direct sum of irreducible operators. Furthermore, N is not the topological direct sum of (SI) operators. i

Proof

For the first part of the proposition, if N = Yl

(BNi,l
i=i

where NiG(RI), then Ni is self-adjoint and the the dimension of the space Hi, on which Ni acts, is just 1. Thus o~p(N) ^ 0. It is a contradiction. For the second part of the proposition, if PGA'(N) is a nontrivial idempotent, then by the spectral theorem of self-adjoint operators, there exists a orthogonal projection P' such that ranP = ranP' and P'GA'(N) [Putnam I.[l]]. If N is the topological direct sum of sum (SI) operators, then by the fact stated above, we can find an orthogonal projection P'GA'(N) and N\ranP' is irreducible. By the argument used in the first part we conclude that crp(N) j= 0. It is also a contradiction. Although it is not every operator to be the direct sum of irreducible operators, we have the following proposition. Proposition 4.1.2 Every operator T&C(7i) is the direct integral of irreducible operators. Proof In fact this is a corollary of Theorem 3.6 of [Azoff, E.A., Fong, C.K. and Gilfeather., F. (1976)]. The weakly closed algebra A(T), generated by

Unitary Invariant

and Similarity

Invariant

of Operators

45

T and / , can be expressed as / A ®A\d[j,(\), where A is a separable metric space, fi is a regular a-finite Borel measure on A and A\ is a weakly closed irreducible operator algebra for almost all AsA. Recall that an operator algebra is irreducible if it has no nontrivial reducing subspace. Therefore, we have T=

/ ® T A ^ ( A ) , T A e A , A e A a.e..

./A

Thus A(T)cfA®A(Tx)diJ,(\)cfA®Axd[i(\) = A(T). This implies that A{T\) = A\ for almost all AeA. Thus the irreducibility of ,4 A implies that T\£(RI). Therefore T = fA®T\dn(\) is the asserted irreducible integral decomposition of T. For C*-algebra A and natural number n, let M„(A) denote the set of all nxn matrices with entries in A. The following theorem gives the number of reducing subspaces of an operator. Theorem 4.1.3 The number of reducing subspaces of any operator Te£(W) is either finite or uncountably infinite. The former case occurs if and only ifT is the direct sum of finitely many irreducible operators, i.e., T = Ti©T2© • • • ®Tn, and Ti,Tj are not unitarily equivalent ifi^j. In this case, the number of reducing subspace is 2™. Theorem 4.1.3 has a similar pattern but in a different context with the following result of [Ong, S.C. (1987)]. Theorem Ong Given T € £ ( C n ) , the number of invariant subspaces ofT is either finite or uncountably infinite. The former case occurs if and only if T has a cyclic vector. To prove Theorem 4.1.3, we need four lemmas. The first is a structure theorem for two orthogonal projections, and has been quoted in many literatures before. The reader is referred to [Halmos, P.R. (1968)]. Lemma 4.1.4 Given two orthogonal projections P and Qg£(W), there is a unitary operator U such that /i0 ®h®h®0®0 0 0

U*PU and U*QU =

A B

B ©/2©0©/ 4 ©0 h-B

46

Structure

of Hilbert Space

Operators

under the decomposition of the space H = Hi®H2®H3®Hi®Hs, where is a positive contraction and B = [A(Ii — A)] 2. We may assume that 0 < A<^Ii and A is unique up to unitary equivalence. Using Lemma 4.1.4, we can prove the next lemma.

J4G£(WI)

L e m m a 4.1.5 IfTGC(H) has countably many reducing subspaces, then A'(W*(T)) is abelian. Proof Let P , Q&A'(W*(T)) be two orthogonal projections represented as in Lemma 4.1.4. Since PT — TP and QT = TQ, a simple computation 5

indicates that T = T\@T2® ^ ©T, under the decomposition of the space i=3 5

and T\A = AT\. For each complex number A, denote M\ = {\BX®X®Q®Q®Q :

XeHi}.

It is easily seen that M\ is a reducing subspace of T, and if Hi ^ {0}, then Mx ± My

(A ^ A').

Since T has only countably many reducing subspace, Hi = {0}. Thus P = / 2 ©/ 3 ®o©0 and Q = /2©0©/ 4 ©0. This implies that PQ = QP. Since von-Neumann algebra A'(W*(T)) generated by the projections in it, A'(W*(T)) is abelian.

is

Recall that a projection p in a C*-algebra is minimal if there is no projection q other than 0 and p such that pq = q. L e m m a 4.1.6 Let PeA'(W*(T)) and only ifT\ranpG(RI).

be a projection, then P is minimal if

The proof of this lemma is an easy consequence of the definition of minimal projection. L e m m a 4.1.7 Let A, BGC(H)C\(TZT), then A and B are unitarily equivalent if and only if there exists a nonzero operator X such that XA = BX and XA* = B*X.

Unitary Invariant

and Similarity

Invariant

of Operators

47

Proof If XA = BX and XA* = B*X for some nonzero X, then kerX and ranX are reducing subspaces of A and B. If kerX ^ {0}, by the irreducibility of A kerX = Ti., i.e., X = 0, a contradiction. Thus kerX — {0}. Similarly, we can conclude that ranX = "H, i.e., X has a dense range. Let X = UP be the polar decomposition of X, where U is unitary and P = (X*X)2>0. Since X*X,4 = X * £ X = AX*X, PA = AP. Therefore UAP = UP A = XA = BX = BUP. Since P is also range dense, UA = BU. Thus A=B. We are now in a position to prove Theorem 4.1.3. Proof of Theorem 4.1.3 Assume that T has countably infinite many reducing subspaces. By Lemma 4.1.5, A'(W*(T)) is abelian. Thus A'(W*(T)) is generated by some Hermitian operator A [Radjaval, H. and Rosenthal, P. (1973)]. Note that cr(A) can not be finite. Otherwise, n

A = Y^ ®*Ji i=l

and n

W*(A) =

{J2®aiIi:ai€C}. »=i

This implies that W*(A) = A'(W*(T)) contains only finitely many projections, and contradicts the assumption. Thus a{A) can be decomposed into countably infinitely many pairwise disjoint Borel subsetslcrj}?^, each of which has a strictly positive spectral measure. Since a (A) has uncountably many different decompositions, so is the spectral projections of A. Therefore, there are uncountably many orthogonal projections in W*(A) = A'(W*(T)). This is also a contradiction. Thus the number of reducing subspaces of T can not be countably infinite. Now we assume that T has finitely many reducing subspaces. By Lemma 4.1.6 A'(W*{T)) is abelian. Let Pi,--- , P n be minimal projections in

48

Structure

A'(W*(T)).

of Hilbert Space

Operators

Since PP,- = P , P , it is not difficult to prove that PiPj = 0 n

n

for i ^ j , and ^ P* = /. Let T — S 0 ^ 1=1

under the decomposition

2=1

n

H = Yl ®ranPi, where T{ = T\ranPi- By Lemma 4.1.7, Ti£(RI).

We are

i=l

now to prove that T is not unitarily equivalent to Tj for i ^ j . Otherwise, if Ti=Tj, there is a unitary operator £/ such that UTi = TjU, where l < i < j if A 7^ A' and A4\ is a reducing subspaces of T. Note that there are infinitely many of .MA'S, this contradicts our assumption of T. Conversely, if T = £ ®Tt on H = £ ©"Hi, T^RI) i=l

and Tj is not

i=l

unitarily equivalent to Tj for i ^ j . Let P = {Py}"., = i be orthogonal projections commuting with T. Then PyTj = TiPij for all l
p y T/ = P ^ T ; = (TjPjy =

(P,^)*

= T ; P ^ = i-p,-.

Since T, and Tj are irreducible and not unitarily equivalent for i 7^ j , by Lemma 4.1.7, Py = 0 and so Pji = 0. Thus Pjj is an orthogonal projection commuting with T,, which implies that Pa = 0 or I-nt (i = 1,2, ••• , n). Therefore T has only 2™ reducing subspaces. In the following, we will characterize the decomposibility of an operator into direct sum of irreducible operators in terms of C*-algebra language. For TG£(H) and an integer n, l
operators (i.e., T = ^ © T ;

(

•)

is the direct sum of irreducible

, l<Tii
i=l

wise not unitary equivalent) if and only if A'(W*(T))

is ^-isomorphic to

n

^2 ©M n i (C). Moreover, the (RI)-decomposition

ofT is unique in the sense

i=l m

.

of unitary equivalent. Precisely, if T = Yl ©^fc

.

is another direct sum

fc=i

of irreducible operators {Sk}™=1, which are pairwise not unitary equivalent, then n = m and there exist a permutation n of {1,2, • • • ,n) and a unitary operator U in A'(W*(T)) such that n, = rnw^ and UTi = Sn^U,i =

Unitary Invariant

and Similarity

Invariant

of Operators

49

1,2,--- ,n. Note that [Takesaki, M. (1979)] showed that every finite dimensional C*-algebra is *-isomorphic to the direct sum of finitely many full matrix algebras. Thus we have the following corollary. Corollary 4.1.9 T is the direct sum of finitely irreducible operators if and only ifdimA'(W*(T)) < oo. To prove Theorem 4.1.8, we need the following lemma. Lemma 4.1.10 If T is irreducible on H and X££(H) satisfying XT = TX and XT* = T*X, then X = dl for some deC. Proof Since X*X commutes with T, T commutes with any spectral projection P of X*X. Since T&{RI),P = 0 or P = I. Thus a{X*X) is a singleton {a} and X*X = al. On the other hand, it follows from XT = TX and XT* = T*X that kerX is a reducing subspace of T. Since T€(RI),kerX = {0} or H. Similarly, ranX = H or ranX~ = {0}. This implies that X = 0 or X is injective with dense range. Thus

where U is unitary. If a ^ 0, then UT = TU and UT* = T*U. Repeating the arguments above, we have U = /3I or X = dl, where d = \/a/3. Thus, the proof is complete. Proof of Theorem 4.1.8

Let T = £ ®T}ni)

satisfy the conditions of

i=l

the theorem. For any X€A'(W*(T)),

by Lemma 4.1.10,

n

X =

J^(BXi,Xi€A'(W*(T^)). i=l

Let

then

YjkzA'(W(Tt)). By Lemma 4.1.10, Yjk = ^)kh, where 7j is the identity on Wj.Thus

Structure

50

of Hilbert Space

Operators

n

It is obvious that the mapping J n ^ ®[^}fc-^]?fc=i

defines a *-

i=l

isomorphism from A'(W*(T))

onto £) ©M n i (C).

Conversely, if 4> is a ^-isomorphism from A'(W*(T))

onto A

:=

n

X)©M n j (C) and let Etj denote the element 0® • • • ®etj® • • •ffiOin A, i=l

where e^ is an riiXTii matrix whose (i,j)-entry is 1 and the others are 0. Then 4>~l(Eij)£A'(W*(T)) are pairwise orthogonal minimal projections with £_1(.EV,) = ln- Obviously, by Lemma 4.1.7, ^{E^H is one of •J

the reducing subspaces of T with Ta :=

TU-HEii)nZ{BI)

and

T = £©!;,. Since Eij^Eik

for all (j, k), Tij^Tik

and T ^ £

®TJni).

»=i m

.

In order to prove the uniqueness, let T = X) ©S^.

.

be another ex-

pression of T, where {Sk}™^ are pairwise not unitary equivalent and m

.

H — ^2 ®L(™k • Let Pki be the orthogonal projection from H onto the fc=i

Z-th subspace of £j,

> then F ^ = {Pki) are pairwise orthogonal in

n

.4 := £ ® M n i ( C ) , and Y,Fki = ^ Since T | r a n p u e ( i ? / ) , P H is minimi fc,i mal. Thus Fki is minimal in A and there exists an integer n, such that Fki€Mni(C) with rankFu = 1. Therefore X).FM = 7 ni and so mk = rii. I

Prom ^.Fjfez = / , we can conclude that m — n. The remainder of the theorem can be proved directly from the *-isomorphism 4>. Next we will consider when two operators have isomorphic reducing subspaces lattices. [Conway, J.B. and Gillespie, T.A. (1985)] solved this problem in the case of normal operators. Using their result, we can characterize isomorphism of the reducing subspace lattices of two operators if they can be expressed as direct sums of irreducible operators.

Unitary Invariant

Proposition 4.1.11

and Similarity

Invariant

of Operators

51

Let

j=i

anc?

* = £«*£" fc=l

where Aj,Bk&(RI), {Aj} and {Bk} are pairwise not unitarily equivalent, l{A,J}"_ 7 - =1 induces a lattice isomorphism from RedT^ onto Redln. Proof of Proposition 4.1.11 By Lemma 4.1.7, RedA is isomorphic to ^ © i Z e d A ^ 0 , and by Lemma 4.1.12 £@#e
morphic to RedB, then n — m and there is a permutation 7r of {1, 2, • • • , n} such that rij = ^ ( j ) [Conway, J.B. and Gillespie, T.A. (1985)]. This proves the necessity part. The sufficiency is obvious. Proposition 4.1.13 IfT^ is a direct sum of irreducible operators, then so is T. Proof Without loss of generality, we assume that n t=i

{Tj}™=1 is a sequence of irreducible operators, which are pairwise unitarily inequivalent. Then there are pairwise orthogonal projections Pj,j =

Structure

52

of Hilbert Space

Operators

1,2, • • • , n, each of which commutes with T^

and J2Pj

=

sucn

!•>

tnat

r^fc^Irani3,,i = 1,2, • • • , k are pairwise unitarily inequivalent. By Lemma 4.1.7, Pj = 2_^ ®Qij, where Qij£A'(T>

) , {Qij} are pairwise orthogonal, Y^®Qij

=

^

anc

^

j

Tt \ranQij, j = 1,2, • • • , A; are pairwise unitarily equivalent. Therefore we need only to prove that if A^^B^, l
\ranQ-

Proof Set W = U\ranp- Then W is a unitary operator from ranP onto ranQ and satisfies W(T\ranP) = (T\ranQ)W. Lemma 4.1.16 Let TeC(H)

and

\J(A'(W*{T)))^N{fl]®(N+u{<^}){k2\ 0
Proof

Assume that T| ran f> is irreducible and [P] = J2 ©aje,, where a* is

Unitary Invariant

and Similarity

Invariant

of Operators

53

an integer, CKOJJ < oo. If there are at least two nonzero a^s, say a\, a? ^ 0. Then / = a.\e\ and g = J2 ®aiei

a

re nonzero elements in

\J(A'(W*(T))).

i=1

Thus there exist a natural number m and mutual orthogonal projections Q and RGA'(W*(T^)), such that [Q] = f and [R] = g. Set S = Q + R, then /

[5] = [Q] + [R} = f + g = J2 ^id

= [P].

Therefore S is unitarily equivalent to P © 0 ( m - 1 ) in A'(W*(T^m^)), denotes the zero operator on H. By Lemma 4.1.15, r

r(rn)]

-t

'\ranS

where 0

„ /^> /T>(m)j

= J-

^

"ranP©0("-!)

T\ranP€(RI).

This contradicts the fact that T\ranp€(SI). Thus [P] = a»ei. Similarly, we can prove a* = 1. Conversely, assume that [P] = ei and T\ranp is reducible. Then there are nonzero projections Q^GA'CW^T^)) such that QR = 0 and P =

<2 + P. Let

[Q] = 5 3 ®aiei i=l

and

[P] = 53©ftei, i=l

where 0
and

©(a, + &>)<*

Structure

54

of Hilbert Space

Operators

for all i>2. Therefore ax = 0 and j3x = 0 and on = # = 0 for i>2. This implies that [Q] = 0 or [R] = 0. A contradiction. Therefore T\ranp£(RI)Lemma 4.1.17 Let AGC(H) be a direct sum of irreducible operators and B&C{K) have no reducing subspace on which B is irreducible. If there exists an operator X such that XA = BX, XA* = B*X, then X = 0. oo

Proof

Without loss of generality, we assume that A = £ ®An with n=l oo

respect to H = £ ©W„, where Ane(RI).

Then X* = [X{, X | , • • • ]*. Now

n=l

we prove X\ = 0. In fact, from XA — BX and XA* = 5 * X , we have XXB = AtXi and XXB* = A*X. Thus {X^^Ai = Ai(XiX{) and ( X i X j * ) ^ = A\(XiX{). Since ^ is irreducible, by Lemma 4.1.10 XXX{ = XIHl for some AeC. If A ^ 0. Let U = \~lXi. Then UU* = IHl and Q := U*U is a projection in C{K.) with Q-B = BQ. Set p = IHl®0,

Q = 0©Q,

then p, ge£(Wi © £ ) • Set p' = pffiO, ?' = q®0, where

Set C = J 4 I © B . We claim that p' and g' are unitarily equivalent in A'{W*(C^)). To prove this, we define v=^QU^££(H1+JC). It is easy to see that v is a partial isometry, vv* = p and v*v = q. Then the assertion follows from the Proposition 5.2.12 of [Wegge-Olsen, N.E. (1993)]. By Lemma 4.1.15, C^\ranp^C^\ranql. But C^\ranp, = A^RI). Thus C

\ranq'=B\ranQe(RI),

which contradicts our assumption on B. Hence X\ = 0. By the same arguments we can prove that Xn = 0 for n > 2 . Thus X = 0. Proof of Theorem 4.1.14 The necessity follows from the analysis above. We only prove the sufficiency. Assume that \/{A'(W*{T)))^l)@{N+U{oo}Yk2\

0
Unitary Invariant

and Similarity

Invariant

55

of Operators

Let P be a projection in Mk(A'(W*(T))) = A'{W*{T^)) such that [P] is a free generator of \/(A'{W*(T))). By Lemma 4.1.16, T^\ranPe(RI) (here we embed A'(W*(T)) into Mk(A'(W*(T))) with the embedding \A 0] map A>-» ). Using Zorn's lemma, we can find a maximal family in A'(W*(T^)) of pairwise orthogonal projections {Pj}n=l, l
will prove that Q = 7^fc', the identity operator on H^. Otherwise, set 7\ := T^\ranQ,T2 := ^ (fc) | r <m(/«-Q)- Since
Thus

V(^'(W/*(T(fc))))^V(-4'(iy*(:ri)))®\/(-4'(W'*(T2))) (Isomorphism Theorem). Let R be a projection in A'(W*(T2 )) for which [i?] is a free generator of \J(A'(W*(T2))). By Lemma 4.1.16, T^m)\ranRe(RI). By the similar argument above, we find a nonzero projection Q iGA' (W* (T2 )) such that T3 := T2 |ranQi is the direct sum of irreducible operators and T4 := T;j; | r an(j-Qi) n a s n o reducing subspace .M with T4|^€(i?7). Using Lemma 4.1.17 again, we have A'(W*(T^m)))

=

A'(W*(T3))®A'(W*(T4)).

Thus Q\ commutes with every operator in A'(W*(T2 )) and QiGA'(W*(T2 )) by von-Neumann double commutant theorem. Therefore Q = S(m\ where S is a nonzero projection in W*(T2), and rp

<7-t( m )|

jrn |

\(m)

Since T3 is the direct sum of irreducible operators, so is T2\TanS by Proposition 4.1.13. This contradicts the assumption on T2. Thus Q = 1^ and T(fc) is a direct sum of irreducible operators. By Proposition 4.1.13, T is also a direct sum of irreducible operators.

Structure of Hilbert Space Operators

56

By Theorem 4.1.14 and its proof, we have the following theorem. Theorem 4.1.18

Let A,B£C(H). i

,

operators, i.e., A = J2 (BA),

n

If A is the direct sum of irreducible

,

, 1 < / < O O , l
i=l

are not similar if and only if there exists an isometric isomorphism (j) such that (j>{\J{M{W*{A®B))))

= N^ fcl) ©(N + U{oo})( fc2 \

Q
i

and (f)(1) = 2^2,riiei, where {e,} are the generators

of\J(A'{W*(A))).

i=l

Note that for A£C(H), if A is a direct sum of infinitely many irreducible operators, then KQ(A'(W*(A))) = 0. Thus, in this case, i^o-group can not describe the unitary equivalent relation between operators. But we have the following theorem. Theorem 4.1.19

Let A,Be£.(7i).

Assume that A is the direct sum of i

,

,

finitely many irreducible operators: A = ^Z ©Aj™ , 1
oo, i = 1, 2, • • • ,1. Then A=B if and only if there exists an isometric isomorphism <j> such that cj>(K0(A!(W*(A®B))))

= N^}

and <j>(I) = 2 ^ n^ei, where {e,} are the generators of

Ko(A'(W*(A))).

i=i

Note that for A€Bn(Q.). Since A is a direct sum of finitely many irreducible operators, we can get the following theorem by Theorem 4.1.19. Theorem 4.1.20 Let A,B£Bn(fl), then the following statement hold. (i) A has a unique (SI) decomposition up to unitary equivalence. k

(ii) IfA=

J2®A\ni>,Aie{RI)

and{Ai}ki=l

are pairwise unitarily in-

2=1

equivalent. Then A=B if and only if there exists an isomorphic map such that (K0(A'(W*(A®B)))) = Z(IA'(W(A®B))) \j(A!(W*(A))).

=

^Ylniei'

where {e;} are the generators of

Unitary Invariant

4.2

and Similarity

Invariant

of Operators

57

Strongly Irreducible Decomposition of Operators and Similarity Invariant: K0-Group

It is more convenient in terms of i^o-group to describe the unitary invariant of operators which are direct sum of irreducible operators. The proof of it depends strongly on the tools of C*-algebra theory. We give, in terms of semigroup theory, a necessary and sufficient condition to that an operator can be expressed as a direct sum of irreducible operators. In this section, we first prove the following theorem. Theorem 4.2.1

Given Te£(H),

the following are equivalent: n

(i)

T is similar to ^2 ®A\

under the decomposition of the space

»=i k

.

H=J2 ®KT

.

, where

kn

,i

< oo, Ai£(SI),

Ai^Aj

for i ^ j and each T(n>

i=l

has a unique (SI) decomposition up to similarity; (ii)

\J(A'(T))= N(k>, and this isomorphism 4> sends [I]—miei + n 2 e 2 H

h n^,

where {ej}f=1 are the generators of N^ and N = {0,1,2, • • • }, n, ^ 0. From Theorem 4.2.1, we obtain the following corollary. Corollary 4.2.2 Let T i , T 2 e ( 5 / ) , T = Ti®T2. If \f(A'(T))*N, then Ti~T 2 . Moreover, if T^ has a unique (SI) decomposition for all natural numbers up to similarity, then Ti~T 2 if and only if Ko(A'(T))=Z, where Z := {0, ± 1 , ±2, • • • }. Before we prove Theorem 4.2.1, the following lemmas are needed. Lemma 4.2.3 Given A, BGC(H). Assume that ip is an isomorphism from A'(A) onto A'(B). {Pi}^ i is an (SI) decomposition of A if and only if {y>(P;)}"=1 is an (SI) decomposition of B. In particular, if A~B, then A!(A)^A!(B). Proof Since

J2 tp(Pi) = I- We need only to show that B\v^Pi)H£(SI),

i = 1,2, • • • , n.

Otherwise, there exist two nonzero idempotents Qi and Q2&A'(B) such that Q2Q1 = Q1Q2 = 0 and Qi + Q2 =
Structure

58

of Hilbert Space

Operators

If A~B, there exists an invertible operator X such that XAX 1 = B. Define Tt-^XTX'1 for all T^A'(A). This mapping is an isomorphism from A'(A) onto A'(B). Lemma 4.2.4 Let Te£(H) and Pi,P2£A'(T) be idempotents. If Pi~A'(T)P2> then T | P 1 - H ~ T | P 2 7 ^ , where ~A'(T) means similarity in A'(T). Proof Since Pi~A'(T)P2, there is an invertible operator X£A'(T) such that XPiX-1 = P2. Thus XranPi = ranP2, Xran(I - Pi) = ran(I - P 2 ). Set X\ = X\ranp1, Then X = X\+X2,

X2 =

X\ran(i_pxy

the topological sum of X\ and X2, and

XiCGLiCiPtH,

P2H)),

- P{)H, (I -

X2GGL(C((I

P2)H)),

where GL(A) is the set of invertible elements in the algebra A. Note that

T=

Ti 0 0 T2

PiH (I-Pi)H

where Tx = T\PlH,T2 = T\{l_Pl)H,T[ A simple calculation shows that T[ 0 0 Ti

Xx 0 0 X2

T[ 0

p2n

on

{1-P2W

= T\P2H and T2' = T\(I_P2)n.

Xx 0 0 X2

Ti 0 0 T2

ThatisT|PlW~T|p2>i. Lemma 4.2.5 Let TeC(H) and let {Pi}?=1 and {Qi}?=1 be two (SI) decompositions ofT. If there exist Xi&GL(C(PiH,QiH)) such that Xi(T\PiH)X-1=T\Qin,i

=

l,2,.-

then X = X!+X2+

Proof

•••

+XneGL(A'(T)).

Since

H = ranP\+ranP2-\

\-ranPn = ranQ\+ranQ2+

• • • +ranQn,

Unitary Invariant

and Similarity

Ti

pxn

Invariant

of Operators

QlH

T{

;

J

-n

where Tt = T\PiH,T[ is invertible. Lemma 4.2.6

PxH

= T\QiH,i

59

0

Ti

•>

QiH

= 1,2, • • • ,n. Clearly, XT = TX and X

Let \ - * l i ' ' ' ) * m > * 771+1•>' ' ' i -* n /

and •,Qn} 6e two spectral families in A'(T), where TGC(H). If there are X,YGA'(T) and a permutation n of Sn = {1,2, • • • , n) satisfying (i) XPiX'1 =Qi,l. Moreover, {Pr'} is exactly a rearrangement o / { P r } " = m + 1 . Proof Given Qr,m < r
^ r, such that

XYQrY^X^^Qj,. By (ii), YQjxY~l

= PJ2 for some j 2 . If m < J2
If

1<72<TI,

it is obvious that j i ^ j 2 . Otherwise, Qh = Y~lPhY

= Y~lPhY

= Qr,

which contradicts m < r
60

Structure of Hilbert Space Operators

Similarly, j 3 g {31,32}- If m < j3
=

^/r2^ir2^T2 •

Without loss of generality, we may assume that Si>S2- If s\ > S2, then ^r2

Note that Z~2Zri

ZriQnZri

ZT2 = Qr2^\Qm+li

' ' • ,Qn}-

= X F - • - X F (X appears j S l — j S 2 times). Set R = YXY- • -XY,

where X appears j S l — j S 2 — 1 times. By the procedure of the choice, we have RQrxR-1e{Pi,P2, • • • ,Pm}. Thus XRQriR-1X-1e{Q1,Q2,--But XRQriR~

X~

=

Z~2 ZriQriZ~i

,Qm}ZT2

= <2r26{Qm+l> • • • ,Qn}-

A

contradiction. Thus s x = S2- But if s\ = s 2 l we can easily prove that Qrx = Qr2, which is also a contradiction. This completes the proof of our Claim and the lemma. By the similar argument of Lemma 4.2.6, we can prove the following result. L e m m a 4.2.7 Letre£(H) i-'l)'*' I M H I I ' "

and let j P-m.k-i — 1) ' ' ' i-• mjt; •• mfc + 1) ' " '

i*nf

and i V l ) ' * ' i V m n ' " i V m i b _ i - l ) " ' j Wmk > Wmk+li " " " j VnJ

be two spectral families in A'{T).

If there exist

X1,X2,---,Xk:YeGL(A'(T)) and a permutation IT of sn such that XiPjX~l

— Qj,mi

+ l<j<mi+i,

i = 0,1, • • • , k - 1, mi = 0

Unitary Invariant

and Similarity

Invariant

of Operators

61

and Y~1PjY = Q-K(J), l < j < « - Then forVr,mk < r •' • > Qm, Qm+i, ••• , Qn} be two spectral families in A'(T). If the following conditions are satisfied: (i) For each Pi, there is an Xi&GL(PiH,QiH) XiT\PinX-1

such that

= T\QiH, l
(ii) there exist a Y&GL(A'(T))

and a permutation ir of Sn such that

Y~~ PtY = <2„-(,),

then given Qr, r£(m + 1, • • • , n), we can find r'G{m + 1, • • • , n} and Zr€GL(QrH,Pr
r(T\Qrn)Zr

=T\piH-

Furthermore, if r\ ^ r2, then r{ ^ r'2. Proof Given r£{m + l,--- ,n}, by (ii) of the lemma there exists an operator Pj1€{Pi}?=1 such that YQrY~x = Pj1. If m < ji
62

Structure of Hilbert Space

Operators

Let {-Pi}£Lx and {Qi}^ be two (57) decomposition of T\PH, and let {Pi}" = T O + 1 be an (SI) decomposition of T\^_P^n. Then {Pi •• l
l
are two (57) decomposition of T. By the uniqueness of the decomposition, we can find an operator YGGL(A'(T)) such that {YPiY-1}

= {Ql,.--

,Qm,Pm+u...

,P„}.

By Lemma 4.2.6, we can find Zi&GL(C(QiH,PiH)) of Sn, such that Zi{T\QiH)Z^ Set Z r = I\pkn,k>m

= T\p^n,

and a permutation TT

l
+ 1 a n d Z = Zi-i

i-Zn. By Lemma 4.2.5,

ZGGL(^'(T)) and Z P W e G L ( ^ ' ( r ) | p W ) . Note that (Z|PW)Qi(Z|p„)-1 = PT(i),l<»<m. The proof of the lemma is complete. Lemma 4.2.10 Let TG£(H). IfT has a unique (SI) decomposition up to similarity. If P and Q are two idempotents in A'(T), then the following are equivalent: (i) P~A>(T)Q; (ii)

T\PH~T\QH-

Proof (i)=^(ii) is a consequence of Lemma 4.2.4. (ii)=Ki). By Lemma 4.2.9, T\pH,T\QH,T\{I_P)n and T\{I_Q)n all have a unique (57) decomposition up to similarity. Since T\PK~T\Q-H, there exists an operator X £GL(C(PH, QH)) such that X(T\pU)X~

=T\QH.

Thus if {Px,P2, • • • ,Pm} is an (57) decomposition of T\PH, then (XP1X-\XP2X-1,--^PmX-1} is an (57) decomposition of T\QH. Assume that {P m +i, • • • , Pn} and {Qm+i, • • • , Qn} are (57) decomposition of T|(/_p) W and T | ( / _ Q ) ^ respectively. Then {Pj}™=1 and {XPiX~l

: \
l
Unitary Invariant

and Similarity

Invariant

of Operators

63

are two (SI) decompositions of T. By the uniqueness of the decomposition, there exists an operator YeGL(A'(T)) such that {YPiY~l}f=1 is a 1 rearrangement of {XPjX" : l
and r[ = r'2 if and only if n = r%. Set Z = Zi+ • • • +ZneGL(A'(T)).

Since ZPZ"" 1 = Q, by Lemma 4.2.5,

P~A'{T)QLemma 4.2.11 Let T££(H) and let P and Q be idempotents in A'(T). IfT\pn is not similar to T\Q-H, then P©0W<» is not similar to <5©0W(n> in A' (T^n+1^) for each natural number n. Proof If there is an XeGL(A'(T(-n+1'>)) satisfying X(P®OnM)X^

= Q®0Hin)

for some nGN, then by Lemma 4.2.4, T(n+1)

l(Peo w ( „,)«(»+» ~ T ( n + 1 ) |(Q e o K(n) )W(»+D •

But we have r( +1)

"

l(p©o w ( „ ) )w("+ 1 )- r |pw

and T(n+1)

\{Q®On{n))H<-"-+V-T\Q-H-

Therefore

T\PK~T\Q-H-

A contradiction.

Lemma 4.2.12 Given Te£(H). IfT^n) has a unique (SI) decomposition up to similarity for each natural number n, then for two idempotents P, Q in A'(T), P~A>{T)Q if and only if [P] = [Q] in \/(A'(T)). Proof The "if" part is obvious. Assume that [P] = [Q], then there is a natural number k such that P©0W(io~_4,(T(k+i))<3(B0H(fc). By Lemma 4.2.4, P~A'(T)Q-

T\P-H~T\Q-H-

By Lemma 4.2.10 we conclude that

64

Structure of Hilbert Space

Operators

Proof of Theorem 4.2.1 (i)=S-(ii). Let P be the orthogonal projection from H onto Hi, and let E be an idempotent in A'(T^). Since T<") has a unique (SI) decomposition up to similarity, by Lemma 4.2.9 T^\E-H(„) and T(n}\,I_g)-H(n) have unique (SI) decompositions up to similarity. If {Qi, ••• , Qa} is an (SI) decomposition of T^\EnM and {Qa+i, ••• , Qb) is an (SI) decomposition of T^\^I_E-)H(n), then {Qi, • • • ,Qb} is an (SI) decomposition of T K Since {P 1 ( n n i ) ,.-- ,Pfc(rmfc)} is also an (SI) decomposition of T^n\ by the uniqueness of the decomposition, there is an XeGL(A'(T^)) such that XQjX-1 Since E = Qi + Q2 + • • • + Qa, XEX'1

= Pt. = X P}mi) • Define a mapping

\J(A'(T)) -> N
then by Lemma 4.2.12, F~E~

E P ^

k

in Moo(A'(T)). U F~ E P ( m i ) in

i=l

i=l

Moo(^'(T)), then h([F]) = h([E}) and F ~ E . Thus /i is one to one. For a &-tuple (mi, • • • , mfc) of nonnegative integers, we can find ma number n i i such that mi
E

pr} to

i=l

(mi, • • • , mfc) and is onto. Thus

\J(A'(T))^N^\ By the construction of h, we know that h([I]) = (n\, • • • , n^). (ii)=*-(i). Suppose that V(-^'( T '))=N (fc) a n d h is the isomorphism. Then there exist a natural number r and k idempotents Qi,--- ,Qk in A'(T^) satisfying h([Qi]) = eu\
Unitary Invariant

and Similarity

Invariant

of Operators

65

fc fc

Assume that h([P}) = £

A e

* i = £ A;/i([<3i]), A^eN. Set w = r J2 <**,

i=l

i=l

find a natural number n > w such that fc t ,, P©o w( „-i)~ i4 , {r ( n ) ) ] T QlA,)©o " -w- ( n - « 0 .

1=1

By Lemma 4.2.4, T< n >|,p« „ W W r~ -T W |. n . „_„)«(») l(P©o M(

- (Ai, (E Q l ' ®°„(«-»))w(n)

Thus (E

Note that i.e.,

T|P-H€(S7),

Q^i^n^i

thus only one Aj equals 1 and the others are zero, h([P}) = e*

for some i. (b). For arbitrary idempotents P and Q in „4'(T<">), if h([P]) = h([Q]), then T\PH~T\Qn. Repeating the arguments in (a), we get (b). Let {Pi, Pi, • • • , Pm} be a spectral projections of T and assume that fc P

K[ i\) = J2XiJeJ^iJ^N.7=1

Then m i=i

m

k

i=i j=i

fc ra fc fc fc

From h(I) — Y2 n^e*, we have J ] J3 ^»j — 12 n»- Thus m < J^ m. This i=l

i=l j=l

*=1

i=l

implies that the number of elements in each set of spectral projections of T is finite and T is the direct sum of only finitely number of (SI) operators. Furthermore, let {P 1 ; P2, • • • ,Pi} be an (SI) decomposition of T, then fc

E j2 ^-

h(j2iPi}) = H[i]) = i=l

i=l

n

66

Structure of Hilbert Space

k

By (a), t = ^2 ni

aQ

Operators

d for each i, l
i=l

Pil,---,Pinie{P1,P2,-.-,Pt} such that h([Pil]) = -- = h([Pitli]) = ei. By (b), T\PijU~T\pikH,

l<j,k
Denote A, = T\Pi.H, l<j
»=i

Assume that {P{, P2,-- • i P's) ls another (SI) decomposition of T. Then k

repeating the above arguments, we have r = J2ni

an

d f° r

eacn

h l
there are n, idempotents in {P[, P^ • • • , P^} such that ft maps each of them k

to <*. By (b), if h{[Pi\) = h([P$),l
£ ni: then T\Ptn~T\p.n.

By

j=i

Lemma 4.2.5, T has a unique (SI) decomposition up to similarity. The proof of the theorem now is complete. Proof of Corollary 4.2.2 Note that if T W has a unique (SI) decomposition up to similarity, then by Theorem 4.2.1 Ti~T2 if and only if \J(A'(TX®T2))^N. Therefore, if Ti~T2, then KQ(A'(T1®T2))^Z, since \J(A'(T1®T2))^N. Conversely, if K0(A'(Ti®T2))^Z, by Theorem 3.2.1, V(-4'(Ti©T 2 ))SN( fc \fc<2. Since K0(A'(T1eT2))^Z, V M ' ( T i © ? 2 ) ) = N . Thus Ti~T 2 . This completes the proof of the corollary. The following proposition tells us that besides normal operators, a class of analytic Toeplitz operators whose unitary invariant and similarity invariant are same. In fact, we have stronger result. Proposition 4.2.13 Let ipi and tp2 be two univalent analytic on the unit disk D. Then the following are equivalent: (ii) kerTTvl<Tie2 # {0} and kerrTv2tTvl Proof (i)=>(ii) is obvious. oo

(ii)=>(i).

functions

¥= {0}.

Assume that ifi(z) = £} AJz J ',zeZ?,i = 1,2. Since
Unitary Invariant

and Similarity

Invariant

of Operators

univalent, X\ ^ 0 for i = 1,2. If there are X, Y£C(H2)

67

such that

Denote fli = <^i(D),J72 = ^(D). Clearly, fii and Q2 are simply connected and T^eB^flt^T^eBiiQ^). If fi2 ^ fi2, by Lemma 4.2.3 kerrTipi)Tv2

= kerrT^Tify

= {0},

this contradicts our assumption. Thus we may assume that Q.i = Q2 = ^ and o^T^) = cr(TV£,2) = fi. Without loss of generality, we assume that Oefi and v?i(0) = 0, (^2(20) = 0,zo£D. Then there exists a Mobius transformation X:£>->£>

satisfying x (0) = z0. Thus y 2 (x(0)) = 0. This implies that TVtM*TV2UW). Therefore we may assume that ^2(0) = 0. Note that T*VT*2 have the following matrix representations with respect to the ONB {1,2, z2, z3, • • • }: 0 X1 A 2 A 3 • • • 0

A; A 2

••.

0 X1 A 2 A 3 0 Af Al

rp#

0 A?

0 Ai '••

.0

.0

Prom T*XY* = Y*T*2, computation indicates 2/11 2/12 2/13 • '

y* __

2/22 2/23 • '

A2 n-1 and t/nn = [-n"] 2/ii, A

n

2,3,

i

0

We claim that | ^ | < 1 . Otherwise, since Y* is bounded, y n = 0 and ynn = 0 for n = 2,3, • • • . Similarly, j/y = 0 for i, j>l. This contradicts Y* j^0. Similarly, | j t | < l . Thus |Aj| = |A?| = A. Denote 6/j = a^ff-f-

and f/j = diag(l,e

>,e^idj

• ).*' = !. 2-

Structure

68

of Hilbert Space

Operators

Obviously Uj is unitary. Denote 0 A *" 0 A

R^UjT^U^

0 '••

'

0

Since UjT*U* = e~ie'T*, RjSA'(T*),j = 1,2. Thus, there exists a function gjGH00 such that Rj = T*.. Since T*.^T*.,gj(D) =
= 0,^(0) = A > 0 , j = 1,2.

It follows from Riemann mapping theorem that gi = gi and therefore •L
In the following we will compute the i^o-groups of some Banach algebras in terms of Theorem 4.2.1. Theorem 4.2.14 K0(H°°)^Z, V(#°°)=N. Proof Consider the analytic Toeplitz operator Tz. It is well known that

A'(TZ)^H°°. By Theorem 4.2.1, we need only to show that T = TJ has a unique (SI) decomposition up to similarity for each natural number n. Since T*£Bn(D), there are only finitely number of elements in each spectral family of T*. Thus we need only to prove that if P is a minimal idempotent in A' (T), then T\P(H2)M~TZ. Note that T is an isometric isomorphism and P(H2)^ is an invariant subspace of T, it follows from the famous von-Neumann-Wold theorem T\P^H2^)—TZ. This completes the proof of the theorem. Corollary 4.2.15 K0(H°°(n))^Z and V(#°°(fi))=N, where ft is a nonempty bounded simply connected domain. Proof Since ft is nonempty bounded and simply connected, there exists a univalent analytic function

n

T

N(fc) and

\/M'(5Z® ^))»=i

Ko(A'(J2®T^-zik)i=i

Unitary Invariant

and Similarity

Invariant

of Operators

69

Furthermore, T — ^ ®TVi has a unique (SI) decomposition up to similarity. Proof

£ ®(H2)n*. Thus it follows

By Proposition 4.2.13, A'(Y, ®TVi)^ i=i

j=i

from the isomorphism of if-groups and Theorem 4.2.14 that

VCA'(E ®^j)- £

© V((#2))K)=N(fc)

and n i=l

To summarize the facts discussed above we have the following corollary. Corollary 4.2.17 Let
(Hi) 4.3

lVl=llfi2;

K0(A!(TVl®TV2))^Z.

(SI) Decompositions of Some Classes of Operators

Using Lemma 2.2.4 and Theorem 4.2.1, we get the following theorem. Theorem 4.3.1 Let Ai,A2,---

, Ak€(SI)n£(H)

A'(Ai)/radA'(Ai)^C,

satisfying

i = 1,2, • • • , k.

Then the following statements hold: (i) Ai~Aj if and only if K0(A'(Ai®Aj))=Z; (ii) LetT = J2 ®A
^ j , then

i=l

\J(A'(T))^N(-k\K0(A'(T))^Z{kl Furthermore, T has a unique (SI) decomposition up to similarity. A unilateral weighted shift T on Hilbert space Ti. is an operator that maps each vector in some ONB {e„}^L0 into a scalar multiple of the next

70

Structure of Hilbert Space

Operators

vector, i.e., Ten =

anen+i,aneC

for all n. If an ^ 0 for all n, T is injective. Lemma 4.3.2 Let A££(H) be an injective unilateral weighted shift and power bounded, i.e., ||A n ||<M < oo for a constant M > 0 and all n>l. Then there is an injective unilateral weighted shift B with ||J3||<1 such that A~B. Proof Suppose that

A =

0 0 ai 0 a2 0

eo ei e2*

0 It is well known that A is unitarily equivalent to the unilateral weighted shift with weighted sequence {|a n |}. Thus we can assume that a „ > 0 for all n. If an 1}. k

If Y\ aj>l for all k>ni, then set mi = oo. Otherwise, set j=ni k

mi = min{k>ni

: 1 [ ctj < 1}, j=n1

then mi < oo and ami < 1. If mi < oo, consider the set {k > mi : a^ > 1}. If it is empty, let n^ = oo. Otherwise let ri2 = min{k > mi : a^ > 1}. If k

Y[ Oj>l for all k>ri2, set mri2 : k

J2 ctj < 1}. Similarly, we define finitely many or countably many rij's and mi's inductively satisfying (1) m0 = 0 < ni < mi < n2 < m2 < • • • < rik < mk < rifc+i < • • • (the last one is oo if the sequence is finite); (2)oy
(3) [ ] ai if nk<j<mk,k i=nk

= 1,2, ••• ;

Unitary Invariant

and Similarity

(4) atj
Invariant

of Operators

71

= 1,2,

Define 0<j < n\

fl ai nk<j < mk

Xj

mk<j Denote M = sup{\\Ak\\ Xe£(H) by

: k>l},

< nk+1.

then l<Xj<M,j

OO

OO

3=0

j=0

= 1,2,---. Define

for all J2 oijejCH. It is easily seen that X is invertible. Set B =

X~1AX,

j=o

then B is still an injective unilateral weighted shift. If the weighted sequence of B is {bj}^1, then bj — Note that 0 < bj = <x,
j = nk, k>l, since m fc _i

bj = (\\

ai)~1aj{

xJ1&jXj-i,J>l= Xnk-i if j-i

\[ at) = 1 if

nk < j < mk,k>l. If j = mfc, fc>l, X j - i > l , Oij < 1, thus 0 < bj =

l-ajX-lx
IfTO*< j < rafc,fc>l,0 : H°°—>A'(A) satisfying (1) If h is a polynomial, then (/i) = h(A); (2) If 4(h) =0, thenh = 0; (3) Given h£H°°, there is a sequence {pn}^=i of polynomials such that {Pni-^)}^! converge in strong operator topology. Denote 4>{h) =

lim

pn(A).

sot n~+oo

(4) For each h£H°°, ||ft(i4)||<||h||oo, where ||,

H°

is the norm of h in

72

Proof

Structure of Hilbert Space

Operators

Denote R = H{oo) and A

0

{I-A*A)i

0 / 0

T =

Ho

n2-

0

where Hi = H,i = 1,2, • • • . Clearly, T*T = IR, thus T is an isometry. Since 'A* (I-A*

A)*

0 / 0 I

0 T* =

Wo

0

and since {ker(A*)k

: k>l} = H0,H0c\J{ker(T*)k 0

0 " eo

«i 0

A =

: k>l}. If

ei

a2 0

e2

For xfc = 0©0 © • • • © 0 © efc©0© • • • € # , obviously (T*)fc+2a:fc = 0 for all Ar>0. Hence HiC\/{ker(T*)k : k>l}. Similarly, Hjc\J{ker{T*)k : k>l} for all j>0. Therefore T is a pure isometry and for each heH°°,h(T) is well-defined,

ii^cmi = iwioo and there is a sequence {p n }^Li of polynomials such that h(T)=SOT-

lim p„(T). n—>oo

Let Po be the orthogonal projection from i? onto Ho. Define

$(h) = PoHT)\no. Then it is not difficult to see that $ is a homomorphism from H°° to .4'(A) satisfying (l)-(4). Lemma 4.3.4 Lei A be a power bounded injective unilateral shift with o~e{A) = {z : \z\ = 1}, then given T£A'(A), there exists an h£H°° such that T = h(A).

Unitary Invariant

and Similarity

Invariant

of Operators

73

Proof By Lemma 4.3.2 and Lemma 4.3.3, h(A) is well-defined for each h£H°°. Since 0^ae(A) and kerA — {0}, there exists an injective unilateral weighted shift B such that A*B = I. Since ae(A) = {z : \z\ = 1}, ae(B) = {z : \z\ = 1}. Since B has no eigenvalue and is not invertible, o~(B) = {z : |,z|
For |A| < 1, denote f\ = J2 XnBneo,

then

71=0

A* f\ = A/A and < eo,f\ >= 1. It is obvious that /^ is a vector-valued analytic function for X&D. Given TGA'(A), define /i(A)==<e0,T7x>. Thus h is analytic in D. From fj ^ 0, (A*-X)fj = 0 and dimker(A*-\) = 1 for X&D, there exists an a€a(T*) such that T*fj = afj because T* commutes with A*. Thus h(X) =< Te0, fx >=< e 0 , T*fj > = < eo, afj >= a. Since \h(X)\<\\T\\,h€H°°. Note that h(A)* fj = M*)/x = T * / A for all Ae£> and V{/x = 1*1 < 1} = W, we have T = h(A). Using Lemma 4.3.2, Lemma 4.3.3 and Lemma 4.3.4, we have the following proposition and theorem. Proposition 4.3.5 If A is a power bounded injective unilateral weighted shift with ae(A) = {z : \z\ = 1}, then A'(A) is isomorphic to H°°. Theorem 4.3.6 If A is a power bounded injective unilateral weighted shift with o-e(A) = {z : \z\ = 1}, then A ( n ) has a unique (SI) decomposition up to similarity for each natural number n. Furthermore, \/(A'(A))=N and K0(A'(A))^Z. Example 4.3.7 Let A be an injective unilateral weighted shift with increasing weight sequence {wn}^=1 and let r = lim |iu n |. Then A/r is a n—>oo

power bounded injective unilateral weighted shift and ae(A/r) = {z : \z\ = 1}. By Theorem 4.3.6, \J (A'{A))*?N, K0(A'(A))^Z and A™ has a unique (SI) decomposition up to similarity for each natural number n. Example 4.3.8 Let T be an injective unilateral weighted shift with decreasing weight sequence {wn}^=1 and lim wn = 0. A simple compu-

74

Structure of Hilbert Space Operators

tation shows that A'(T)/radA'(T)^C. By Theorem 4.3.1, \/(A'(T))^N, KQ{A'(T))=7i and T^"' has a unique (SI) decomposition up to similarity for each natural number n. In the following we will discuss the class of bilateral weighted shift. An operator AGC(H) is called a bilateral weighted shift if there is an ONB {e n }^L_ 00 of the space H and a sequence { a „ } ^ = _ 0 0 of complex numbers such that Aen = anen+i. If an ^ 0 for all n, then A is injective. It is wellknown that A is unitarily equivalent to the bilateral weighted shift with weight sequence {|«n|}^L_oo- ^n w h a t follows we always assume A is an injective bilateral weighted shift with (real) monotone weight sequence and A is not invertible. Denote

0

ax 0

e2

a0

ei

0

eo

a-i

A = 0

a_ 2 0

Q!_3

e_i e-2 e-3

0

S R 0 T

\fiej : j > 0 } = H+ \Z{ej :j<0} = H--

We assume that a,j>aj+\ > 0 and

lim an = 1. Since A is not invertible,

lim an = 0. n—* + oo

For polynomial p(.z) = £) OjZJ', denote j=0

PW = Then MT^WpW^

p(S) i? p 0 p(T)

and |b(S)||<|W|oo- Let Q and P* be the orthogonal

Unitary Invariant

and Similarity

Invariant

of Operators

75

projections from H onto H- and , respectively,

V w := \A A e * : A e C >' k =°- ± 1 > ± 2 ' • • • • Note that when /c>0, &—1

71

A;-Rp = ^ Q j P f c ^ Q = j=0

E

(otj J J

fc<j
ai)ek®ek-j.

i=k—j

Thus t.

1

„

L.

1

fa

^

\\pkRPh<{ n «o( E «?)*<( n «onpii2<(n aoiHioo, i= —1

j=fc+l

i= —1

i= — \

where ||Pfc7?p||2 is the Hilbert-Schmidt norm of PkRp, \\p\U is the norm of p in H2(dD). Therefore, +oo

P P II 2 < E n*wi2< E( fc=0

n ai)iipn- < +°°-

+oo fc—1

fc=0

i=-l

are

If {pn}n°=i uniformly bounded in H°° and converges to some h£H°° uniformly in every closed subset of D, then {pn(S)}^=i and {pn(T)}n°=1 converge to h(S) and h(T), respectively, in strong operator topology, and {Rp^n^i converges to some operator Rh in the Hilbert-Schmidt norm. Thus for each h£H°°, we get an operator 'h(S) Rh ' 0 h{T)\ ' denoted by h{A). more,

Obviously, H/i^U^U/iH^ and h(A)eA'(A).

Further-

(/ii + h2){A) = hi(A) + h2(A), hxh2{A) = fti(i4)fea(4) for all

h1,h2£H°°.

Theorem 4.3.9 A'(A)/radA'(A^H00, where A is an injective bilateral weighted shift with decreasing weight sequence {oin}t^-oo' ^m an = 1 n—> — oo

and A is not invertible. Proof By the analysis above, we need only to show that for each XeA'(A), there exists an h£H°° such that X = h(A) + Q, where

76

Structure of Hilbert Space

QGradA'(A).

Operators

Assume that

Xn

-Xio

-X^l-1 -X^l-2

^oi

-Xoo

Xo-1

Xo-2

ei

eo

X = X_i_i X_i_2

X_21 X_20

e-i e-2

A"_2-l -X"-2-2

For each integer k, denote Ak(X) = (£ijXzj)i,j, where e^ 1 i—j =k . That is, Afc(X) is the operator, which reserves the fc-th 0 i-jj£k diagonal in the matrix representation of X and let the other entries be zero. It is obvious that X commutes with A if and only if A commutes with every Ak(X). Moreover, computations indicate that Ao(X) = xool, ^k(X)A^ commutes with A and A0(Ak(X)A^) = Ak(X)A^ for k < 0. Thus there is a constant a such that Ak{X)A^ = al. If a ^ 0, we have

{a~lAk{X)A\k^l)A

= A{a~1^k(X)A^-1)

= I.

This contradicts that A is not invertible. Therefore a = 0 and Afc(X) = 0 for all k < 0. Thus

X =

Xi X 2 0 X3

and X commutes with T. A simple computation shows that Xi X2 eradA'{A). 0 0 Since ae{T) = {z : \z\ = 1}U{0} and ||T|| = 1, we can find heH°° and QeradA'(A) such that X2 = h(T) + Q by Lemma 4.3.4. This completes the proof of the theorem. Theorem 4.3.10 Let AECCH) be an injective bilateral weighted shift with monotone weight sequence. If A is not invertible, then \f(A'(A))^,

K0(A'(A))^Z

Unitary Invariant

and Similarity

Invariant

of Operators

77

and A^n> has a unique (SI) decomposition up to similarity for each natural number n. Given two power bounded injective unilateral weighted shifts A~{ak}'j?Li and B~{/3fc}^=1, by the arguments used in the proofs of Lemma 4.3.3 and Proposition 4.2.13, we have the following proposition. Proposition 4.3.11 A~B {0}.

if and only if kerTA B ¥" {0} and kerrg A ¥"

Note, if A is not similar to B, then kerTA,B = {0} or kerrs,A Assume that kerTA,B = {0}. Define

= {0}.

B 0

.° A It follows from kerTA,B = {0} that A'(T) = {

TB

TBA

0

TA

:TBeA'(B),TAeA'(A)

By Proposition 4.3.5, A'(T)/radA'(T)^H00®Hco. lowing proposition.

and

TBA&kerTB,A}.

Thus we have the fol-

Proposition 4.3.12 Let A, BeA'(T) be two power bounded injective unilateral weighted shifts, then the following are equivalent: (i) A~B if and only if A'(A®B)/radA'(A®B)^M2(H°°); (ii) A is not similar to B if and only if A'(A®B)/radA'(A®B)^H°°®H'x; (Hi) A~B if and only if K0(A'(A®B))^Z; (iv) A is not similar to B if and only if KQ(A'(A®B))=Z®Z. Let S be the unilateral shift on H2 given by Sf = zf(z), feH2. Let 9GH°° be a nonconstant inner function and Pg denote the projection of H2 onto H(6) = H2eOH2. The Jordan block S(9) is defined by S(9) = PBS\H(6) [Bercovici, H. (1988)]. In the following, we will prove that for the singular inner function 9 with S(6)e(SI),S(6)(-n^ has a unique (SI) decomposition up to similarity. Applying this result we get that V^ has a unique (SI) decomposition up to similarity. Theorem 4.3.13 Let 9GH°° be a singular inner function such that S(9)£(SI), then for each n>l,S(9)^ has a unique (SI) decomposition up to similarity.

78

Structure

of Hilbert Space

Operators

Proof Applying the six-term exact sequence of if-theory [cf. [Taylor, J. (1975)]] and the short exact sequence of Banach algebra O-^0H°°

- U H°° -^

Hoo/0H°°

— • 0,

we obtain the following exact sequence of groups Q^K0(H°°)

- ^ Ko(H°°/0H00)

-£+ KiiOH00)

-±> Ki[H°°),

(4.3.1)

where 7r* (resp. i») is the induced homomorphism of -K (resp. i) on KQ{H°°) (resp. Ki(9H°°)) and d is the connected homomorphism. It is proved [Tolokommokov, V. (1993)] that Bar((9H°°)+) = 1, where Bar((0H°°)+)=min{n:(alr--

,am+1)T<=Lgm+1((0H°°)+)

such that (aj + b\, a m +i, • • • , am + bmam+i )TeL<7m((0tf°°)+) for some {6 i }^ 1 c(6'ii' 0 0 ) + ,V m>n}, and n

Lgn((0H°°)+)

: aie(0Hoo)+,

= {(ai,---,anf

(i = 1,2, • • • , n ) , ^ a ^ = 1 2=1

T

+

for some (&!,••• ,bn) ,bi£(6H°°) (i = 1,2,-•• , n ) } . From this i(0H°°)+ is an isomorphism [Wang, Z.Y. and Xue, Y.F. (2000)]. Thus for any a£Ki(9H°°) with n(a) = 0 in Ki(H°°), there is an / = 0g + leGLrd&H00)-*-) for some geH°° such that [a] == [/] in Ki(0H°°) and i»([/]) = 0 in ^ ( t f 0 0 ) . Thus we can find heH°° such that f = eh so that e ' W = TT(/) = 1. Noting that A'(S(0))S*H°°/0H°° and since 00 S(9)G(SI), we have that H^/OH contains no non-zero idempotent. Therefore ir(h) = 2km for some integer k [Taylor, J. (1975)] and hence he(eH°°)+. So / = eh£(6H°°)+. This means that d = 0 by (4.3.1). Finally, we conclude from (4.3.1) and [Wang, Z.Y. and Xue, Y.F. (2000)] that Ko(H°°/9H00) = {n[l] : n e Z } . Let P be a nontrivial idempotent in A'(S{0)W) = Mn(A'(S(0))). Since g
ueHco}^H°°/9H°°,

and k with l < f c < n - 1 such that P =

Xdiag(Ik,Q)X-1.

Unitary Invariant

Set Ti = X\

diag(ikfi)H<-n)

and Similarity

an

d T2 — X T1T2 =

Invariant

1

\PH(n).

of Operators

79

It is easy to verify that

IpHW

and ^2^1 =

Idiag(Ik,0)HM

and T a S ( 6 l ) < n W . o r i = diafl(S(0),O) on #( fc >e0 which is similar to S{6)W. Thus S(<9) (TI) has a finite decomposition. Suppose that {7 J i }£I 1 C.4'(S'(#) ( " ) ) a r e idempotents, m i=l

and

s(e)^\PiHin)e(Si)(i

= i,2,---,m).

By the above arguments, S(9)^\P.H(n)£(SI) operator Xi€GLi(A'{S(6))) such that Pi = Xidiag{IH2,0)Xr1, m

implies that there exists an

(i = 1,2, • • • , m).

m

Thus n[/# a ] = [ £ P.] = £ [ # ] = m[IH2] in 7f o (,4'(S(0)))=Z. Since [J*,] i=l

i=l

is the only generator of Ko(A'(S(6))), we get that n = m. Therefore we can choose YieGLi(A'(S{0)^)) such that Pi = Yidiag(0,0,---

, 0 , 7 ^ , 0 , •• • . O ) ^ " 1 = YieiYr\

(i = 1,2,- • • ,n).

n

Set V = J ] PiYi. Then it is easy to check that FeGL 1 (^'(5(0)(" ) )) with F - 1 = £) y . " 1 ^ and y - ^ - y = e*. That is 5(0)) has a unique i=l

finite (SI) decomposition up to similarity.

80

Structure of Hilbert Space

Operators

Let V be the Volterra operator on H = L2([0,1]) denned by

Vf(t)= [

Jo

f(s)dsjen,te[0,i}.

Then ||V|| = ^,a(V) = {0} and A'(V) is the weak closure of the algebra generated by IH and V. Put Tv = (IH - V){IH + V)'1. Then A'(TV) = A'{Tv) and Ty is unitarily equivalent to 5(e) by [Bercovici, H. (1987)] and hence A'(V)^H°°/eH°°, where e(z) = exp(^). Corollary 4.3.14 Ve(SI) and 0 n > has a unique finite (SI) decomposition up to similarity for each n>\. Proof We have supp(e) = {1} and every nontrivial divisor of e has the form et(z) = Aezp(*f±i),0 < t < 1, |A| = l,zeD. Take zn = 1

, n = 1,2, ••• . n Then lim e t (z„) = lim (e/e t )(z„) = 0, i.e., ( e t , e / e t ) T £ Lg2(H^). n—»oo

So

n—>oo

5(e) G (57), since it is not difficult to see that S(9) 0 (57) if and only if there exists a nonconstant division B\ of 0 such that (6i,6/9i)T£Lg2(H(-00^). Now suppose that {P^^dA'(V^) is an idempotents, PjP, = 0(i ^ j ) m

and ^ Pi= I. Let Q be an idempotent in .4'(Ty |p.W(„)). Then gPiG^'(2v n ) ) = ^ ' ( ^ ( n ) ) is an idempotent and moreover Q^A'(V^\PiH(n)). Thus lA™)|p.-H(„)€(57) ) means that Q = 0 or Pu i.e., T^" | Pi H(«)e(57), so that {Pi}^ is an (57) decomposition spectral family of Ty . By Theorem 4.3.13, n = m and there is YzGLi(A'(V^)) such that P< = YeiY~l,(i = 1,2,--- ,n). The corollary follows.

4.4

The Commutant of Cowen-Douglas Operators

As indicated in Theorem JW2 of Chapter 1, Cowen-Douglas operators have very rich contents, including many of hypernormal operators and subnormal operators. In order to characterize the similarity invariant of CowenDouglas operators, we will discuss the commutant of Cowen-Douglas operators in this section, which is the preparation for characterizing the similarity invariant of Cowen-Douglas operators in Chapter 6.

Unitary Invariant

and Similarity

Invariant

of Operators

81

In this section, we always assume that TeS n (fi), thus V ker(T — z) = H, where H is a complex, separable, infinite dimensional Hilbert space. Note that B1(Q,)c(SI) and for every TeBi(fi) we can easily prove the following result. Proposition CD [Cowen, M.J. and Douglas, R. (1977)] A'(T) isomorphic to a subalgebra of H°°(Q).

is

Proposition CD indicates that if Tei3i(f)), then A'(T) is commutative. In the finite dimensional space C", J n ( A ) s £ ( C n ) , and A'(Jn{X)) is commutative. For Volterra operator V, A'(V) is also commutative. In Section 4.5, we will see that for multiplication operator Mf on the Sobolev disk algebra, MjeBi(D) if and only if A'(Mf) is commutative. In Chapter 5, we will show that the set of strongly irreducible operators, whose commutant is commutative, is dense in the set of all Cowen-Douglas operators in the norm topology. Example 4.4.1

Let Xk = \ and Wk = Xk + J-i 0WX 0 0 W2 T =

Afc 1 0 Afc

Define

C2 C2 °° 2 C2<E£(J>C l

Proposition 4.4.2 Suppose that TG.C(T(.) where H = 8 C 2 . Then (i) Te(SI); (ii) A'{T) is not commutative; (Hi) A'(T)/radA'(T) is commutative. Proof (i) Suppose that PGA'(T), then P i i A2P13 • • • P22

P23 • • P33 ••

c 22 c C2

is given in Example 4-4-1>

(cf. [Jiang, C.L. and Li, J.X. (2000)])

0 and Pfc+i fc+1 = Ek 1PuEk for all k > 1, where Ek = W1W2- • -Wk. Compu-

82

Structure

1 fe+i fc! 2(fc-l)!

tation indicates that Ek = [_

Then Pk+lk+l=Ek

0

Operators

an

. Assume that P n

h

a 12

0-21 0,22

fc!

PnEk an

0

of Hilbert Space

A;!

\_ fc+i fc! 2(fc-l)!

a12

0-21 0,22

a

n - <»2i2^\y. k l

(an - a 2 2 ) # I J T * ! + °ia ~ a

^2U^vk]

0.21 „ fc(fc+l) Oil ~ Q21 2

+ 0,22 fc2(fc+l)2 4

/ \k(k+l) . Q ( a n - a22j V 2 + 12 ~ a 2 1 fc(fc+l) . «21 + «22 2

A21

«2I(2^I)T)2(^)2

Since k(k + 1) —> oo as k —> oo, a,2i = 0 and a n = 022It is easy to see that if P 2 = P , then P = 0 or 7, i.e., Te(SI). (ii)

° *

Denote ^4Q =

00

Since WkA0 = A0Wk for all k>l,

A := diag(i4 0 , A), • • • )zA'(T).

Denote B02

10 00

Set B fcfc+2 = Ek-1B02W3Wi-

fc!_M^i)fc,

-Bfcfc+2 =

0

jfc!

10 00

1 3-4-5

(fc+2)

/•I i(*+B)-fc (fc+2) "" 3-4-5 (fc+2)

0

0

fc(fc+5)

(fc+l)(fc+2)

2(fc+l)(fc+2)

0

0

3+4+5+--- + (fc+2) 3-4-5 (fc+2)

0 U

3-4-5

• -Wk+2 for fc>l. Then

1 3-4-5

(fc+2)

Unitary Invariant

and Similarity

Invariant

83

of Operators

Thus ||5fcfc+2|| < 2 for all k>l and

B:=

00Bo2 0 0 0 0 0 0 0 B13 0 0 0 • • • 0 0 0 0 5 2 4 0 0 • • €C(H).

Computation implies B£A'(T). But since A0B02 ^ therefore A'(T) is not commutative. (iii) Suppose that BGA'(A), then Bu

BQ2AO,AB

^ BA, and

B\2 5 i 3 • • • B22 B23 • • •

5 =

B33 • • •

0 Similar to the proof of (ii), we have

Bij =

Let

0 4

{ek}kLi be an ONB of H = ®C^ 'OWi

such that

0

0 0 W2 0 ••• 0 0

respect to {efc}^=1, where Wk =

0 W 3 ' •.

*1

We may rearrangement {ek}kxL1 denoted by {fk}kx'=i such that 4-

"yli_S" 0 Ai W 2 '

84

Structure of Hilbert Space

oo

Operators

oo

where Hi = V {f2k},H2 = V {/2/t-i} and A;=l

fc=l

0 10

h h /ka ' A l

00±

Ai =

000

=

0 1 0 ••• 00± 0

/3

000

/5"

1

/l

S72fc-i = /2it-2 and 5 / i = 0. Note that Bij

0 b£

Thus 5 i B12 0 52

5

W2'

Without loss of generality, we may assume that Hi —H^- Then H H'

Ax S 0 Ai

A:

where S us an injective back shift with power sequence {1}. By former argument, we have B1B12

B =

0

where Bi&A'(Ai),i = 1,2. Suppose that B,TeA'(A). B =

•Bi B12 0

B2

,

V

BGA'(A),

Then

ft

£2

T=

TiT 1 2 " H 0 T2 H

Furthermore, BT - T B it follows that A'(Ai) commutative.

0*" H 00 -;H'

is commutative. It shows that A'(A)/radA'(A)

is

Unitary Invariant

and Similarity

Invariant

of Operators

85

In the chapter 6, we will show that the class of operators C := {TeCH)D(SI) : A'{T)/radA'(T) is commutative} is dense in (57) operators in norm topology. Therefore we have the following conjecture. Conjecture

Given TeC(H)n(SI),

A'{T)/radA'(T)

is commutative.

If Te6 n (fi)n(S7), we can give a confirmative answer to the conjecture. Theorem 4.4.3 Let AGBn(Q.)n(SI), then there exists a natural number m, 0 < mi where /(A) is a function analytic in 4>. Furthermore, A'(A) jrad A' (A) is commutative. Before we prove Theorem 4.4.3, the following definition and proposition are needed. Definition 4.4.4(Minimal index)

Let

TGBn(n),AeA'{T) and A(z) := A| f c e r ( T _ z ). If a(A(z)) is disconnected at zo£Q, then there exists a positive number 6 such that cr(A(z)) is disconnected in each point in D(zo, 5) := {z : \z — ZQ\ < 5}. Thus we can find a positive number s such that a(A(z))r\D(\(zo),s) = \(ZQ),Z&D(ZO,5), where A(zo) is an eigenvalue of A(ZQ). Set P(z)=

{A{z)-X)-1dX.

f dD(\(z0),e)

P(z) is said to be a holomorphic idempotent defined on D(\(zo), e) induced by A'(T). If each holomorphic idempotent P(z) induced by A'(T) satisfies dimker(T\

v

r a n (P( 0 ) _ Z o ) )

< n,

then n is called the minimal index of T or we say that T has the minimal index n. Example 4.4.5 IfTGBi(Q),

the minimal index ofT is 1.

Example 4.4.6 Assume that f(t) = z(z~ | ) , then Tj£B2(Q)n{SI),OeQ. 2 is not the minimal index ofTi. But there is a connected open set fii such thatT^B^VLx).

86

Structure of Hilbert Space

Operators

Example 4.4.7 For the analytic Toeplitz operator Tzi, T*2€B2(D). ple computation shows that T*2 has no minimal index.

A sim-

By the definition, we have the following proposition. Proposition 4.4.8 Given A£Bn(Q,), n is the minimal index of A if and only if there is no B£Bm(Q),m < n, such that BGA'(A). According to the literature of [Cowen, M.J. and Douglas, R. (1977)] (Chapter 3), we have the following proposition. Proposition 4.4.9 T£Bn(Q)r\(SI), there is a natural number m,0 < m
A2 =

(A*\ni)

Then Ai

A12

Hi

0 A2 By Lemma 1.2 of [Herrero, D.A. (1990)], erp(A*) = 0. Thus dimH{ Since A — z is right invertible for zefl, let B =

Bi

B12

Hi

0

B2

Hi

= oo.

be a right inverse of A — z, i.e., (A - z)B

~A\-z 0 ~IHI

.°

.4l2 'B1B12 A2-z_ 0 B2 0 " IH

±.

Thus (A2 — z)B2 = Iftj-, and Ai — z is right invertible for zef2. It is not difficult to prove that B2&Bi{Q) for some I < n. Let ir be the canonical mapping from £(H) to £(H)/JC(H), where £(H)/fC(H) is the

Unitary Invariant

and Similarity

Invariant

of Operators

87

Calkin algebra. Then we have n(B)n(A

- z) =

ir(I W l )

0

0

7T(/ W x)

This implies that ran(A\ — z) is closed. Denote (ei(?/),••• ,ek(y)) holomorphic frame of ranP(y), then for l<j
=

= (\{y) -

\J ranP(y),ran(Ai

the

z)ej(y).

— z) = H\ for each zG<&.

Thus

Let m = n — I, the proof of the theorem is complete.

Lemma 4.4.11 [[C owen, M.J. and Douglas, R. (1977)] Let f : ri—+Gr(n,H) be a holomorphic curve, where fi is a connected open set. Let Ef be a vector bundle defined on fi and let ri(z),--- ,rn(z)) be the holomorphic frame of Ef. If ri(zo),-- • ,rn(zo) form an ONB of f(z0), then there exists a holomorphic frame rj", • • • ,7^. of Ef, defined on some open set A containing ZQ, such that ri(zo) =n{zQ),i

= 1,2, ••• ,n

and < ^k)(zo),rj(zo)

>= 0, l
1,2, • • • .

Proof of Theorem 4.4.3 Without loss of generality, we assume that n is the minimal index of A and D c O . If there is a B£A'(A) such that a(B(0)) = {Ai, A2}, Ai =^ A2, then since B(z) is holomorphic for z€.Q, there exists an £ > 0 such that <x(B(z)) = {X1(z),\2(z)},zeD(0,e)

= {z : \z\ < e},X1{z) ? X2(z),Xi(0)

and A2(0) = A2. Since B(z) is holomorphic on Q, we can find an £1 > 0 such that D(Xue1)na(B(z))=D(X1,e1)na(B(z))

= {Ai(z)}

D(X2,e1)n
= {X2(z)}.

and = D(X2,e1)n<j(B(z))

= Ai

Structure of Hilbert Space

Operators

Set P(z)=

(B(z)-\y1dX,zeD(0,e1).

J SD(Ai,ei)

Then I-P{z)=

{B(z)~\)-1d\,zeD{0,ei).

f dD(X2,El)

Denote M =

\J

ranP(z). By Proposition 4.4.10, we may assume that

«e£»(0,ei)

Ai := A\M&Bk{Cl). Denote A2 = {A*\Mi.)*,Bl Me{LatA)n(LatB) and A =

Ai A12 0 A2

=

B\M,B2

=

M M x. B

(B*\Mx)*.

Bi B\2 0 B2

Note that

M

where LatA denotes the lattice of invariant subspaces of A. Let {e\(z), • • • ,ek(z)} be a holomorphic frame of {ek+i(z), • •• ,en(z)} be a holomorphic frame of E2 = {(x, z) : x€(I - P(z))ker(A

- z),

ZGD(0,

EA1

and

e)},

then {ei(z),--- , ek(z), ek+i{z), • • • ,en(z)} is a holomorphic frame of EA-I on£(0,e). Since AB = BA,B\A\ = A\B\. By Proposition 4.4.10 it follows from Ai€6fc(fi) that A2£Bn-k{Sl). Denote Hi = kerA,H2

= kerA2GkerA,

• • • ,Hm = kerAmekerAm'1,-

By Theorem CD in Section 1.4, we have Wi©W2® • • • ®Hm = V k 0 ^ 0 )

:

!<*<», 0 < j < m - 1},

and 0 Al2 A13 • • 0 A 2 3• •

0 '•

Bn B\2 B\z H2

B22

B

B23

B-33

n2

••.

Unitary Invariant and Similarity Invariant of Operators

Denote A = kerA1,L2 = kerA\ekerAi,• • ,Cm = kerAfekerAm~1,-• • . By Theorem CD again, Cii ®Cm = V R 0 ^ 0 ) : l
Bn

0 A12 A13 0 A'23 A, =

C2

0

B'33

0 Since

B12 B13 B22 B23

0 Therefore

BGA'(A),B1€A'(A1).

Bk+i,k+i~Bk,k

and B'k+i,k+i~B'kk,

k= 1,2,-•• .

Since B'n =- B l\/{ei(0),-,e f c (0)},

°{B'n)

= {Ai} and
= 1,2,

Set B'w •••

B'lm

£1

{B\)m =

B'mm_ r

. ° then

a ( ( S ! ) m ) = {Ai}. Since Bu = B| f c e r^ff(5ii) = {Ai,A a }. Thus a(5 fcfc ) = {Ai,A2},fc = l , 2 , - - Set B\\

• ••

B\m

Bm — v

-^mm _

Hm

A

c3-

Structure of Hilbert Space Operators

90

then a(Sm)

=

{Xl,X2}.

Define Pm =

(Bm - A)-xdA

f M>(Al,£l)

and Pm = Pm®0 £ ®Uk • k>m.

Then Pm is an idempotent and PmAm 0 A12

•• •

o

= AmPm,

Air,

where

Hi

••

"lm-1

o Denote Af = \J{ranPm

fc>m

Tim

: m = 0,1,2, • • • } , then N&{LatA)C\{LatB)

and

JW{0}. OO

{Claim 1} Af = M = V f o C O . - " ,e n (z) : z€D(0,ei)} = V AnSince V { e P ( 0 ) : l
- l } c V / R ^ C 0 ) : 0
- 1},

A © • • • ©AnCfti© • • • ©W m , A © • • • ®Cm£LatB and Cx@---®Cm£LatBm. Note that fcer(S! - A)n = ker{Bx ~ Ai)* = kerAx = \/{ei(0), • • • , efc(0)}. A simple computation indicates that dimker(Bm

- \i)mn

= mk.

For each x£C\® • • • ©An, we have (Bm - Ax) m "x = ((ST) m - X^x

= 0.

Unitary Invariant

and Similarity

Invariant

of Operators

- Xi)mn

=

91

This implies that A © • • • ®£mCker(Bm

ranPm.

Since dim{Ci® • • • ®£m) = mk, C\® • • • ®Cm = ranPm Denote K = \J{ek+l{z),---

and M = M..

,e„(z):ze£)(0,e1)},

then JCe{LatA)r\(LatB). Define Qm=

f J SD(A 2 ,ei)

(Bm - A)-^A and Qm = Qm®0

E

k>m

@7ik.

Similarly, we can get K. = \J{ranQm : m = 1,2, • • • }. Since {e\(z),- • • ,ek{z),ek+i(z), • • • ,en(z)} is a holomorphic frame of EA, we can obtain an ONB of kerA by Gram-Schmidt orthogonalization. Without loss of generality, we may assume that {ei(0),--- ,e fc (0),e fe +i(0),--- ,e n (0)} is the ONB of kerA. Since \\(z) ^ \2(z),z£D(0,ei), by Lemma 4.4.11, Theorem JW 1 in Chapter 1 and the argument of Claim 1, we have the following Claim 2. {Claim 2} There are two subspaces E\ and E\ of Wj, i = 1,2, • • • , satisfying (i) Hi = E[+Ei,E\C\Ei1 = {0}(ii) --- ,5fc>fffc+i,--- ,9n} olHi,i = 1,2,--- . By (i), (iii) of the claim and a simple calculation, we have 0 A12 Au 0 A23 A =

0

Bn

W2 W3'

B\2

Bis

B22 B23

B =

B.33

n2 w3'

Structure of Hilbert Space

92

Operators

where "11 "12

Aij —

0

a\\

Bij =

0 4

{Claim 3 } K + M = H and K,C\M = {0}. By Claim 2, we can assume that V{ e j

(0) : l ^ J < n } = 'Hm+i- Set Bn

0 A 1 2 0 ••• Wi 0 A23 • • •

X(d)

0

0

Wl

B22

0

n2 Hz-

0 "-.

B33

then ,4(d)€#„.(fii), where OeftiCft. Since AB = BA, A{d)B{d) = B{d)A{d). Since a(Bn)

= {Ai,A 2 } and since Sfc+1:fc+1~Bfc,fc, a(B(d)) = {X1,X2}

and (B(d) - Ai)*(5(d) - A 2 )"- fc = 0. Set P =

(B(d) — X)~1dX. Then P i s an idempotent. To complete

J 3£>(AI,EI)

the proof of Claim 3, we need the following Claim 4. {Claim 4 } r a n P = M. Set > n

•••

0

: , Pm(d) =

Bm(d) 0

•Bn

and Pm(d) = P m (d)®0 £

Hm &-yik.

J

(Bm(d) - X)^dX

aD(All£l)

Then {P m (^)}m=i

are

uniformly bounded.

fc>m

By Banach-Alaoglu Theorem, Pm{d) converge to P in weak operator topology. It is easy to see that ranPl{d)=ranP1

= {ei(0),--- ,e fc (0)}.

Unitary Invariant

and Similarity

Invariant

of Operators

93

Note that \/{ef(p) : l<j
- z)cker(A

- z)

and dimker(A — z) = n < +oo, (B(z) - f{z))nker{A

- z) = 0, zGQ..

Since B(z) = Ax{z)A2{z) - A2{z)Al(z), f{z) = 0. Thus B(z)nker(A-z) = 0. It follows from \J{ker(A - z) : zefl} = H that Bn = 0. Therefore BEradA'(A) and the proof of second part of Theorem 4.4.3 is now complete. By Theorem 4.4.3 and its arguments, we have the following corollary. Corollary 4.4.12

Let AeBn(Q,)n(SI), radA'{A) = {BeA'(A)

then :Bn=0}.

Corollary 4.4.13 Let AeBn(£l), then AG(SI) if and only if A'(A)/r adA'(A) is isomorphic to a subalgebra of H°°. Proof Since H°° contains no nontrivial idempotent, A'(A)/r adA'(A) has no nontrivial idempotent, thus nor does A'(A). This proves the sufficiency. The necessity follows from Theorem 4.4.3.

94

Structure of Hilbert Space

Operators

By Factorization Theorem / = \Q f° r e a c n function f£H°°, where x is an inner function and Q is an outer function. If \ is a finite Blaschke product, then by the knowledge of Toeplitz operator Tf£Bn(Q). Thus we have the following corollary. Corollary 4.4.14 Let feH°°. If there is a XQ&D such that inner part °f f — /(-^o) is a finite Blaschke product, then Tf^(SI) if and only if A'(Tf)^H°°. Proof We first prove that if Tfe(SI), then radA'(Tf) = {0}. Without loss of generality, we assume that / = \Qi where x is a Blaschke product of order n and Q is a outer function. It follows from [Cowen, C.C. (1978)], that A'{Tf)cA'(Tx). But Tx^n). Thus A'(Tx)^Mn(A'{Tz)). Since radMn(A'(Tz)) radA'(Tf) By Corollary 4.4.13, A'(Tf) A'{Tf)^H°°. 4.5

= {0}, = {0}.

is commutative.

Since

H°°
The Sobolev Disk Algebra

Let Q be an analytic Cauchy domain in the complex plane and let W22(Q) denote the Sobolev space W22(ti) = {f&L2(Q,,dm) : the distributional partial derivatives of first and second order of / belong to L2(Cl,dm)}, where dm denotes the planar Lebesque measure. For f,g€W22(Q,), we define

IDafD^dm,

:= Y, |a|<2

7

then W 22 (fi) is a Hilbert space and a Banach algebra with identity under an equivalent norm. By Soblov embedding theorem, f&W22(Q) implies that feC(Ti) and ||/|| C m)<M||/||n/22 ( n) for some M. For f£W22(Cl), the multiplication operator Mf on W22(fl) is denned as Mfg = fg,

geW22(n).

Let

W(n) := {Mf : f£W22(Q)},

Unitary Invariant

and Similarity

Invariant

of Operators

95

then W(Q.) is a strictly cyclic operator algebra with strictly cyclic vector e(s,t) = 1. It has been proved that a(Mz) = aire{Mz) = Q. and A'(MZ) = W(Q). Let Aa(Mz) be the algebra generated by the rational functions of Mz with poles outside Q and R(Q) := Aa(Mz)e. Since W(Q) is strictly cyclic, there exist positive numbers N and K such that for each / G W / 2 2 ( 0 ) ,

N\\f\\w>Hn)<\\Mf\\
k*Z0(Mf(n)) =

f(zQ)=

for some kZo€R(Q), and kZo£ker(Mz(Q.) — z0)*; (Hi) A'(MZ(Q)) = Aa(Mz{Q)) = {M/(fl) : feR(Q)}, where M,(fi) := Mf\mn). Therefore A'(MZ(U)) is strictly cyclic and Mz(£l) is rational strictly cyclic. Proof (i) If 20 £ Q, then

(*o -

z^eRiCl),

M(*o-*)-i = (ZQ - M , ) " 1

and M ( ^_ x ) -i(fi) = ( z 0 - M , ( f i ) ) - 1 . Therefore, a(Mz(Cl))(Zil. Assume that z0 6 (T and for all g£R(£l), the function (z0 - Mz(Q,))g = (zo - ^)s(z) vanishes at z0. Thus zo - M 2 (fi) is not onto and a{Mz(Q))cCl. Let z 0 efi, then for /ei?(fi) there exist a number (5 > 0 and a function h analytic in T such that

f(z)-f(z0)

=

(z-z0)h(z),zeY,

96

Structure of Hilbert Space

where T := {zeC : \z - z0\ < 6}cTcCl.

Operators

Denote

1

A

E := Q. - - F = n - {zeC : |z - z 0 | < - } and define

«.)-{*<*> /(z)(z-z )

-1

0

-*e£

Then f(z) — /(zo) = (z — zo)<7(z) for all zefi and g is analytic in fi. Note that f Qf 9 y \9\2dm<—J |(z-Zo)0(z)|2dm<-^||/-/o||2y22(f1), i.e.,
[ / (

^

/ ( 2

°

) ]

= /'(*) = g(z) + (z-

z0)g'(z)eL\E).

We have (z - z 0 )s'(*)€L 2 (£;). Thus jf \g'{z)\2dm< 1 ^ |(z - z 0 ) 5 '(z)| 2 dm< J ^ l / - / o | | U ( n ) That is g'eL2(E).

Since

a 2 [ / ( z ) - / ( z 0 ) ] _ o „ , , ,_ , „ - 2,

2 5 ' + (z - z 0 ) f l "eL (£),

ds2 we have (z — zo)g"&L2{E) and

^|5"W| 2 rfm<|||/-/ 0 || 2 v 2 2 ( n ) . Therefore, 5 e W 2 2 ( £ ) . For k = 0,1,2 and z e f r = {zeC : |z - z 0 | < §<J}, by Cauchy formula,

I^WI-ISi/K^*1^!*1"01«esr 2

Therefore, (?,(?', 5 "eZ, (§r) and # e W 2 2 ( § r ) , Since geW22(E) at the same time, we have g£W22(Q,). It follows from / ( z ) = /(ZQ)+(Z—zo)g(z) that the

Unitary Invariant

and Similarity

Invariant

97

of Operators

codimension of ran(Mz(tt) - ZQ) is 1. It is obvious that ker(Mz(Q) — ZQ) = {0}, thus z0Gps-F(Mz(n)), ps-F(Mz(n)) = fi and ind(Mz(Q) - zQ) = - 1 . (ii) and (hi). Denote A := {M/(fl) : feR(Q.)}. It is easy to see that A is a commutative algebra with identity closed in weak operator topology and Ae = R(fl), that is A is strictly cyclic. By [Lambert, A (1971)], the adjoint space A* = {g* : there exists a function g£R(Q) such that g"(Mf{Sl))=<Mf{Sl)e,g>}. Thus, for homomorphism k*0 : k*ZQ{Mf(Q)) = f(zo),f£R(Cl), kZa£R(il) such that

there exists a

K0(Mf(n)) = f(z0) =< Mf{Q)e,kZ0 >=< f,kZ0 > . On the other hand, if p : A —> C is a homomorphism, then p(M,(n)) = z 0 G f f ( M / ( n ) ) = n . For a rational function r{z) with poles outside fi, p(r(Mz(Q))) = r(zo). Since {r(M 0 (fi))} is dense in A, p(Mz(Q)) = f(z0) for each /G.R(fi). Let s€fl(n), then < g,(Mz(n) - z0)*kZo >=< (z - z0)g,kZo >= 0. Therefore, kzoeker(Mz(tt) - z0)*. Assume that T€.A'{Mz(Q)), then for z0ett,

Mz(n)*T*kZ0 = T*Mz(nykZ0 = ztr*kzo. Since nul(Mz(Q)

— ZQ>)* = 1, there is a complex number t(zo) such that T*kZ0

=t(z^)kZ0.

Thus ( r / ) ( z ) = < Tf,kz

> = < /,T*/e z > = t(z)f(z),

Because t(z) = (Te)(z)eR(Sl),T

f£R(Q),

ZGQ.

= Mt(il). This proved that

A'(MZ(Q)) = A =

Aa(Mz(Q)).

Proposition 4.5.2 Let Vi be a bounded simply connected Cauchy domain, then the set of polynomials is dense in R(£l). Proof Given a positive number e and an f&R(Q), let r(z) be a rational function with poles outside fl such that ||/ —7i|iy22(fi) < £• For this r, there

98

Structure of Hilbert Space

Operators

is a bounded Cauchy domain fii such that QiDfiiDfiiDfi and the poles of r(z) are outside fii. By Mergelyan theorem, there exist polynomials pn such that pn converge to p uniformly on fii. If follows from Cauchy formula that p'n—>r' and p'^—>r" uniformly on fi. In conclusion, pn^>r in W22(Q,) and there is an integer N such that ||pjv — ^||iy22(n) < £ - Therefore, WPN - f\\w™(si) < 2e. Because of the special definition of the inner product in Sobolov space for general fi, it is very difficult to discuss the properties of the space R(fl) and the multiplication operators further. But if we choose a better fl, many results can be obtained. When fi = D, the unit disc, we call R(D) Sobolev disk algebra. For simplicity of symbols, in what follows we will denote Mf(D), the multiplication operator with symbol f&R(D), by Mf. Proposition 4.5.3 (i) Hilbert space R(D) has an ONB {e n }^L 0 , where e

n

=

Hnz

i Pn

[(3n<'-7r i +2n+l)7r] 2 > n

=

°> *> 2 > ' ' ' !

(ii) R(D) is a functional Hilbert space, the reproducing kernel of which oo

is given by k(u,v) = J2 / ^ u " ^ n - For zoSfi, n=0

K = f>^)*n; (Hi) If f(z) = Yl fnzn

is analytic in D, then fsR(D)

if and only if

n=0

E-"=o l/n3

2 Hn

If fGR(D),

then Sn = J2 fkzk

< +oo.

converge to f uniformly in D;

fc=o

(iv) Mz is an essentially normal unilateral weighted shift, Mzen ,n = 0,1,2, ••• , and \\MZ\\ = (v) Assume that f(z) 52fnzneR{D), then

Qn^n+l)

®n —

/o Mf =

0

/iff f° h%f\% /3

/33

j2

03

/98. 15'

eo ei

/o Jl

03

e j0

2.

e3

=

Unitary Invariant

and Similarity

Invariant

of Operators

99

Proof (i) Computation shows that zn = (s + it)n,n = 0,1,2,-•• ,is n2+2i an orthogonal system and ||z n || 2 = (,a»*-n*+2n+i)* ^B y p r o p o s i t i o n 4 5 2 , n+1

{e„}£°=0 forms an ONB of R(D). (ii) For z0£D,\f{z0)\<\\f\\cl5<M\\f\\w*2{D) for each f&R{D). R(D) is a functional Hilbert space with reproducing kernel 00

00

n=0

n=0

Then

k(u,v) == £e n («)i>) = £/£« n l/ n . l n

Since < f,J2(3W z >= f(z0) for feR(D), kZa = E ( ^ o n ) ^ n . (iii) Assume that f(z) is a function analytic in D with the Taylor expansion / = ^2fnzn, then / = £) J£en£R(D) if and only if X) laH 2 < +°°then «„(«) = £ fkzk

If feR(D),

= f ) £ e f c converge to / in W 22 (£>).

fe=0 fc=0

By Sobolev imbedding theorem, sn(z) converges to f(z) uniformly on D. (iv) Since Mzen = zf3nzn — — — e n + 1 = a n e n + 1 , n = 0,1,2, • • • , Pn+l

MZ*M2 - M 2 M ; = diag(al, a\ - a2,, a2, - a?, • • • ) . Since a2n+1 - a 2 - ^ 0 , M.GC1. (v) For / = £ / „ * " = Zfcen&R(D), ^ Ji/f

^

V^

t

Pn

^

I fm-nTS2-

< Mfen,em

>=< yfk-3 e n + f c ,e m > = < _ ~ Pn+fc [0 Thus M / can be represented as a matrix in (v). Proposition 4.5.4 Let f be analytic in D, then f£R(D)

m>n

^ m
f'eH2,f"eL2(D). Proof If f€R{D) and f(z) = *£fnzn. By Proposition 4.5.3, E 1 ^ < +00. Thus ]>> 2 |/„| 2 < +00 and f'&H2. From the definition of R(D),f"€L2(D). Conversely, if f'eH2,f"eL2(D). A theorem in [Duren, P.L. (1970)](Theorem 3.11) asserts that function / is analytic in D, continuous to C = dD and absolutely continuous on C if and only if f'£Hx. Thus f£C(D) and feR(D). For f&R(D), denote fr := f(rz),z£~D,0

< r
100

Structure of Hilbert Space

Operators

Proposition 4.5.5 (i) fr converge to f in R(D) as i—>1; (ii) fr(Mz)—>Mfasr->l. Proof (i) Since f(z) is uniformly continuous in D, f(rz)—>f(z) formly in D. Thus / \f(rz)-f(z)\2dm-,0

uni-

(r—,1).

JD

Given a positive e, since / ' and f"€L2(D), we can choose ro such that 0 < r 0 < 1 and / \f'(z)\2dm < e and / \f"(z)\2dm < s. D\(2r0-1)D

D\(2r0-1)D

Thus fD\rf'(rz)-f(z)\2dm =

IDVOD

<

WDVOD

k / ' ( « ) - f'(*)\2dm \rf'{rz)\2dm

+ froD \rf'(rz)

-

+ j D V < j D \f'(z)\2dm}

f(z)\2dm

+ JroD \rf'(rz)

f(z)\2dm.

-

Similarly, JD\r2f"(rz)-f"(z)\2dm <

WDVOD

\r2!"{rz)\Hm

+ fD^D

\f"(z)\2dm]

+ JroD \r2f"(rz)

-

f"(z)\2dm.

Since rf'(rz) and r2f"(rz) uniformly converge to f'(z) and f"(z) respectively in any closed subset of D, we can find a r% such that when r > r j , 1. If |2| 2r0 - 1. Thus, if r > n, we have I Vf'irz)

- f'(z)\2dm

< 4e + ns2

JD

and

I

' \r2f"(rz)

- f"(z)\2dm

ID

Therefore, fr—>/

(r—>1) in R(D).

< 4e + ne2.

Unitary Invariant

(ii) Given

and Similarity

Invariant

of Operators

101

g£R(D),

fr(Mz)g=^

f(rm-Mzrlg{z)d£

f

= ^l I /(rOte-*)- 1 ^)^ ICI=J

= f{rz)g{z), Assume that f(z) = ^2fnzn,

zED.

then fr(z) — Y^fnfnzn.

Since

By Proposition 4.5.3, fr&R(D). Thus fr(Mz) = Mfr. Since A'(MZ) strictly cyclic, fr—>f in -R(D) is equivalent to Mfr = fr(Mz)—>Mf. T h e o r e m 4.5.6 Assume that f£R(D), then (i) a(Mf) = f(D); (ii) ae(Mf) = aire(Mf) = f(C). If z0eD and f{z0)
- f(z0))*

is

then

= -»,

where n is the number of zeros of f(z) — f(zo) in D, including multiplicity. Proof (i) Let ZQ^LD, then the value of the functions in the range of Mf — f(zo) is zero at z0 and Mf — f(zo) is not onto. This implies that a(Mf)Df(D). On the other hand, if u>o^f(D), it is easy to see that [f{z)-w0)-^R{D) and [Mf - t«o]M (/ _ 10o) -i = M ( / _ W o ) - i [ M / - w0] = I. Thus a{Mf) = f(D). (ii) By Proposition 4.5.1, ae(Mz) = cr ire (M z ) = C. By [Conway, J.B., Herrero, D.A. and Morrel, B.B. (1989)], ae(Mfr)

= ulre(MfT)

If z 0 e C , f(rz0)eaire(Mfr).

= o-e(fr(Mz))

= fr(C) = {/(*) : |«| = r } .

Since

Mfr-f(rzo)—+Mf-f(z0) f(z0)&aire(Mf)

and

as r —» 1, f(C)Caire{Mf).

102

Structure of Hilbert Space Operators

Conversely, if there is a z0GD such that / ( z o ) ^ / ( C ) , then f(z) — f(z0) has only finitely many zeros in D, denoted by {zo,zi,--- , z n }c.D. From the proof of Proposition 4.5.1(i), we know that f(z) - f(z0) = {z- z0)k°(z -

k

Zl)

> • • • (z - zn)k»g(z),

geR(D)

and g(z) ^ 0 for z&D. By Proposition 4.5.1, Mf - f(z0) = (M,_ Zo ) fco • • • ( M 2 _ , J f c " M s n

is a Fredholm operator and ind(Mf — f(z0))

= - ^ ^

= — n. It is easily

i=0

seen that nul(Mf — f(zo)) = 0 and therefore nul(Mf — f(zo))* = n. Proposition 4.5.7 Let feR(D), then (i) IJZQ&D andf(zo) & f{C), thenMj£Bn(Q,), where Q. is a component of pa-p(Mf) containing /(zo) andn is the zeros of f(z) — /(zo) in D; (ii) Mf is an essentially normal operator. Proof (i) It needs only to prove

\J{ker[{Mf - f{zQ)Y]k : k>l} = R(D). In fact, if yeR(D)e\J{ker[(Mf

- /(z 0 ))*] fc : fc>l},

then for each A;, ye[ker[(Mf - / ( z 0 ) ) 1 Y

= ran{Mf - /(z 0 )) f e .

Thus we can find a function h€R(D) such that y=[Mf~

f(z0)]kh

= (z - z0)kok

• • • (z -

zn)k"kgkh.

Note that ZQ is the zero of y of order kko. Since k>l can be any natural number, ZQ is an essentially singular point of y. It is a contradiction. (ii) By Proposition 4.5.3, there is a sequence of polynomials pn converging to / in R(D). Since A'(MZ) = {Mg : g£R(D)} is a strictly cyclic algebra, MPn—>Mf (n—KX>). By Proposition 4.5.3, Mz is essentially normal, and n(Mz) is normal in the Calkin algebra £(R(D))/K{R{D)), where TT is the canonical mapping from

103

Unitary Invariant and Similarity Invariant of Operators

C(R(D)) to C(R{D))/K(R(D)). so is

Therefore 7r(pn) are normal elements and

TT(M/) = lim 7r(M p J, n—>oo

i.e., Mf is an essentially normal operator in C(R(D)). Example 4.5.8 f(z) = z3 + z2, g(z) = ^ £ - £ j £ , M < 1, |/?| < 1. an(

/ ( C ) separates f(D) into three components: fii,^' and i e 0 3 - M;GBi(ni)nB 2 (Q 2 )nS 3 (n3).

^ ^ 3 , l e Q i , —16^2

M;eB2(D).

In the following, we will discuss the commutant of the multiplication operators. Lemma 4.5.9 Let feR(D),

z0eDx := / _ 1 ( f i ) <""*

M}eBn{fl),

/(2) - / ( z 0 ) = (z - zo)'11 (2 -

h

Zl)

h

> •••(z-

Zl)

1+1

where {zi}i=1
are pairwise distinct, ^2 hi = n,gZo(z)

^gZ0{z), — ^ 0,z£D. Then

i=i

there exist n linearly independent vectors is — ru 1.1 . . , uhi-i i. n-za • — \H-zo,^Zot

iKz0

i^u

kh2~l . . . I'WSI

i

hht+1~i\ i^zi

}

such that

kerM}_f{zo) = \jKZo. Proof

Choose kl0,k%0,--- , k ^ _ 1 £R(D) such that M* 1.1 _ u A/f* 1.2 _ JL1 z - z 0 K z o — "201 J W 0 - z o K 2 o — K2o>

Jw

Then for every usR(D)

11/* juhi-l_i.hi-2 ' Mz-z0Kz0 — Kz0

and 0<j
< u, M}_f(zo)kl0 > = < ( / - /(*«,)«, *£, > = < (z - z0)hi-j{z

- Zl)h*

•••(z-

zi)h^gz0u,kZo

>

= 0. Thus k{oekerM*_f(zoy

Similarly, choose k\.,--- ,kz*+1~2eR(D)

M*z_Ziki = ki;\o< Then

keekerM}_f{zo).

i
such that

Structure of Hilbert Space

104

Operators

If there is a sequence of complexes {c^ : 0
— 1} satisfying „ , we get

- 1). Thus kZQ is linearly

= \/^o-

= / _ 1 ( n ) . For z 0 e D i ,

Given feR(D),M*feBn(n),D1

/ ( 2 ) - /(^o) = (z - 2o)(^ - zi) • • • (z - z„_i)5 Z0 (z), # Zo (z) ^ 0 for ze£>. (4.5.1) Let iV~z0 = {zi}2=Q, the numbers in Nzo can repeat. Denote T = Li{NZo :

ZQ£DI,

there is at least one Zi(0
/ ' ( * ) = 0}. Lemma 4.5.10 T is an at most countable subset of D\ and ZQGDI\T and only if the numbers in DZo are pair-wise distinct. Proof If zkGNZo with f(zk) = 0 for some k, 0
= (z - Zi) • • • (z - zn-i)gZ0(z)

if

+ •••

+ (z - z0) • • • (z - Zfc_i)(z - zk+1) • • • (z - zn-i)gZa(z)

-\

+ (z-z0)---(z-zn_i)^0(z). Let z = Zfc, we have (zfc -

Z0) • • • (Zfc -

Zfc_l)(Zfc -

Zfc+l) • • • (Zfc -

Zn_l)g20(Zfc) =

0.

Thus there exists at least an i ^ k with z; = z&. The converse is obvious.

Lemma 4.5.11 Let f€R(D), Mj?eZJ„(fi), then there exist an open subset Acf~1(Q)

and analytic functions ai{z),a2(z),---

,an(z)

and Zi(z),Z2(z),..-,Zn_i(z)

Unitary Invariant

on A such that for each

and Similarity

Invariant

105

of Operators

A&A'(Mf),

(Ag)(z) = a1{z)g(z)+a2(z)g(Z1{z))

+- •

•+an(z)g(Zn-1(z)),z€A,g£R(D).

Proof Denote Dx = / _ 1 ( 0 ) . If the set {z&D^ : f'(z) = 0} is finite, then T is finite. Let A = Di\T. If the set {zeD\ : f'(z) — 0} is a countably many set. Choose an open subset D2CD1, then D2C\T is finite. Let A = D2\T. For zGA, by Lemma 4.5.9 and Lemma 4.5.10, kerMf_f(zj

= y\kz,

kZl, • • •

and the numbers in Nz are pairwise distinct. Since

,kZnlj A&A'(Mf),

A*kz&kerM*f_f(z). Thus there exist n complex numbers cei(z), 02(2), •• • , an(z) such that A*kz = ai{z)kz

+ a2(z)kZl +

\- ^ ( z ) ^ . ; .

Therefore, (Ag)(z) =
>

=
>

= ai(z)g(z)

+ a2(z)g(zi)

+ ••• + an(z)g(zn_i).

(4.5.2)

For Nz = {z, z\, • • • , zn-i}, we choose ui, u2, • • • , un-\&D\ such that (i) the n open balls B(z,e), • • • ,B(zk,£)(l 0; (ii) {wfc}fc~i are pairwise distinct and n-l

{uk}nkzln[B(z,e)U(\jB(zk,e))}=
Set fk(Z, Zi, • • • Zn-x) = (uk-Z)(uk-Zi)...

(uk-Zn-.i)gz(uk)-f(uk)+f(Z),

where (k = 1,2, • • • , n — 1) and ZeA.. Computations show that l A l = Me*[|^|(z,2 1 ,...,z n _ 1 )=(*,«i > -,*»-,)]! = I[I1 («i - «i)][Il (* - ^)][ n ff(«i - z)][U gz(ui)]\ ± 0 iytj

ijij

t=l

1=1

106

Structure

of Hilbert Space

Operators

By the implicit holomorphic function theorem [Griffiths, P. (1985)] (Theorem 9.6), there exists a 6 > 0 and analytic functions Zk = Zk(v), (k = 1,2, • • • , n — 1) in B(z, S) such that for v£B(z, S), f(uk)-f(v)

= {uk-v){uk-Z±{v))

• • • {uk-Zn-i{v))gv(uk),

k = l,2,---

,n-l.

Thus for vGB(z,S), V,ZI(V),- •• , Z n _i(v) satisfying (4.5.1) and the analytic functions Z\(v), ••• , Zn-\{v) can be extended analytically to A satisfying f(u) ~ f(z) = (u- z)(u - Zi{z)) • • • (u -

Zn_1(z))gz(u).

From (4.5.2), we get (Ag)(z) = a1(z)g(z)+a2(z)g(Z1(z))+---+an(z)g(Zn-i(z)), Set g = e,z,z2,-

V

g£R(D),z€A. (4.5.3)

• • ,zn~1 respectively and denote h\ = Ae, h2 = Az, • • • , hn = Az7 1 - 1

then cti(z) + zoti(z) +

a2(z) -\ Zi(z)a2(z)

+

H

an(z) = hi(z)

+ Zn-i(z)an(z)

= h2(z) zeA

{ zn-lai{z)

+ Z^-1(z)a2(z)

+ • • • + ZZll(z)an(z)

= hn(z)

Since the coefficient determinant J(z) is a Vandermonde determinant,

J(z) =

1 z Zn-X

1 Zn-\

1 Z\ Zn-l

.. .

^0.

Zr

By Cramer's rule, a\(z), • • • , an(z) are analytic in A. Since Ag is analytic, (4.5.3) determines the vector Ag and describes AeA'(Mf). Corollary 4.5.12 J / M ; e B i ( f i ) , then A'(Mf)

= {Mg : gGR(D)}.

Unitary Invariant

and Similarity

Invariant

of Operators

107

Let n>2 and w be the n-th root of 1, i.e., weC, wn = 1. Let A n denote the Vandermonde determinant of order n: 1 1

1

1 w

1 wn-1

w

n-l

w(n-1]

•••

For l
if and only if for each

n i=l n

where cn(z) = £ ^ ( 2 Proof

A»J'57^T)

and {hj}nz=1

are n functions in R(D).

For f{z) = zn,M*neBn(D), A = {zeD : z + 0}. If 1 z, z0eA, zn -Z% = (Z- Z0)(Z - wz0) •••(z-wn zQ),

then Zi{z0) = wl

L

z0

Let AeA'{Mzn), denote hk = Azk~1

(fc = l , 2 , - - - ,n).

Using Theorem 4.5.11, computations indicate that

Anz

2

^=1

A

L n

On the other hand, for arbitrary hi, hi, • • • , hn€.R(D),

^ 3=1

set

••'2J-

108

Structure

of Hilbert

Space

Operators

For gGR(D), formally, set n

i=l

{Claim} A is bounded. For each integer m, let m=m(modn)

and 0 < m < n — 1. Assume that

oo

g{z) = J2

9mZm

'

m=0

then (Ag)(z)=Zai(z)g(wi^z) n

n

n

oo

. .

oo

1 £ iEKE = l j=\ A„M^)( mE = 0 s^^'- ^*")] n

s r E [ E ffn(E Afc-u/"*-1 V n _ J ' + 1 ) ] i=l m=0 oo

j= l n

n

= £ E
i=l j = l n n

E 5m[E(E A y ^ C - ^ ^ z " - ^ 1 ) ] .

= £

TTI=O

i=i

*=i

It is easy to see that

gAy^^^l i=i

An 0

m=j - 1 m ^ ' - l .

Therefore, (Aff)(z) = E (Qknz^hiz)

+ gkn+1zknh2(z)

k=0 n

oo

= E(E i = 0 fc=0

Qkn+i-lZ^hi).

+ ••• +

gkn+n^zknhn)

Unitary Invariant

and Similarity

Invariant

of Operators

109

For each i, l
£

gkn+i-iz

hi\\a

k=0 < M\\hi\\2\\

£ fc=0 00

9kn+i-lZ fcn||2

a

fc=0

= Af ||M 2 E |f^±^i| 2 |%±i=i|2 fc=0


Thus ||j4<7||2
(2) ai(z) = ^

+ 1 + E «2fc+i22fc+1 andai(z)ai(-z)

=

a2(z)a2(-z);

= £ bnzn, h2(z) = £ cnzn,

then one of

fc=0

(Hi) If hi,h2&R(D),hi(z)

00

00

n=0

n=0

the following two must be true: (1) zhi(z) = h2(z) and P = I or 0; (2) b0 + ci = \,b2k = -c2k+i (k>l) and hi(z)h2(-z) = hi(-z)h2(z).

110

Structure of Hilbert Space

Proof (i)=>(ii). -z (z ± 0). Since P 2 = P , a\(z)g{z)

Operators

If 0:1(2;) and 0:2(2) correspond to P , then ^1(2:) =

+ ai(z)a2(z)g(~z)

= <xi(z)g(z) + a2(z)g(-z)

+ a2{z)a1(~z)g(-z)

+

a2(z)a2(-z)g(z)

(z ^ 0)

for each g£R(D). When g = e, a\{z) + ax(z)a2{z)

+ a2(z)cti(-z)

+ a2(z)a2(-z)

= ax(z) + a2(z).

- a2(z)ai(-z)

+ a2(z)a2(-z)

= 0:1(2:) -

When g = z, a\(z) - ax(z)a2(z)

a2(z).

Simple computations show that a21(z) + a2(z)a2(-z)

= a1(z)

(4.5.4)

[ai(z) + ai(-z)]a2(z)

= a2(z)

(4.5.5)

Since a i and a2 are analytic functions, a2(z)=0 or 0:1(2:) + a\{—z)=l. If a2(z)=0, then ai(z)=l or 0 and P = I or P = 0. If ai(z) + a i ( - 2 ) = l , if follows from Proposition 4.5.13 that 0:1(2:) is analytic in -D\{0} with a pole of order 1 at z = 0. Therefore, 0:1(2:) can be expressed as 0:1(2:) = ^

+ \ + E a2k+1z2k+K

By (4.5.4),

fc=0

a2(z)a2(-z)

= ai(z)[l - 0:1(2:)] OO

fc=0

= ai(z)«i(-z).

Unitary Invariant

and Similarity

Invariant

of Operators

111

(ii)=Ki). {P2g){z) = a\(z)g{z)

+ a2{z)a2{-z)g{z)

= a\{z)g(z) + a1(z)a1(-z)g(z) = cti(z)g(z)[ai(z) = ai(z)g(z)

+ a2{z)g(-z)[a1(z) +

+

ax{-z)}

a2(z)g(-z)

+ <*i(-2)] + (*2{z)g{-z)

+ a2(z)g(~z)

= (Pg)(z),

g€R(D).

Similar computation shows that (ii)<£=>(iii). Example 4.5.15

Operator P is defined by

{Pg)(z) = (-+

sinz)g{z) -{-

+ sinz)g(-z),

z ^

0,geR(D).

and P2 = P.

Then PeA'{Mz2)

In the following we will discuss the commutants of the multiplication operators on R(D) with the symbol of the form f(z) — znh(z). Proposition 4.5.16 Let f£R(D),f'(0) ^ 0,T£A'(Mf), then T = Mv for some (psR(D) if and only if T admits a lower triangular matrix representation with respect to the ONB {en}^LoP r o o f If T&A'(Mf) admits a lower triangular matrix representation T = (tijhj^ij

=0(j

>i).

Let f{z) = J2 Uzn with fx ^ 0. Compare the (2.1) entries of T M / =

MfT,

n=0

ft

A> _ f + ^° Pi Pi

Since f\ ^ 0, t n = t22. Compare the (3.2) entries, we get t22 = £3. In general, tu = t22 = • • • = CQ. Compare the (3.1) entries, we get / / P° -r J. JiHz— f t @° —- ]2tt\x— * ^° +_L f Jit +2i—. &1 i\Hi-zPi Pi Pi P2 Since £33 = *n, £32 §7 =t21@±. Thus t32 = cx@±, where cx = jf^*2i- Suppose that £(+1,; = ci^f^

(l
Pfc-2 . t . Pk-2 j. . Pk~2 . t . Pk-1 t . Jl£fc+l,fc"3 r /2tfc+l,fe+l—;5— = 72lfc-l,fc-l—5 1" Jltk,k-1—5—• Pfe-l Pk Pk Pk

112

Structure of Hilbert Space

Operators

It follows from tk+i,k+i = *fc-i,fc-i that tk+l,k-Z

0k-2 Pk-X

.

Pk-X

Pk-2 Pk-X

- t f c , k - l — 5 — = C\Pk Pk-X

— Pk

and tk+i,k = c i ^ r 1 - By the mathematical deduction, tk+i,k = c i ^ i p 1 f° r all k>l. Similarly, by the same arguments we can prove that if tk x = Ck-x S2—, '

t h e n tk+i,i+i

= Cfc_i-

r, i = 0,l,2,

0

Co Cl^

Since
Pk — 1

Therefore,

Co

by Proposition 4.5.3, y> = ]T c„z™ and T = Mv. n=0

Conversely, if T = Mv, by Proposition 4.5.3, T admits a lower triangular matrix representation with respect to the ONB {e n }^L 0 . Lemma 4.5.17[Deddens, J.A. and Wong, T.K. (1973), Lemma 2] Let N be a nil-potent on H, X0 = A + JV,0 ^ AeC. If B,A0,Ai,-••££(?{) satisfying ||,4 fc ||<M and AkX0 = X0Ak-i + B (k = 1,2, • • •), then A0 = Ai = A2 = • • • . L e m m a 4.5.18 Let T£A'(Mz~)nA'{Mf), f&R(D), f = zrg, l
G21 G22

0 and

M,T

=

0 Wx 0

0

0 W20

with respect to the ONB {efc}j?L0, where Gij,Wk are rxr matrices. Wk is invertible and since g(0) 7^ 0, Gu is also invertible (i,j, k = 1, 2, • • •). Since TMZ- = MZ~T,

Unitary Invariant

and Similarity

Invariant

'Tn

of Operators

113

0

T21 T22

with respect to the ONB {ek}kX3=o, where Tij is nxn matrix (i, j = 1,2, • • •). {Case 1 } If r is a divisor of n, n = pr, r > 1 and s = r. Suppose that

nv

'Vft Vi12fc ^ J j 2 - *£

Tkk

Vk Vk • • • Vk L- pi p2 PP

where Vk- is a r x r matrix (k,i,j = 1,2,- • •). Denote m = (k — l)p,k = 1,2, • • • . It follows from TMzrg = MzrgT that TkkFkk = FkkTkk and Fkk equals to 0 0

Wm+l^m+l.m+l Wm+2Gm+2,m+l

0

W/m+2Gm+2,m+2

. *^m+p— l ^ m + p — l , m + l **m+p— l^m+p—

l,m+2 ' * ' "m-l-p—l^m-f-p—l,m-f-p—1 " m

Compare the (1, p — 1) entries of TkkFkk = FkkTkk, we get * / l p ^ m + r - l G m - ) - p _ i i m - ( - p _ l = 0.

Since Wm+r-i and G m + P _ i ) m + P _ i are invertible, V^ = 0. Similarly, Vjj — Oif j > i,k = 1,2, ••• . Thus, Tfcfc admits a block lower triangular matrix representation and Vu

T=

0 ^22

where Vkk is a r x r matrix. By the arguments used in the proof of Proposition 4.5.18, T&A'{MZT). {Case 2 } If r is not a divisor of n. Find positive integers p and q such that qr — pn = s. Since TMzqrgq = MzqrgqT,TM2.+»tJ1 = Mzs+ni,gqT. Because TM zP » = M z P -T,

114

Structure

of Hilbert Space

Operators

TMzsgq = MzsgqT. Note that s is a divisor of n and g9(0) ^ 0. Repeating the proof of Case 1, we get T£A'(Mzs) and complete the proof. Using Lemma 4.5.17 and Lemma 4.5.18, we get the following proposition. Proposition 4.5.19 Let f(z) = znh{z)&R(D),h(z) ^ 0 for z&D and n > l . Then A'(Mf) = A'{Mzn)C\A'{Mh) = A'(MZ*), where s = (n,ni,ri2,Proof

••),/i(.z) = ao + a\znx + a,2Zn2 -\

Assume that f(z) = hnzn + hn+izn+1

H

.Set 0

Pnk — n

fink-

Bk

.

n+1

Pnk-

and hnk hnk+1

Hk

- -

ftnfc-1 hnk

.hnk+n-l

'

fonfc-(n-l)

•• • /lnfc-(n-2)

hnk+n-2

••"

/ijifc

where hi = 0 if i < n. Then 0

Mf =

i=2i

0 0

-?31 -F32 0

where Fk+iik = B^HiB^ Since h(z) ^ 0 for z£D, n/c—1

ker(M})k = \ / {e,}i=l

Suppose that TeA'(Mf).

It follows from T*M}

nk—1

T*( V {ei})cker(Mf)k i=0

=

M*fT* that

nfc—1

-

V {ei} i=0

an

^ hence T* admits a block upper

Unitary Invariant

and Similarity

Invariant

of Operators

115

triangular matrix representation with respect to the ONB { e j } ^ 0 , i.e. Tn

0

T21 T22

Compare the (2, 1) entries of TMf = MfT, we get F2xTu = T22F2i. It follows from F2\ = B^H^x

and B^lHiBxTx

HiBiTuBi

= TnB^ExBx

= B2T22B2

that

Hi.

Compare the (i,i — 1) entries, we get

i.e.,

TuBi

HiBi-i

= Bi

H\Bi-.\Ti-\,i-\.

Thus HiBi^iTi-iti^iBi_1

= BiTuB~

Bi-iTi-i^-iBi^

= BiTaB~

Hi.

By Lemma 4.5.17, ,

i.e.

Bi

Bi-\Ti-iti-i

= TaBt

-Bj_i.

Thus Ti+1,i+iWi where Wj = B^Bi. that

= WiTH,

Similarly, by mathematical deduction we can prove

Bk+i-iTk+i+i,i+iBi+1

= Bk+iTk+iiiBi

,

i.e.,

Tk+i+iti+iWi

= Wk+iTk+i,i,k

= 0,1,2,-••

This implies that TMZ-=M^T.

,t = 1,2,3, ••

116

Structure of Hilbert Space

Operators

Since TMz~Mh =

Mz~MhT,

MznTMh =

Mz„MhT.

This means that zn(TMhg)(z)

=

zn(MhTg)(z)

for g£R{D) and z&D. Thus TMh = MhT and T&A'(Mzn)nA'{Mh). Conversely, A'(Mzn)nA'(Mh)cA'(Mf)

is obvious. Thus,

A'(Mf)=A'(Mzn)nA'(Mh). II TeA'(Mf)

= A'(Mzn)nA'(Mh). h(z) = a0 + alZni

Suppose that + a2zn2 + ••• ,

where at ^ 0(z = 0,1, 2, • • •). Then h-h(0) = znklhlt where nj = for ki,riGN and n < n and /ix = z r i g i . Since TeA'(Mh),

Since

TMznklMhl

=Mx^lMhlT.

MznklTMhl

=MznklMhlT,

T&A'(Mzn),

i.e.,

for geR(D).

Thus TM fcl =

MhlT.

By Lemma 4.5.18, T e , 4 ' ( M z n ) , where si = (n,ri) = (n,ni). Similarly, if n 2 = n/e2 + r2 for /c2, r 2 e N and r 2 < n, then a 2 z " 2 + a £ 3 + --- = z nfc2 /i 2 ,

nki+ri

Unitary Invariant

and Similarity

Invariant

of Operators

117

where /12 = ^Qi- By the same argument Te^4'(M z » 2 ) for si = (n, ni,ri2). In general, T€LA'{MZ,), where s = (n,ni,ri2, • • •). Conversely, if r e ^ ' ( M z . ) , then it is obvious that TeA'(Mf). Thus ^ ' ( M / ) = >4'(M z n)a4'(M fc ) = A'(Mz.),a

= (n,n 1 ,n 2 , •• •)•

Corollary 4.5.20 Suppose t/iat feR(D),f(z) = znh(z),h{z) zG-D and for each k > 0, /i(z) is not a function of zk, then A'(Mf)

=£ 0 /or

= A'{MZ) = {M 9 : g&R(D)}.

In the following, we will discuss the strong irreducibility of multiplication operator M/ on R(D). Theorem 4.5.21 Given fGR(D), the following are equivalent. (i) M ; e B i ( n ) ; (ii) A'(Mf) = {Mg : g£R(D)}; (Hi) A'(Mf) is commutative; (iv) If M*f € S„(fii), then for each AeA'(Mf), (Az)(z) = zhxiz), {Az2){z) = z^z),

• • • , (Azn'l)(z)

=

z^h^z),

where hi = Ae\; (v) Mf£{SI). Proof (i)=Kii) Corollary 4.5.12. (ii)=^(iii). Obvious. (iii)=>(iv). If there an operator AeA'(Mf) such that (Azk)(z0) =£ Zohi(zo) for some k, 1 < k
= Z^ZQ)

£ (Azk)(z0)

=

(AMzke)(z0).

Thus MzkA ^ AMzk. But A and M 2 t are in A'(Mf). A contradiction. (iv)=^(v). For z€A, since ai(z) + a 2 (z) + • • • + an(z) = h\{z) and

118

Structure

of Hilbert Space

Operators

hk+i = zkh\ for l
- z)an{z)

=0

+ (ZUz) - z*)a3(z) + ••• + {Z*_x{z) - z*)an(z)

=0

(Z?-\z) - z"-i)a2(z) + (Zr\z) zn~l)a3(z) l + ... + (ZZll(z)-z"- )an(z)=0 Computations indicate that the coefficient determinant is still a Vandermonde determinant:

V =

Zx-z Zl - z2 7TI — 1 _

z?~l -

(-1)

Z2-z Z22-z*

n —1

yn—l

1 Z\

1 Z2

Zn-\ — zLi - • ~n—1

••• 1 • •• Zn-\

yn-i

1 z

zl zl ... zu z*

n-1

yn—l l

Z

yn—l 1 "•

L

n-1

_

^0.

yn—l -n—1 n-1 Z

Z

an(z) = 0 and A'{Mf) = {Mg : Therefore a2(z) = a${z) = • • geR(D)}. Since A'(Mf) does not contain nontrivial idempotent, Mf€(SI). (v)=^(i). Suppose that the minimal index of MJ is n and n > 2 . By Theorem 4.5.11, Ag = ot\{z)g{z) H + an(z)g(Zn-i(z)) for A£A'(Mf), geR(D) and zeA. Since M*eA'{M*f),M*ku - uku and M*zkzx{u) = Zi(u)fc21(u) for uGA. By Theorem 4.4.3, the spectrum of M*\ker(M}-f(u))

is connected. Thus u = Z\{u). This contradicts z ^ ^1(2) when zeA. Therefore n = 1 and M;&Bi(fi). Proposition 4.5.22 Mz™e(S7) j / and on/?/ i / n = 1. Proof If n>2, define an operator P by

Pf = \\J{?) + / M + • • • + /K" 1 *)]

/e/W

Unitary Invariant

and Similarity

By Proposition 4.5.13, PeA'(Mzn).

Invariant

119

For each i, l
= £[/(w
Pf(w'z)

of Operators

+ ••• +

fi^-^z)}

= £[/(*)+ /("*) + "- + /("n-1*)]Thus p2f

tt"n-lz))\

= £ [ " ( / ( * ) + / ( " * ) + ••• + = ![/(*) + / H

+ • • • + /(a;"" 1 *)] = P / .

It is obvious that P is nontrivial and

Mzn$.{SI).

P r o p o s i t i o n 4.5.23 Let feR(D) with f(z) = znh(zm), h(z) ± 0, zeD and n>\. If for each k > m, h(zm) is not a function of zk, then Mf£(SI) if and only if (n, m) = 1. Proof By proposition 4.5.19 A'(Mf) = A'(Mzs), where s = (m,n). By Proposition 4.5.22, A'{Mzs) contains no nontrivial idempotent if and only if s = 1. Thus Mfe(SI) if and only if s = 1. Proposition 4.5.23 requires n>l. One may asks the question: Is the conclusion of Proposition 4.5.23 true when n = 0? In fact the answer in general is negative unless / is an integral function. oo

E x a m p l e 4.5.24 Let f(z) = 2z - £ ^zn,

then

Mfg(SI).

n=l

Proof It is easy to see that f(z) is analytic in 2D = {zEC : \z\ < 2}. Thus f&R(D). Define an operator P as follows: (Pg)(z) = \g{z) + \g{2 + - A _ ) ,

g£R(D).

Then

Simple computations show that P2 = P and f(z) = / ( 2 + j r j ) . Thus PMfg(z)

= \f{z)g{z)

+ 1/(2 + ^)g(2

= lf(z)g(z)

+

= MfPg(z),

+ ^ )

y(z)g(2+^) for all

geR{D).

120

Structure of Hilbert Space Operators

Set g(z) = 10 + f(z), then g satisfies the requirement of Proposition 4.5.23 with n = 0. But Mg£(SI). Proposition 4.5.25 A'{Mz2)/radA'{Mz2) is noncommutative. Proof By Theorem 4.5.21 A'(MZ2) is not commutative. Assume that R is a left ideal of A'(MZ2). Set £1 = {Ae : AeR},R2 = {Az : A&R}. {Claim(i)} Ri,R2 are ideals of R(D), and if hi£Ri,h2€R2, then hl{z) hl{ z)

/n(-*)efli,

~ ' eR1,h2(-z)eR2

and h2{z) -

h2(-z)

eR2.

Let A&R and hi = Ae. Denote h2 = Az£R2. For arbitrary denote B^A'(MZ2) determined by / i , / 2 - For g£R(D),

fi,f2&R(D),

(BA)g(z) =

r/i(z) h l ( 3 ! ) + h ' ( ^l+/ 2 ( z )'' l ( ^-^ ( - )

r

+

/ l ( z )

h2(«Hha(-.) +

h { z )

/.2(»)-h2(-»)

/ 1 ( 2 ) h t ( ^ + " ' ( ^ + / 2 ( Z ) , ' l ( j ) - ^ ( - ^ _ /1(Z)ha(j)+',8(-)+/2(Z)hi»(,>-^(-')1

+ I

22

2

Choose f2 = zfi, i.e., B + M/j, then (BA)g(z) = [ * < f ! ^

+

* ( ^

which means /i/ii€i?i and Ri is an ideal of R(D). Set A = e, / 2 = 0, i.e., (B
then

h2(z)+ ( z)

£ - ]g{z)

_|_ rfei(z) + >ti(-z) _ h2(z)+h2(-z)-i

/ ,N

which means /ii(z) + /ii(—z)Gi?j and /ii(—z)Gi?i. Set A - 0, / 2 = e, i.e., (5ff)(z) = £[
then

>»i(«)-hi(-»)

(BA)
1 hl(«)-hl(-«)

+ [—^

^('l-^C-*)

+

£

]fl(*)

fc2(')-''2(-»)

&—]$(-*),

,

J^V

,

Z

>-

Unitary Invariant

and Similarity

Invariant

of Operators

121

which means €-Ri- Similarly, we can prove that R2 has the same z properties. {Claim(ii)} For arbitrary z\, z2£D, z\ ^=Q,z2^ 0, Set Ri(zi) = (z2 - zf)R(D), R2(z2) = (z2 -

z2)R(D),

then Ri(zi) is a "maximal" ideal which has the properties: if /ii£i?i(zi) then hi(-z)eR1(z1) and hl(z)~^l(~z)eRi(zi). Similar properties hold for #2(22)- For each hxeR1(z1) and h2GR2(z2), ^ n d -^-A'C^z 2 ) with

(A9)(z) = (*£> + *£>),(*) + (*£> - ^ ) 5 ( - z ) , z ^ 0. Let R(z\, Z2) be the set of all such operator A. Simple computation shows that R(zi,Z2) is an ideal of A'(MZ2). If there is an ideal S of A'(MZ2) such that R(zi,Z2)CS, then Si :— {Ae : AGS}DRI(Z\) is an ideal and by (i) and the "maximality" of Ri(zi), Si = R\{z{). Similarly, S2 := {Az : AGS} = i?2(^2)- Hence S = i?(zi,Z2), i-e-, R{zi,z2) is a maximal ideal of A'(MZ2). {Claim(iii)} radA'(Mz2) = {0}. Let i2 = n{i2(zi, z2) : 21, Z2G£>, ZX ^ 0, z 2 ^ 0}, then .Ri = {Ae : AeR} = n{Ri(zi) = {heR{D)

: h(z) = h(~z) = 0,zeD}

R2 = {Az : AGR} = n{R2(z2) = {heR(D)

: zi&D,zx

^ 0} = {0},

: z2e£>,22 ^ 0}

: h(z) = h(-z) = 0,zeD}

= {0}.

Thus R = {0} and radA'(Mz2) = {0}. Since A'(MZ2) = {0} is noncommutative, A'(Mz2)/radA'(Mz2) is noncommutative. In the last part of this section we will consider in invariant subspace of Mz on R(D). Proposition 4.5.26 Let feR(D). (i) IfBa(z) = f^,a£D, then MfoBa~Mf; (ii) Mf~Mz if and only if f(z) = \fE=£, |A| = 1 and aeD.

122

Structure

of Hilbert Space

Operators

Proof (i) It is obvious that foBa€R(D). Define an operator Sa as follows: Saf = foBa, /G-R(D). Computation indicates that there are positive numbers Mi, M2,M$ and M 4 such that / \foBa\2dm<Mi

I \f\2dm,

JD

[ \(foBa)'\2dm<M2

JD

[

JD

\f'\2dm

JD

and f \(foBa)"\2dm<M3 JD

[ |/'| 2 dm + M 4 f JD

\f"\2dm.

JD

Thus | | / o 5 a | | < M | | / | | for some number M and all feR(D), bounded. Note that for each g£R(D), SaMfg

= Sa(fg) =

i.e., Sa is

f(Ba)g(Ba)

and MfoBaSag

= MfoBag(Ba)

=

f(Ba)g(Ba).

Thus SaMf = Mf0BaSa. Since Ba is an invertible analytic function, Sa is one to one and onto. Thus Sa is invertible and Mf0sa = SaMfS*1. (ii) If f(z) = \f5£, then Mf~Mz by (i). On the other hand, if Mf~Mz, by Theorem 3.5.6, f(D) = a(Mf) = a(Mz) = D. Thus f(D)cD. Since / is continuous on D, by maximal module theorem f(D)cD. For arbitrary XeD, since \&a(Mz)\ae(Mz) = f(D)\f(C), Xef(D) and Dcf(D). Therefore / maps D onto D. Note that for each AeD, nul(\ - M}) = nul{\ - Mz*) = 1, this implies that f(z) — A has only one zero. Thus / maps D one to one onto D. It must be a Mobius transformation up to a coefficient of module one. Lemma 4.5.27 Assume that / i , /2, • • • , fn are n functions in R(D) without common zeroes in D. Then there are functions gi,g2, • • • ,gn in R{D) n

such that ]T] fkgk = e. fc=i

Proof Denote J = {gifi + g2f2 + • • • + 9nfn • GR(D)}. It is obvious that J is an ideal of R(D). If J is a nontrivial ideal, it must be contained in a maximal ideal. But this is impossible, because

Unitary Invariant

and Similarity

Invariant

of Operators

123

the maximal ideal space of R(D) is D and / i , /b, • • • , fn have no common n

zeroes in D. Thus eej

and ^ /£<% = e for some 51,52, • • • ,5nSi?(J5). fc=i

Theorem 4.5.28 (i) Subspace M with finite common zeroes {zi, ^2, • • • , zn} in D, multiplicity included, is an invariant subspace of Mz if and only if M =

(z-z1)---{z-zn)R{D);

(ii) If M is an invariant subspace in (i), then the projection onto M. is PM = Mx{M^Mx)-lM^,

where X=f[

f5^i=l

Proof (i) If M = (z — z\) • • • (z — z„)R(D), it is obvious M is an invariant subspace of Mz with common zeroes {xj}f=1 in D. Conversely, if MeLatMz with common zeroes {z,}" = 1 . Denote N := {geR(D)

: (z -

Zl)

• • • (z -

zn)g£M}.

Then it is not difficult to see that N is an invariant subspace of Mz. For each weD, since {zi}?=1cD, there is a function fw£N such that fw(w) ^ 0. Since fw is continuous in D, choose a neighborhood U(w,e) such that fw does not equal zero in U{w,e). Thus we can find a finite open cover U(wi,ei),--,U(wk,ek) of D and functions fWl,fW2,--,fwk in N such that fWi has no zero in U(iVi,Ei) (l
Y^fwigi

= e.

Note that since polynomials are dense in R(D), each invariant subspace of Mz in fact is an ideal. Thus eeN and N = R(D). Thus {z-z1)---{z-zn)R{D)dM. Since M c ( z - z i ) • • • (z-zn)R(D) is obvious, M = ( z - z i ) • • • (z-2„)i?(D). (ii) Since M* is a Fredholm and kerMx = {0},M*M x is invertible. It is obvious that Mx{MxMx)~lM* is a self-adjoint idempotent. Thus it is an orthogonal projection. By Proposition 4.5.7, M* is a Cowen-Douglas operator of index n, and ranM* = R(D). Thus ran\Mx{MlMx)-x

Mx] = ranMx = xR(D) = M,

124

Structure of Hilbert Space

Operators

i.e., PM is the projection onto M. It is well-known that in Hardy space H2 Mz similar to the restriction of it on any nontrivial invariant subspace M. The following result indicates that this statement is valid in Sobolev disk algebra if and only if M has only finitely many common zeroes in D. Proposition 4.5.29

Let M&LatMz, then

MZ~MZ\M

if and only if

M=(z-Zl)---(z-zn)R(D), where Proof

{zi}f=1cD. Assume that M = (z — z\) • • • (z — zn)R(D). P=

Denote

{z-zi)---(z-z„).

Define Tp : R(D)—>M by Tpf = pf,feR(D). maps R(D) one to one onto M. Since T;^Mz\MTpf

It is easy to see that Tp

= zf = Mzf,

MZ =

feR(D),

T~1MZ\MTP.

On the other hand, if there is an W : R(D)—>M such that MZ =

W-1MZ\MW,

then WMZ = MZ\MW. Denote h = We. Computations indicate that Wzn — znh. Thus Wp = ph = Mhp for each polynomial p. By Proposition 4.5.2, Wf = Mhf for all feR(D). This implies that W = M/, and Mh has a closed range. By Theorem 4.5.6, O0/i(C). Since Mh is essentially normal, h has finitely many zeroes in D. By Theorem 4.5.28, M = (z — z\) • • • (z — z„)R(D) for

Theorem 4.5.28 and Proposition 4.5.29 describe the structure of invariant subspaces of Mz with finitely many common zeroes in D. The following example is an invariant subspace of infinitely many common zeroes. oo

E x a m p l e 4.5.30 Then

feR{D).

Let f(z)

= ( J ] f$£)(z

- l ) 5 , where an = 1 - 4j-.

Unitary Invariant

Proof

and Similarity

Invariant

of Operators

125

By a result in [Gardner, B.J. (1989)], f(z) is continuous in D. oo

Let B(z)= n bak, where bak = f ^ . fc=i

{Claim}

B'(z) = £ (^^

ft

bak(z)).

Let G = {z£C : \z\ < r < 1}, choose r so that zeG. Assume that N

oo

an > r when n > N. B(z) = ( fj 6„J( f] 6 0 J. Thus fc=l fc=N+l JV

oo

B'(z) = (Ubak)'( fc=l

N

oo

I] &«*) + ( II *«J( n fc=JV+l

N

fe=l

&«J'

k=N+l

N

= ( I l * . J ' B ^ ) + (ni.JBj»W. fc=l

wher

oo

e BN(z)=

fc=l

ft

bak.

k=N+l

Note that 5jv(z) 7^ 0, z£G. Thus OO

In(Bjv(z))= ]T

M f ^ ) -

k=N+l

But ^ ( 2 )

5

iv(^)

£

K - 2 )( 1 ~ akz)

k£^+l

converges uniformly in G. Thus k=N+l

v

~

akZ

'

n^k,n>N+l

Note that

|S'(z)(z - 1)5| = I £

ia

fe=l

\:ll{k7f

V

n *«„(*)! n^fc

<16E(l-afc)ll^fll2 fe=i

< 64 £ (1 - at) < 00, fc=i 5

thus B'(z)(z — l ) is bounded on Z). It is easy to see that B(z)(z — l ) 4 is bounded on D. Therefore (B(z)(z — l ) 5 ) ' is bounded.

126

Structure

of Hilbert Space

Operators

Similarly, (B(z)(z - l ) 5 ) " is bounded on D, these imply f(z)&R(D). Denote M{an} := {g£R(D) : g(an) — 0,n = 1,2,•••}. It is obvious that M{an} is an invariant subspace of Mz with infinitely many common zeroes and f£M{an}. For a given set M, [M] denotes the minimal invariant subspace of Mz containing M. If M ^ {0} is an invariant subspace of Mz on Hardy space H2, Beurlig Theorem asserts that dim(MQzM) = 1 and [MQzM] — M. Comparing to this, the invariant subspaces of Mz on Bergman space L\ are more complicated. [Apostal, C , Bercobici, H., Foias, C. and Pearcy, C. (1985)] proved that if n is any positive integer or +oo, there is an invariant subspace M in L\ such that dim(MQzM)

— n.

For Sobolev disk algebra R(D), we have the following result. Proposition 4.5.31 Let M ^ {0} is an invariant subspace of Mz in R{D), then dim(MQzM) = 1. Proof [Richter, S. (1987)] studied a Banach space B of analytic functions, which satisfies the following conditions: (i) For each XQD, the point evaluation functional is continuous; (ii) If feB, then zfeB; (iii) If f£B and /(A) = 0, then f = (z - X)g for some g&B. S. Richter also proved that if the above Banach space B is an algebra and M is its closed ideal, then dim(MQzM) = 1. It is obvious that Sobolev disk algebra and each nonzero invariant subspace M of Mz satisfy these conditions. Thus dim(MezM) = 1.

4.6

The Operator Weighted Shift

Let {WAJ^LI

De a

sequence of uniformly bounded operator on C". An

oo

operator S in £( 0 C n ) is called a unilateral operator weighted shift with fc=o weighted sequence {Wk}™=1, denoted by 5'~{W/fc}^=1, if S(x0,xi,---

,xk,---)

= (O.WiZo,---

,Wk+1xk,---)

Unitary Invariant

and Similarity

Invariant

+00

for all (xk)eH+

+00

= © C". Similarly, an operator S on H = fc=0

is called

127

of Operators

©

Cn

k~—00

a bilateral operator weighted shift with weighted sequence

W}fcT-oo.

de

™ t e d b y S~{Wb}£T_oo. if

for all (xk)£H In general, unilateral and bilateral operator weighted shifts are both called operator weighted shift, denoted by S~{Wk}, and n is called the multiplicity of S. When n = 1, S is the scalar-valued weighted shift, from which the operator weighted shift is naturally generalized. We must point out here that it is not just a formal generalization. For example, using operator weighted shift, [Pearcy, C. and Petrovic, S. (1994)] proved that an n-normal operator is power bounded if and only if it is similar to a contraction operator. In this section, we will discuss injective operator weighted shift, that is, each Wk is invertible. Given an operator weighted shift S~{Wk}, since S and el9S are unitarily equivalent for each #£[0,27r] [Lambert, A (1971)], cr(S),ae(S) and oy(S) have circular symmetry. It is easily seen that r(S)=

Km k—>oo

n(S)=

(sup\\Wi+k...Wi+1\\)K i

l\m(mf\\Wi+k---Wi+1\\)l, k—>oo 1

where r(S) is the spectral radius of S and ri(S)=

lim

(m(Sk))i,

fc—>00

m(S) := inf{\\Sx\\

: ||z|| = 1}

is the lower bound of S. Since dirnker(S — \)
= {A : |A|
Structure

128

of Hilbert Space

Operators

triangular matrix representation with respect to this ONB. By the basic matrix theory, we have the following proposition. P r o p o s i t i o n 4.6.1 Every operator weighted shift S~{Wk} is unitarily equivalent to an upper triangular operator weighted shift S~{Wk}T h e o r e m 4.6.2 Let S~{Wk}'£L1 be a unilateral operator weighted shift, if cre(S) is disconnected then S^(SI). To prove this theorem we need the following lemmas. Lemma 4.6.3[Herrero, D.A. (1990)] an(AB)

Let A,BG£(H),

= <JI{TAB) = ai{A) -

L e m m a 4.6.4 Let A££(Hi),

then

ar{B).

B&£(H2) and

dirnHi = dimH-zIf oi(A)C\ar{B) = 0, then G := {YGranTAB '• ||V"||<1} is closed in weak operator topology (WOT). Proof Without loss of generality, we assume that H = Hi = Hi. Let {y a } a £ A be a net in G and Y = (WOT) — l i m i ^ . Then there exists a

Xa££(H) such that Ya = TAB{XO). there exists TGC(C(H)), such that

By Lemma 4.6.3, 0 ^ TTAB = I- Therefore,

O-I(JAB)-

Thus

||x a || = ||T(yQ)||<||T||||rQ||<||T|| for each a. Since each bounded closed set in £(H) is compact in weak operator topology [Conway, J.B. (1990)], we can find an operator XG£(H) such that X = WOT — limX a . Therefore, for arbitrary re, y£H, a

lim < (TABXO)X,

y > = lim < AXax, y > — lim < XaBx, = < AXx,

y > - < XBx,

Thus Y = WOT - lim TABXa = TABX.

y>

y >=< (TABX)X,

y > .

Clearly ||F||<1, therefore Y<=G.

L e m m a 4.6.5 Let ^4~{^4fc}j£Li and B~{Bk)V=i ^e unilateral operator weighted shifts with multiplicities n andm respectively. If ai(A)(lo-r(B) = 0

Unitary Invariant

and Similarity

Invariant

of Operators

129

and if C is an operator of the form 0

0"

cj 0 C2 0

C

0

c3 0

cm cm cm Cm

,

c™ c n c n then CEranTA,BProof Let M = sup{||c fc ||}, k

0 ci 0 c2 0

0 00 0 0

Fk =

and Ek = cfc 0

Cfc 0

0 0

0 0

Then \\Fk\\<M,Fk

= £ Ej and C = WOT - limjFfc. By Lemma 4.6.4 we

need only to prove that each Ek&anTA,B- Set Xk-i Xk-2

= A^ Ck, = (AkAk-i)~1CkBk-i,

X0

(Ak- • •A2)~1ckBk-v (Ak---A1)-1ckBk-V--Bi.

• -B2,

Define X0

X

cr Xk-i

Cr' Cr'

r~in /~m /"m r~*n

A straightforward computation shows that AX — XB = Ek-

130

Structure of Hilbert Space

Operators

Let Hk be the k-th subspace in the orthogonal direct sum H+ = 0

C".

k=0

Then for each unilateral operator weighted shift S~{Wk}kxLi, ^k can be regarded as an invertible operator from Hk-i to Hk- In what follows we will always identify x&Hk with (0, • • • ,0,x,0, • • • )GH+, the k-th of which is x. L e m m a 4.6.6 There exists an ONB {ef'}f = 1 of Hk {k = 0,1,2,---), such that the weighted Wk of S is of the form (fc)

Jfe-i)

wi Wk =

wkk)

(4.6.1)

Sk~i)

„(*) and di>di+i (i = 1, 2, • • • , n — 1), where di=]im(f[\WiU)\)i.

(4.6.2)

3=1

P r o o f For convenience denote Mk = WkWk-i- • -W\ and MQ = / . Choose an ONB {e^0), ef\ ••• , e{°}} of H0- Set dx =max{TInr||M f c ef ) ||* : \
Without loss of generality, we may assume t h a t d i = lim ||M f cM\ ef J ||s. Set k—>-oo

=(°) l|Mfce<°>|| Then (fc-i)

Wke\'

(0),

||M fc el'

„(*) . =

(0)|

\\Mk-A

,mjk)

= u)K 'e

(4.6.3)

and S i := {e[ f c ) }^ 0 is ONB of H+. Let P[k) be the projection from Hk onto the subspace [\j{e{k)}]x and Pj (0) = / . By (4.6.3), P^WkPf-V

= P[k)Wk

(4.6.4)

Unitary Invariant

and Similarity

Invariant

of Operators

131

Suppose that we have found an orthogonal set Bj of H+,

Bj = {ef] : l<j0} and dj = '^^\\PJt\Mkef)\\i, fc—>oo

J

where P, W (l<j0) is the projection

J

J

from H+ onto (\/{e[k\ ••• , e
d\>d2>

• • • >df,

(h) PJk_\Wke?-V = WJk)ef\ (j = l,2,.--,i). (hi) Pf ]WkPf ~l) = P$k)Wk, 0 = 1,2,-.- ,i). Without loss of generality, we may assume that

(4.6.5) (4.6.6)

d- Hi+1- =l - ^ limll^Mfee^l ^ » ' W

( 0 )

fc—>oo k—*oo

Thus di>di+i. Set (k) _

M

^i

kei+i

f^efi + l l

-eftfc.

By (4.6.6), Fi

Mk&i

^

-

-

WP^M.-^w J "° M f c e ^ 1 1 . c(fc) ||if- 1 , M, k -ieS"'iir + 1

(fc)

(4-6-7)

(fc)

Besides, Si+i = {e^ : l < j < i + l , fc>0} is still an orthogonal set of H+. Let P i + { be the projection from Hk onto the subspace (V{ e i > • • • i ei+i})" 1 a n d i^+i = / . It follows from (4.6.6) and (4.6.7) that P^WkPfc1* = P$\Wk. Repeating the procedure above, we can find {rfi}?=1 and ONB {e[k)}?=1 oiHk such that d i > d 2 > • • • >dn and {e(k) : l0} is an ONB of H+. From the choices of e\ and w\ ', each wk has the form (4.6.1) and fc

~ n 5 " ( i n « W =TS"||i'i(*)iMtei0)||* =di(» = l ) 2,.-- ,n). fc—»oo

-*•-*•

fc—>oo

i=i

By Proposition 4.6.1, the following lemma can be easily proved.

132

Structure

of Hilbert Space

Lemma 4.6.7 Let S~{Wk}kLo

Operators

be an operator weighted shift, then

S\ S\2 • • •

Sin

s2 '••

Hi

!

n2

' *~*n — l , n

'•'rin

o whereH+ = W ^ ' . ^ ^ K ^ l f c l i is an injective unilateral weighted shift and S^ is a scalar-valued weighted shift (not necessarily injective). Without loss of generality, we may assume that Tii = I2. Lemma 4.6.8 Let S~{Wk} be an operator weighted shift of the form given n

in Lemma 4-6.7. Then cre(S) = \J ae(Si). Proof We only prove the lemma when n = 2, and let S =

Si S\2 0 S2

Let n be the canonical map from C{H+) onto C{H+)/K{Ji+).

Then

wi(Si)7ri(5i2) 0 7Ti(5 2 )

n(S)

where TTI : C{Hi)—+C{Hi)/)C{Hi). We need only to show that (T(TT(S)) = O-(-ITI(SI))UO-(ITI(S2)). By Gelfand-Naimark-Segal Theorem, we need only to prove that a(S)=a{S1)Ua(S2). If \€p(S),

then (S - A ) - 1 = (Aij)2x2 S\ — A 0

Sl2 S2-\

satisfying An AX2 A2i A22

Thus (S2 - A)^22 = I- Since dimker(S2 (S2 — X) has a left inverse and \£p(S2). Therefore

- A)<1, by Atkinson Theorem, Since (52 — X)A2i = 0,^21 = 0.

(Si-\)Au=Au(Si-\)=I. This implies that Aep(5i). Thus

\Gp(S1)np(S2).

Unitary Invariant

and Similarity

Invariant

133

of Operators

Conversely, if AGp(5i)np(5 2 ), set X

(51-A)-1-(51-A)-15(52-A)-1 0 (S 2 - A)- 1

Then X{S - A) = (5 - X)X
I. This implies XGp(S) and a(S)

=

Now we are in a position to prove Theorem 4.6.2. n

Proof of Theorem 4.6.2 By Lemma 4.6.8, ae{S) = \J ae(Si). Since ae(S) = an(S)c{\GC

: r!(S)<|A|
there exist e and S, 0<e < <5, such that fl := {A : e < |A| < 5} is a bounded component of PF{S). Therefore, for each Si, at least one of the following holds: (a) «5}. Denote io = max{i : cre(Si)cVg}, then by the definitions of r(S) and ri(5) and Lemma 4.6.6, ri(Si0) io, Si does not satisfy (b), thus a(Si)cUe. Note that {A : |A| = e or ^}ccr e (5). We have l < i 0 < n. io

n

Now denote Mi = © Hi and M2 = M^ = i=l

0

Hi. Then

i—io-\-l

s=

AC 0 B

M2'

where A and B are operator weighted shifts with multiplicities io and n — io respectively, and C is of the form given in Lemma 4.6.5. By Lemma 4.6.8, io


\J
By Lemma 4.6.8 and its proof ar(A) = a(B) =

\J

a(Si)cUe.

oe{A)nar(B)=$. By Lemma 4.6.5, there exists an operator X such that AX - XB = C.

Therefore

134

Structure of Hilbert Space

Operators

Set Y =

IX' 0 /

Then YSY'1 and

=

A 0 0 B

Sg(SI).

Next we will discuss when the adjoint of a unilateral operator weighted shift is a Cowen-Douglas operator. Proposition 4.6.9 Let <S~{Wfc}j£Li be a unilateral operator weighted shift. IfO€ae(S), then S* can not be a Cowen-Douglas operator. Proof If ae(S) = a(S), then S* is not a Cowen-Douglas operator. Thus we may assume that ae(S) ^
=

A 0 Mi = D. 0 B M2

A* 0 M 1 . Since a(B*) = a{B)c{\ : \\\<e} 0 B* M (see the proof of Theorem 4.6.2), B* — X is invertible for all Asft. Thus

So D* = (Y*)~1S*Y*

A;er(Zr-A)oMi. This implies that \J{ker(S* - A) : Aeft} ^ H+. Theorem 4.6.10 Let S^{Wk}^=1 Then the following are equivalent: (i) M := sup ||Wfc-1|| < oo;

be a unilateral operator weighted shift.

(ii) n ( 5 ) > 0 ; (Hi) 0£pF{S); (iv) There exists a connected open set ft containing 0 such that S*eBn{n). Proof (i)=»(ii) and (ii)=^(iii) are obvious from the definitions.

Unitary Invariant

and Similarity

Invariant

of Operators

135

(iii)=4-(iv). It is easy to see that there exists a connected open set fi containing 0 and riCpF(S). Since ap(S) = 0,5* — A is surjective and dimker(S* - A) = ind(S* - A) = indS* = n for all Aeft. Note that \/{ker(S*)k : fc>l} = H+, thus S*eBn(Q). (iv)=>(i). Since 5* is surjective, it has a right inverse T = (Tk,i)%l=0eC{H+). Thus W^Tk>k-i = / and H ^ H = Wn^W^TW for each k. By Proposition 4.6.9 and Theorem 4.6.10, the following corollary is obtained. Corollary 4.6.11 Let S~{Wk}kLi be a unilateral operator weighted shift. Then B* is a Cowen-Douglas operator if and only if sup || VK^-1 j| < +oo. fc

In what follows we consider the backward operator weighted shift S~{Wk} with the form OWi

0

0 W2 0

C" Cn

'•.

on H = 0 C n , n=l

0 and denote Ao(S) =

A'(S)\kers-

Theorem 4.6.12 Let S~{Wk} be a backward operator weighted shift, then the following are equivalent: (i) There exists AoGAo(S) such that card(a(Ao)) = m, (m
multiplicity n-i and ^2 rii = n; i=l

(Hi) S has a spectral family. Proof (i)=>(ii). For convenience, we assume that m — 2. Since A0eAo(S), there is A£A'(S) such that A\kers = A0. Let Aa = diag(A0,Ai,- • • ,Ak,---) be the diagonal part of A and let p(z) be the characteristic polynomial of AQ. Since Ak — Ek~1AoEk,p(Aa) — 0, where Ek = Wv • Wk (fc>l). Thus p(a(Aa)) = {0}. This implies that a(Ao) = {Ai,A2}. Thus there are Jordan domains S7i and 0,2 such that Aiefii,A 2 en 2 and 0[r^h = 0- Denote T* = dtli (i = 1,2). Clearly,

136

Structure

of Hilbert Space

Operators

Ad&A'(S). Thus the Riesz idempotents Pi =

2^ /

(A

" Ad)~ldX£-A'{S)

(i = 1,2).

Note that (A - Ad)'1 Thus Pi = diag(P£\

Pk] = Ek1^

= diag((X - A0)-\-

• • , E^(X

- A0)-lEk,

•••).

• • • ,P{ki] ,• • •), where

I

(A - A0)-1d\)Ek

= E^P^Ek,

(i = 1,2)

and

P f ' p f =0, P^ + P™ =/c»,(* = 0,l,2,...). Denote Mk = P ^ C " , Nk = P?)Cn, Hi = PJi, (* = 1,2). Then OO

W = ®(Mk+Afk) = H1+H2, fc=0

and 5 = S1+S2, where 5i = S\Hl, 5 2 = S| W a . Since

= ni

dimAik = dimMk-i and

dimMk = dimAfk-i — ^2 (&>!•)• Therefore Wk = W ^ - i - W ^ , where ^ e A . M f c . M f c - i ) , ^ 2 ^ / : ^ , ^ - ! ) . 00

Define the unitary operator Ui£C(Hi,

® Alt) as follows fc=0

^1 = ((j/o,0),(yi,0),--- , (yfc)0),---)i-> (y0,2/i,---

,Vk,---)-

Then U\S\Ul = Si is an operator weighted shift with weight sequence {Wjfc }£ii a n ( i multiplicity n i . Similarly, we can define a unitary operator

Unitary Invariant

and Similarity

Invariant

of Operators

137

U2£C(H2, 0 A4) such that U2S2U2 = S2 satisfying the requirements of fc=o the theorem. (ii)=>(iii). Obvious. (iii)=>(i). We still assume that m = 2. Since {Qi,Q2} is a spectral family of 5, Qi,Q2eA'(S). Let

Qf=diag{Q(i\..-,Qf,...) be the diagonal of Qi{i a spectral family of 5.

= 1,2). Clearly, {Qd ,Qd } is also Set Si = S\ „(i) (i = 1,2). Then

5 = S\+S2- Repeating the argument of (i)=>(ii), we can prove that Wk = W^+WP, where W^]eC(Mk,Mk-i), W™eC(NkMk-i) and dimMk = dimMk-i,dimAfk = dimNk-i for all k>l. Choose i±\,ii2&C such that \ni\ < |/n2|Set AQ = HIIMO+MI/SO and Afe = E^xA0Ek = fiilMk+^I^, k>l. Then

||Afc||<|H(IIQi|| + IIQ2||). Thus A:=diag(A0,Ai,---

,Ak,---)eA'{S)

and A0 = ^| fcer s,cr( J 4o) = { / i i , ^ } , i.e., card(a(Ao)) = 2. Theorem 4.6.13 Let S~{Wk}

be a backward operator weighted shift and r

let r = ma,x{card(a(A0))

: A0&Ao{S)}.

Then S~S

= 0 Si, where each r

Si^(SI)

is an operator weighted shift with multiplicity rii and £^ n» = n. i=l

Corollary 4.6.14 Let S~{Wk} be a backward operator weighted shift and A0eAo{S), then a(A0) = aAo{s)(A0). Proof Clearly, a(A0)CaAo(S)(Ao). Assume that Ad = diag(Ao,Ai,---,Ak,---), where Ak = Ek~1AoEk (k>l). By the preceding discussion, cr(Ao) = o-(Ad) and AdeA'(S). If A £ a(A0), AdB = BAd = I for some Be£(H) and BGA'(S). This implies that B0 = B\kers€Ao(S) and A0B0 = B0A0 = J o - Therefore <MO(S)(4>)C
138

Structure of Hilbert Space

Operators

Theorem 4.6.15 Let S~{Wt-} be an operator weighted shift, then the following are equivalent: (i) Se(SI); (ii) cr(Ao) contains only one point for each AoGAo(S); (Hi) Ao(S)/radA0(S)^C; (iv) There is no nontrivial idempotent in Ao(S), Proof (i)=>(ii). It is a straightforward conclusion of Theorem 4.6.12. (ii)=Kiii). Let J = {A0eAo(S) • C by tp[Ao] = XAo, [Ao]eAo{S)/radAo(S),

where

XAoea(A0).

Note that a{Ao) consists of one point and for each Bo£[-<4o], Ao — B0€radAo(S). Thus a(A0 - B0) = {0}. Assume that a(A0) = {A}, a(B0) = {/j,}, then 0 = tr(Ao — B0) = trAo — trBo = nX — nfi and A = v. This implies that

+ XAB-XAXB]

= [A- XA\[B\ +

XA[B-XB],

it follows from a((A - XA)B) = a{XA{B - XB)) = {0} that ([AB]) = ([£]) = A ^ A B and ip is an isomorphism. (iii)=>(iv). Let ip be an isomorphism from Ao(S)/radAo(S) to C. Then
for all Po£Ao{S), where Ao = 0 or Ao = 1. Since ip([PQ - AJc-]) = 0,
= trPo—nX. This implies that P0 = 0 or Po =

Ic-

Unitary Invariant

and Similarity

Invariant

of Operators

139

(iv)=>(i). Assume that PeA'(S) is an idempotent, then Po = P\kers£Ao(S) is also an idempotent. Thus Po = 0 or Po = i o - Without loss of generality, we assume that PQ = 0. From PS = SP,

c nn c

OPoi * 0 Pl2 P =

Since P 2 = P, P = 0. Thus

C"'

0 '••

Se{SI).

Theorem 4.6.16 Let S~{Wfc} be an operator weighted shift, then SG(SI) if and only if J := {A£A'(S) : c(A\kers) = {0}} is a maximal two sided ideal of A'(S) and A'(S)/J is abelian. Proof Suppose that Se(SI). {Claim a } J is a linear space. For A, B&J, (A + B)\kerS = A\kerS + B\kerS = A0 + Bo.

By Theorem 4.6.15, a{A0 + B0) = {A}. Thus nX = tr(A0 + Bo) = trA0 + trB0 = 0. This implies that A = 0 and A + B e J. {Claim b } J is a closed two-sided ideal. Note that for A£j and BeA'(S), AB\kers = (A\kers)(B\kers)&MS)Thus a(AB\kers) = {0} and Claim b holds. {Claim c} J is a maximal ideal. Suppose that J' is a two-sided ideal of A'(S) satisfying J' D J and J' ± J. Choose AeJ'\J, then Ad, oo

the diagonal of A with respect to H = 0 C n , is in A'(S).

Since

A^J,

fc=0

o-(Aa) = o-{A\kers) = {A}, A 7^ 0. This means that Ad is invertible. Denote Ar = A- Ad. Then AreJ. Thus Ad = A - ArSj'. This implies that J' = A'(S). Therefore J is maximal. {Claim d } A'{S)/J is commutative. For A, BeA'(S), Set C = AB - BA. Then Co = C\kerS = (AB — BA)\kerS

= A0B0

— Po^O-

By Theorem 4.6.15, a(C0) = {fi}. Thus n/i = trC0 = tr(A0B0

- B0A0) = tr{A0B0)

- tr(B0A0)

=0

140

Structure

of Hilbert Space

Operators

and fi = 0. Therefore C£j and A'(S)/J is commutative. Conversely, suppose that J is a maximal two-sided ideal of A'(S) and A'{S)/J is abelian. If P is a nontrivial idempotent in A'(S). Set J' = A'(S)PA'(S) + J. It is not difficult to see that J' is a twosided ideal of A'(S) and J'DJ. Since P 0 = P\kerS ¥= 0 , P £ J and J ' ^ J ' . We assert that the identity 7 ^ J ' . Otherwise I = APB + C, where A,B<=A'{S) and C e J . Thus 7 C " = liters = A0P0B0 + C 0 , here ^4o = A\kerS,B0 = B\kerS and C 0 = C|fcers. Since C0Sk7|fcerS,o'(Co) = {0}. This means that J c „ - C 0 = A)P 0 So is invertible. But the determinant detPo of P 0 is zero (P 0 7^ 0). Thus deL4 0 P 0 B 0 = (detA0)(detP0)(detB0)

= 0.

A contradiction. Therefore J'Z)J (but J' ^ J7) is also a maximal proper ideal. This contradicts the assumption that J is a maximal two-sided ideal. Corollary 4.6.17 Let S~{Wk} be an operator weighted shift satisfying that Wk = W for all fc>l. Then SG(SI) if and only if We(SI). Proof By Jordan Theorem, there is an invertible matrix XGMn(C) such that Jnj(^l) l

J = XWX~

0

= Jnt(h)

and ^2 rii = n. i=l

_

Set Y = diag(X, X, • • •), then Y is invertible, and S = YSY^^Wk} is an operator weighted shift, where Wk = J. By Theorem 4.6.15, S£(SI) if and only if I = 1 or if and only if W£(SI). The following example indicates that the condition Wk = W (k>l) can not be omitted. Example 4.6.18

Let W2k+i

11 01

, w2k =

1 -1 0 1

Unitary Invariant

and Similarity

Invariant

of Operators

141

then Wk-

11 01

e(Sl).

Let S be the operator weighted shift with weighted {Wk}. Set 2k+l

=

10 00

11 p2fc

00

and P — diag(Pi,P2, P3, • • •). A simple computation shows that PS = SP andP2 = P, i.e., S#(SI). Now we will discuss the similarity of two operator weighted shifts. Proposition 4.6.19 Let S~{Wk} andT~{Vk\ be two operator weighted shifts. Then 5 ~ T if and only if there is a sequence of invertihle matrices {Xt} such that sup{\\Xi\\, ||Xf i } < 00 and Wt = X i K i X i + 1 1 , (*>1). i

Proof

on C n such that

If there are invertihle operators {Xi}^

supdl^llJX - l l

<

00

and Wi =

XiViX-+\,(i>l).

Set X = diag(Xi,X2, •••). Then X is invertihle and SX = XT, i.e., S~T. On the other hand, if 5 ~ T or SX = XT and X^S = TX~l for some operator X. Note that X can be expressed as Xi X\i

X\z

X2

-^23

X =

where WiXi+i

x3

= XiVi and sup{||Xj||} < 00. Since X ^ 1 ^12 ^13 X2

X~l

=

Y23

x

has the form

142

Structure

of Hilbert Space

Operators

where Y,Wi = ViYi+\ and sup{||Yj||} < oo. Since i

xx-1 = x~1x = i, XiYi = YiXi = ICn, Corollary 4.6.20 Proof

(i = l , 2 , - - - ) .

Let S^{Wk}f=1,T^{Wk}^=2

By Corollary 4.6.11, SeBn(tt)

and S€Bn(Sl),

then

implies that

sup{||W i ||,||W i - 1 ||}
Set Xi = Wi, then Wi = XiWi+iXr^v

By Proposition 4.6.19, S~T.

Proposition 4.6.21 Let S~{Wk} and S~{Vk} be two operator weighted shifts, then S~S if and only if S~q.sS, where "~ q . s " means quasisimilar. Proof We need only to verify that S~q.sS implies S~S. By the quasisimilarity, we can assume that S and S have the same multiplicity n and there exist X, Ye£(H) with trivial kernels and dense ranges such that SX = XS and YS = ~SY. Then X and Y have the form Xx

*

x2

X

Y2

0 Since kerXxckerX

0

= {0},Xi is invertible. If follows from SXY

XYS that XYeA'(S)

= XSY +

and XiYx

* X2Y2

XY = 0

Thus d%ag{XxYi,X2Y2, • • • )£A'(A). From SX = XS, we have X2 = W^XM,.•

• ,Xk+1 = W^XkVk, (fc>l).

This implies that Xk (k>l) is invertible. Similarly, Yk (k>l) is invertible. Thus XkYk (k>l) and diag(X\Yi,X2Y2, • • •) are invertible, and [diagiX^XiYi,

• • • J]" 1 = diag{Y^ X^\Y2~l

X^\

•••).

Unitary Invariant

and Similarity

Invariant

of Operators

143

Therefore, there exists M > 0 such that suP{||yfc-%-1||}<M. k

Since diag(X\,X2,

• • • )£kerrs•§, there exists N > 0 such that sup{||Xfc||} < k

N. Furthermore,

supdin-^D^supdin-^fc-^i.HXfciD^supdiy^Xfc-^D-supfHXfciD^Miv. This implies that diag(Yi,Y2,- ••) diag(Yi,Y2, • • • )€kerTgS. Thus S ~ 5 . Proposition 4.6.22 Let S~{Wk} shifts with multiplicity n. S,S£(SI) Then Ao(T)/radA0(T)^C ® C. Proof Note that

-4'(T) = J

For arbitrary Xi2&kerrSg 'Xx

invertible

and

and S~{Vk} be two operators weighted and S is not similar to S. T = S®S.

XS<=A'{S) Xi2£kerrSg

-^21 -X"s

is

Xg£A'(S) X2i€herrg

s

1 J '

and X21 £kerTg s, we have * '

x2

'Yx

*•

Y2

, X21 =

.0

. 0 Computations indicate that X12X2i&A'(S),X2iX12€A'(S) and -^12-^2l|fcerS = XlYi,

Since S,'Se(SI),

•^21-^12|fc e r 5 =

YiX\.

it follows from Theorem 4.6.15 that
= {A}.

{Claim} A = 0. We assume that Xi and X2 are invertible. Set X' = diag(X\,X2, •••), then kerX' = {0} and ranX' = H. Similarly, if Y' denotes Y' = diag(Yi,Y2,- ••) then kerY' = {0} and ranY' = H. Since X'ekerrs^ and Y'£kerTg s, S~S. A contradiction.

Structure of Hilbert Space

144

Operators

For each En E12 E21 E22 and XGkerTs-g,

eMT)

\&

D =

En E12 0

E21 E22

=

0

By the claim above, cr(E2iX\kerg) that

.0

E2lX\kerg_

= {0}. Thus a(D) = {0}. This implies

0X1\kerS 0

'0EnX\kers

GradA0(T).

0

Repeating the argument above, if YGkerr-g s , then 0 y\kerS

0 0

GradA0(T).

Thus Ao{T)/radA

o(T) = {

Xs + radAo{S) 0

By Theorem 4.6.15, Ao(T)/radAo(T)^C

0 Xg+radAo(S)

XS&A Xs€A' '(S)j-

© C.

Proposition 4.6.23 Let S and S be two operator weighted shifts, S,Se(SI) and S~S. T = S®S, then MT)/radA0(T)^M2(C). Proof

Without loss of generality, we assume that S = S. Then

IfcerS X\2\kerS XijeA'(S),i,j = l,2.|. !\Xn \kerS X22I kerS U*21 Since S£(SI), by Theorem 4.6.15, A0(T)/radAo(T)^C. Thus we have Ao(T)/radA0{T)^M2{C).

MT)

=

Proposition 4.6.24 Let S~{Wk} and 5~{Vfc} be two operators weighted shifts. S,SG(SI) and S is not similar to S. T = S(BS. Then Ao{T)/radA0{T)^C

© C.

Unitary Invariant

and Similarity

Invariant

of Operators

145

Proof Assume that S and S have multiplicities m and n respectively. By Theorem 4.6.15, we need only to prove the proposition in the case of m
since E12E21 GAQ(S) Denote

is not invertible, by Theorem 4.6.15, a(Ei2E2i)

radA'(S) E21

J =

= {0}.

EuEkerTgg 1 E2iGkerT-gS J '

Ei2_ radA'(S)

By the arguments used in the proof of Proposition 4.6.21 we have J radAo{T). Thus Ao(T)/radAo{T)^C 0 C.

=

Summarizing the discussion above, we have the following theorem. Theorem 4.6.25 Let S and S be two operator weighted shifts. S, S€(SI) and T = S®S. Then 1. S~S if and only if Ao(T)/radAo(T)^M2(C)] 2. S is not similar to 5 if and only if Ao(T)/radAo(T)=C © C. As a matter of fact, we have a more general result by Proposition 4.6.21, Proposition 4.6.22 and Proposition 4.6.23. Theorem 4.6.26 m

S~@Sj

Let S and S be two operator weighted shifts and

, where Si£(SI)

is an operator weighted shift and S^Sj

i=i

_

i ^ j (Corollary 4.6.13).

Let T = S®S,

then S~S

if and only if

m

Ao(T)/radA0(T)^

0 M 2fci (C). i=i

Proposition 4.6.27 Let S~{Wk}

k

be an operator weighted shift and

~ |_0 1 J ' fc

then A'(S) is commutative if and only i/sup | ^ Aj| = +00. fc

for

_

i=i

146

Structure of Hilbert Space

Proof

If

Operators

then T is of the form

TGA'(S),

T i i T\i T13 • • • 2~22 ? 2 3 • • '

0

and Tk+1>k+1 = Elk1TnE1,k+i-i

T33'--

(A;>1, Z>1), where k

Elk=WlWl+v--Wk

lEAi

=

i=l

0

1

Set Tu =

*21 *22

then = Hi Hi

Tk+i,k+i — Elk T\iE^k+i^i

i'

t'

'21 r 2 2

where k

t'n = hi — ( E Ai)*2i i=l

k+l-1

k+l-1

t'12=t12 + ( E

Wn

- (Z W22 - (Z \i)( E

i=i

t=l

fc

i=l

i-1

= *i2 + (EAi)(tn-*22) + ( - E A i + i=l

i=l

AOfei

i=i

fc+i-1

E

fc

k+l-1

A0*n-(EAi)( E

i=fc+l

i=l

Ai)t2i

i=l

k+l-1 t

t'22= 22 + ( E

A<)*21.

i=l

If sup | E Ai| = +oo, since t'n are uniformly bounded, £21 = 0. Since fc

»=i

sup{||Wfc||}<+oo,|Ai|<M fc

Unitary Invariant

and Similarity

Invariant

of Operators

147

l-\ fc+J-1 Thus | - E A* + E A»| < 21M. Since t'l2 are

for some number M.

i=i

uniformly bounded, tu = t\i.

»=fc+i

Thus

Tk+l,k+l = -Effc T'll-Bi.fc+i-l = *ll-f + *12«^2(0). Therefore, .4' (5) is commutative. * If s u p | ^ Aj| = N < + o o , set A k

diag(B\,B2,

= diag(A\,A2,---)

and B

=

»=i

•••), where fc fe

•EAi(EAi)2 -4fc+i

fc

EAi i=l

Bi =

4.7

01 00

. Then A,

BGA'(S),

but AB - BA ^ 0.

Open Problem

1. Is every operator in JC(H) is a direct integral of strongly irreducible operators? 2. What is the necessary and sufficient conditions for an operator T€£(H) to have only finitely many Banach reducing subspaces? 3. If Te£(H)fl(SI), is A'(T)/radA'(T) commutative? 4. Does the following statement holds for arbitrary Ti,T2&£(J~t)C\(SI)l T!~Ta if and only if Ko(A'(T1®T2))=Ko(A'(T1))^Ko(A'(T2)). 5. Let T&£(H)r\(SI), then for each natural number n, does T^ have a unique (SI) decomposition up to similarity? 6. If TG£(H) is a direct sum of finitely many (SI) operators, does T have a unique (SI) decomposition up to similarity? 7. What is the "Beurling" Theorem for Mz in Sobolev disk algebra? 8. Given a necessary and sufficient condition for an injective unilateral operator weighted shift S~{Wk} to be strongly irreducible. 4.8

Remark

Theorems 4.1.3-4.1.20 are given by [Fang, J.S., Jiang, C.L. and Wu, P.Y. (2003)]. Theorem 4.2.1, Proposition 4.2.13 and Theorem 4.2.14 are due

148

Structure

of Hilbert Space

Operators

to [Cao, Y., Fang, J.S. and Jiang, C.L.(2002)]. Theorem 4.3.1 belongs to [Fang, J.S. and Jiang, C.L. (1999)]. Proposition 4.3.5, Theorem 4.3.9, Proposition 4.3.11 and Proposition 4.3.12 are proved by [Ji, Y.Q. and Yang, Y.H. (2003)]. Example 4.3.8 is due to [Fang, J.S. and Jiang, C.L. (1999)]. Theorem 4.3.6 is given by [Jiang, C.L. (2004)]. Theorem 4.3.13 and Corollary 4.3.14 are proved by [Wang, Z.Y. and Xue, Y.F. (2000)]. Example 4.4.1 and Proposition 4.4.2 belong to [Jiang, C.L. and Li, J.X. (2000)]. C.L. Jiang also proved Theorem 4.4.3 [Jiang, C.L. (2004)]. The all results in Section 4.5 are proved by [Wang, Z.Y. (1993)], [Wang, Z.Y. and Liu, Y.Q. ], [Liu, Y.Q. and Wang, Z.Y. (2004)], [Liu, Y.Q. and Wang, Z.Y. ], [Jin, Y.F. and Wang, Z.Y.(l)]]. Theorem 4.6.2, Proposition 4.6.9 and Theorem 4.6.10 are given by [Ji, Y.Q., Li, J.X. and Sun, S.L. (2003)]. Theorem 4.6.12-Theorem 4.6.26 are proved by [Jiang, C.L. and Li, J.X. (2000)]. Example 4.6.18, Proposition 4.6.19-Proposition 4.6.27 are given by jia-jin-wan. The reader can refer to [Davidson, K.R. and Herrero, D.A. (1990)] about the (SI) decomposition of some special operator classes and [Behncke, H.], [Yan, C.Q. (1993)] about to the unique (SI) decomposition up to unitary equivalence.

Chapter 5

The Similarity Invariant of Cowen-Douglas Operators 5.1

The Cowen-Douglas Operators with Index 1

The backward unilateral shift is a typical Cowen-Douglas operator with index 1. Denote H = I2 — {(xi,x2,- • •) : X) W 2 < °°}- F ° r («o,ai,- ••)&2, define T*(ao, a\, • • •) = (ai, «2, • • • )• For I'M < 1> w e have r 2 *(l,A,A 2 ,-..) = A(l,A,A 2 ,..-).

This implies that Dcap(T*). Clearly, T*eBi(D) and A'(TZ)^H°°. Note that Tz, the adjoint of Tj*, is an analytic Teolitz operator and also a pure isometry operator. oo

An operator

SGC(H)

is called a pure isometry operator if f] SnH = n=l

{0}.

von-Neumann-Wold Theorem Let S£C(Ti) be a pure isometry operai

tor. Then S^ ® Tz, where I =

dimkerS*.

fc=i

It is easily seen that Tz is also a pure isometry operator for each natural number n. The following is a well-known result. Lemma 5.1.1 Let SGJC(H) be a pure isometry operator, then (i) 5 S T i ° and S*eBt(D), where I = dimkerS*; (ii) S€(SI)ifandonlyifS*eB1(D). Lemma 5.1.2 Let P be an idempotent in A'(Tz), let S = TJ"^ |PW
150

Structure of Hilbert Space

(i) U(PH^)=H^®0^m\

Operators

i.e.,

UPU* =

0

0

(ii) LetV = U\ Pfi(rv), then VSV* = Tzm , i.e., S is unitarily equivalent

toZ

(m)

Proof It is obvious that S is a pure isometry. By von-Neumann-Wold Theorem, S is unitarily equivalent to TJ . Thus there is a unitary operator V : PH{n)

—• H{n)

such that VSV* = T ( m ) . Note that if m < n, W ( n ) ©PW ( m ) is infinite dimensional. Therefore, there exists a unitary operator W : H(n)ePH(n)

—> Hin~m).

Set U = V®W, then U satisfies the requirements of the lemma. Lemma 5.1.3 Let P be an idempotent in A' ((T*)^) and let S = (T*)(n^\p-H(n). If m = dimkerS, then there exists a unitary operator U such that: (i)

U(Pn^)

=

ft(m)©0(n-m\

i.e., UPU* =

(ii) Let V = C/| PH (n), then VSV* = (T*)^m\ alent to (r*)( m >. Proof Let Q = (IHM - P)*, then QGA'((T2)^) (Q(Tzr>Q)*

0

y{n-m)

0

i.e., S is unitarily equivis an idempotent and

(n)/( / „ , - P ) . = (/„(„, - P ) ( T ; ) W H(

Thus

{{T*T%H^Y Since

=

Wl {T;T\i H{n)-P)H^

PeA'((T*)^), dimker((Tz)^\QnMr

=

•

dimker{{T*z)M)\(jH{n)_P)nM

= n- dimker((T*)^)

=n-m.

The Similarity

Invariant

of Cowen-Douglas

Operators

151

By Lemma 5.1.2, there exists a unitary operator XJ\ such that

0

0

and rp(n-m)

U^QT^QUt

=

0

^

ft(n-m)

0

U(m).

Thus UxPU* =

0

0 H(m)

and UiPiT^PU?

0

<^(n-m)

0

* (T*)( m )

W(m)

•

Define U2 : H{n) = H{n-m)®H{m) - ^ H{m)®H(m) by U2{x®y) = y®x onto, {n m) {m) for x€H ~ and y£H . Then t/2 is a unitary operator. Let U = U2UU then U satisfies the requirements of the lemma. For T e # n ( Q ) and zeQ, S(T - zl) = (T - zI)S for all SeA'(T). Sker(T - zI)cker{T

-

Thus

zl).

Define (TTS)(Z) = S\ker(T-zi)- In general, we substitute S(z) for (FyS 1 )^) (see Chapter 3). Clearly, TT is an injective contraction. But for T — (T*)(n^GBn(D), we can easily obtain the following lemma. Lemma 5.1.4 Let T = {T*)^n\ A'{T) onto Mn(H°°).

then TT is an isometry isomorphism from

Lemma 5.1.5 Let H = H2, .4e6i(fi)n£(W) and T = A<-nK For each idempotentPeA'(T), denoted =T\PnM. //Ti<=£ m (ft), thenT^A^l Proof Without loss of generality, we can assume that DcQ. Then we can find W-valued holomorphic functions v(z) and e(z) on D such that (A-z)v(z)

= 0, (T; - z)e(z) = 0,

and v(z) and e(z) can be chosen to be the holomorphic frames of ker(A — z)

152

Structure of Hilbert Space

Operators

and ker(T* — z) respectively. Set «fc(z) = ( 0 , " - , 0 ) v ( z ) , 0 , - . - , 0 ) ) k = 1,2, • • • ,n,z&D. efc(z) = ( 0 , - - - , 0 , e ( z ) , 0 , . . . , 0 ) , Let P{z) = (TTP)(z), zeD, then P{z) = ( P ^ z ^ ^ e M ^ t f 0 0 ) is an idempotent. By Lemma 5.1.4, P(z) is an idempotent in A'((T*)'")). Set

g = P(z) and

s = (r;)(nW>. Since Ti£Bm(fl),

dimkerS = ranP(0) = dimkerTi

= m.

By Lemma 5.1.3, there exists a unitary operator U such that U(PH{n))

= W(m)©0("_m),

i.e.,

UPU* =

0

0


(5.1.1)

and if V = *7|PW(n), then KSK* = (Tz*)(m>. Since l/*(V^ m )©0("- m )) = l/*(ft( m )eO( n ~ m )) = QH™ and VSV* = {T*)(m\ U*ei(z)£ker(S-z)cker((T*)(n'>-z),l
-\

h Xin(z)en(z),

l
>= <% < e(z),e(z) >, l
h A in (z)Aj n (z) = <5y, l
From (5.1.1), UP(z)U*ei(z)

= IH(m)ei(z)

= e^z), l
P{z)U*ei{z) =

(5.1.2) Therefore,

U*ei(z),

i.e., (Pij(z)) n x „(Aii(z),- • • ,\in(z)) where Ki<m

and z e D .

= (\n(z),---

,Aj„(z)),

(5.1.3)

The Similarity

Invariant

Set Wi{z) = Xn(z)vi(z)

of Cowen-Douglas

Operators

+ • • • + \in(z)vn(z),l
=Sij

< v(z),v(z)

>,l
153

Since j
(5.1.4)

by (5.1.2), < Wi(z),Wj(z) > = Sij < v(z),v(z) >,l
- z) — ker(T\ - z),

thus Wi(z)£ker(Ti — z), lker(T1 - z) as follows U(z)vi(z) = Wi(z),

l
It follows from (5.1.4) and (5.1.5) that < U(z)vi(z),U(z)vj(z)

>=< Vi(z),Vj(z) > = S^ < v(z),v(z)

>,l
Since U(z) is a holomorphic isometric bundle map, using Rigidity Theorem wehaveTi^^™). Theorem 5.1.6 Let A&Bi{Q)nC{K), then V ( * 4 ' ( ^ ) ) - N and K0(A'(A))^Z. Proof By Theorem 4.2.1, for every natural number n and idempotent Pe(A'{A^)), if Ax = A(")|p W („), then A^A. This is a straightforward corollary of Lemma 5.1.5. Proposition 5.1.7 Let A,BGBI(£1), then the following are equivalent. (i) A~B; (ii) K0{A'(A®B))^Z. Proof (i)=^(ii). It is a straightforward conclusion of Theorem 5.1.6. (ii)=Ki). We need only to show that if A^B, then K0(A'(A®B))^Z. Otherwise, we assume that K0{A' {A®B)=Z. Since A^B, there exists a maximal ideal J in A'(A®B) such that A'(A@B)/J^C, where I" J' [kerTBtA

kerrAiB A'(B) _

and J' is a maximal ideal of .4'(.A) (see Section 5.2) following separating exact sequence: 0—>J -i-»

n A'{A®B)^A'{A@B)/J A

Thus we have the

Structure of Hilbert Space

154

Operators

and the semi-exact sequence:

Ko(J) - ^

K0(A'(A®B))^K0(A/J). A*

Observing the six-term exact sequence:

K0(J) KX(A\A®B)IJ)

- ^ K0(A'{A®B))

-^

*— K^A'iAQB))

—

K0(A'(A(BB)/J) K^J)

It is easy to see that 3 = 0. Thus we get the exact sequence: 0^Ko(J)

- ^ K0(A'(A®B))

Note that K0{A'(A@B))^K0(A/J)=Z, tradicts that K0 (J) ^ 0. 5.2

-^

tf0(„4/J)^o.

therefore K0{J)

= 0. This con-

Cowen-Douglas Operators with Index n

L e m m a 5.2.1 Let AeBn{fl)n(SI), T = A{l)e£{H{l)) and let P be an idempotent in A'(T) satisfying T|pW(i)€(S'-r). Then A\ := T\PH(i) is similar to A. Proof Without loss of generality, we may assume that D. = D an n is the minimal index of A. For convenience we will prove the lemma only in the case of n = 2. Then T = A® A. Note that P is an idempotent in A'(T), by Theorem 4.4.3 we can find an idempotent P\€A'{T) and B£radA'(T) such that P(z) = P\(z) + B(z), where fn(z) fn(z)

fn(z) /22O)

ker(A — z) ker(A — z)

and B{z) =

Bn(z)B12(z) B2i(z)B22(z)

ker(A — z) ker(A — z)'

fijGH00 and Bij£radA'{A). Set G = -Inm + (2Pi + B). Since B€radA'(T), G is invertible in A'(T) and PG = GP^ This implies that G-1PG = P1eA'(T).

The Similarity

Invariant

of Cowen-Douglas

Operators

155

Without loss of generality, we may assume that P — P\, i.e.,

fn(z) h2{z) ker(A — z)

P{z)

hl{z)

ker(A — z)'

fn(z)

fi2(z)

./2l(z)

f22(z)

ker(Tz* — z) ker(Tz* — z)'

f2l(z)

Set P'(z) =

Since T\Pnm&{SI), tr(P'(z)) = 1 for all z£D. By the arguments similar to that used in the proof of Lemma 5.1.5, there exists an invertible element X(z) in M2(H°°) such that IcO 0 0

X(z)P'(z)X-\z) and

0 0 0/c

X(z)(I-P'(z))X-\z)

Furthermore, X(z)\ranp>^ and X{z)\ran(j-pi(z)) are isomorphic bundle maps from ranP'(z) and ran(I — P'(z)) respectively onto ker(Tz* — z). Set X(z) =

un(z)

ui2(z)

U2l(z)

U22(z)

and X{z) =

Ull(z)Iker(A-z) U2l{z)Iker(A-z)

Ul2(z)her(A-z) U22(z)Iker(A_z)

Then X(z)P(z)X(z)

=

her(A-z) 0

0 0

Note that X(z)ker(T - z) = ker(T - z). {Claim} G(z) = X(z)\ranp^ is an isomorphic bundle map from ranP(z) onto ker(A — z). Note that G(z) = X(z)\ranpi^z-) is an isomorphic bundle map from ranP'(z) onto ker{Tz> — z). Let e{z) be a holomorphic frame of ker{Tz* — z) and ti(z) = e(z)®0,

t2(z) = 0®e(z).

156

Structure

of Hilbert Space

Operators

Then (£i(z), ^(z)) is a holomorphic frame of ker(T^,' — z). Set Ai = Tz. |p/(z)W(2) and Z(z) is a holomorphic frame of fcer(.Ai — z), then i(z) = a(2)t 1 (z) + / 8(z)t 2 (z), where a(z),(3(z) are analytic functions in D. Since G(z) is a holomorphic isomorphic map, we can find a function c(z) holomorphic in D such that G(z)l(z) = c(z)e(z) and |K(z)||2 = (|a(z)| 2 + |/3(z)| 2 )||e(z)|| = |c(z)| 2 ||e(z)|| for zeD. Let (Si(z), • • • , Sn(z)) be a holomorphic frame of ker(A — z). Set Vj(z)

= Sj(z)eO, Uj{z) = OeSj(z), (j = 1, 2, • • • , n).

Then (fi(z), • • • ,vn(z),ui(z), • • • ,un(z)) is a holomorphic frame of ker(T— z). Let fj(z) = a(z)vj(z) + /3(Z)UJ(Z), (j — 1,2,- • • ,n). Then (fx(z), • • • , fn(z)) is a holomorphic frame of ker(A\ — z). Set V{z)fi(z)

=

c(z)Vj(z).

Let K\(z), • • • , Kn(z) be analytic functions in D and g(z) = /fi(z)/i(z) + • • • +

Kn(z)fn(z)

= #i(z)(a(z)vi(z) + )8(z)ui(z)) + • • • + tfn(z)(a(z)t>n(z) + j 8(z)u n (z)). Then G(z) = < Ui(z),Uj(z) >=< Si(z),Sj(z) > for z£D, < g(z),g(z)

| ^ ( z ) | 2 ( | a ( z ) | 2 + |/?(z)| 2 )||Si(z)|| 2

>=t t=i

+ t

Ki{z)Kj{z){\a{z)\'>

+

\(3{z)\'>)<Si{z),Sj(z)>.

The Similarity

Invariant

of Cowen-Douglas

157

Operators

Furthermore,
> = E |#,-(z)| 2 | C (z)| 2 ||Si(s)|| a

+ E

Ki(z)^-W|c(z)|2 < ^ ( z ) ^ ) > .

This implies that ||G(.z)<7(.z)|| = ||ff(-z)|| and it verifies our claim. Similarly, we can prove that -^(z)|ran(/-P(z)) is also a holomorphic isomorphism from ran{I — P(z)) onto ker(A — z). By Rigidity Theorem, we can find two isomorphisms and U2££({I - P)tt ( 2 ) ,0©ft)

UitCiPH™,H®0)

such that X = Ui + U2<=A'(T) and XPX This indicates that Ai~A

-l

0 0

and ends the proof of the lemma.

By Theorem 4.2.1 and Lemma 5.2.1, we get the following theorem. Theorem K0(A'(A))^Z.

5.2.2 Let

AeBn{n)r)(SI),

then

\J(A'{A))^N

<™d

Similar to the proof of Proposition 5.1.7, we have the following result. Proposition 5.2.3 Let A,BeBn(fi)D(SI), ments are equivalent: (i) A~B; (ii) K0(A'{A®B))^Z. 5.3

then the following two state-

The Commutant of Cowen-Douglas Operators n

In this section, we always assume that T = 0 Tk, where TfcSBnfc (0fc)n(S7) fc=i

and

\/ ker(Tk — z) = Tik- By the basic operator theory, we have the

following properties: (5.3.1) A'(T) = {(Sij)nxn\Sij£kerTT algebra.

T .,

l
Structure of Hilbert Space

158

Operators

(5.3.2) kerrT, is a linear space and kerrT, T = A'iTi) is a unital Banach algebra. (5.3.3) Denote eA,(T) the identity in A'(T). Then e

A'(T)

e

A'{T1)i

">t'(T„)-

(5.3.4) If Sij£kerrT. T . and Sjk&kerTT. (5.3.5) If (Sij^neA'iT), then o... 5(«,j)

, then SijSjk£kerrT.

.

T

o ••• o

0 • • • Sij

o...

T

•••

0

£A'(T).

o ••• o

By (5.3.5), we can define a canonical map $ y : A'(T)—>kerrT. follows:

T.

as

(5.3.6) $„• is a linear map and $ i i (5)G^ / (T i ) for all S G ,4'(T). In this section, J denotes a proper two-sided ideal. (5.3.7) Let J be an ideal of A'(T). Define o... Uij — i*~^j • >Jij^.rCGTTrp_ rp_ clIlCl

o ••• o

0 • • • Sij

•••

0

0

•••

0

eJ}

Then (5.3.7.1) Ju is an ideal of A'(Ti) or Ju = A'(Ti); (5.3.7.2) Jij is a subspace of kerrT. T.; (5.3.7.3)

S(i,j)&J

for S = ( S y ) n x n G J .

By (5.3.7), we can define a canonical map from kerrT kerrT, T./$ij{J) as follows: 5jj— > [S'ij] l 7, where kerrT, T./$ij(J) quotient of kerrT T. modulo Qij(J'). If i7 is closed, then • A ' C n / J - {([Sii]j)nxn •

SijGkerr^.}

T.

onto is the

The Similarity

Invariant

of Cowen-Douglas

159

Operators

is a unital Banach algebra. Thus we have a canonical map $j : A'(T) —>

A'(T)/J

as follows: ®j((Sij)nxn)

= ([Sij]j)nxn-

(5.3.8) Let J be a closed ideal of A'(T). ([Sii]j)nxn

=

If

*j(S)€A'(T)/J,

then •Q...

••• 0_

0

0 • • • [Sii\j • • • 0 $j(S(i,mA'(T)/J.

o...

o ••• o

Lemma 5.3.1(Lifting Lemma)

Let T = © Tk and J\ he an ideal fc=i

of A'(Ti), then there exists an ideal J of A'{T) such that §i\(J) = J\. Furthermore, if there is another ideal J1 of A'{T) such that $n(»7') = J\, then JQ-J', where <J?n is given in (5.3.5). Proof Set X={

Rl R^D I A21K3 B21H4B12]

:

Ri£Ji,i

= 1 , 2 , 3 , 4 ; B « , A i j € k e r T ,3i "

and J = {xi + X2 H

h xn : \
{Claim} J is an ideal of A'{T). Clearly, J is an additive group. Set W =

Wn W12 W21 W22

&A'{T) and X =

Ri R2A12 A21R3 B21R4B12

ex-

Then WnRi

{WllR2)Al2

W21R1 W21R2A 12

+

(W12A2i)R3 (W22A2i)R3

(W12B21R4)B12 (W22B21)R4B12

= 1,2},

160

Structure of Hilbert Space

Operators

Since WuRi

(W12A21)R3 (W22A2l)R3

(WuR2)A12

W21R1

W21R2A 12

(W12B21R4)B12 (W22B21)R4B12 ex,

WXeJ. Similarly, XWeJ. Thus J is an ideal of A'(T). Since $n(X)€j1 for all XGJ, eA, $ J. Therefore J" is a proper ideal of A'{T) and $ n ( J ) = J i . If $n(J') = J i for another ideal J' of A'(T), by (5.3.4) and (5.3.7), JcJ'. Corollary 5.3.2

Let T = 0 Tk and J i be an ideal J of A'{Ti),

then

fc=i

there exists an ideal J of A'{T) such that $ n ( J " ) = J\. Furthermore, if $11 (J') = J i for another ideal J' of A'(T), then JcJ'. Corollary 5.3.3 Let T = 0 Tk and J<=M(A'{T)), .4'(Tfc) or $kk{J)£M{A'{Tk))7k

then $kk(J)

= 1,2, • • • ,n.

Lemma 5.3.4 Let T = 0 Tk and S = (Sij)nxneA'(T).

If for each

fc=i

Rji£kerrT, T ., RjiSij = 0, £/ien S(i, j)£radA' Proof For each

Rn

(T).

Rln

R tlnl

' ' ' *Mi

0 ••• RuSij RS(i,j)

=

Corollary 5.3.5 LetT= §kk{radA'{T))

••• 0

0 • • • RjiSij • • • 0 0 • • • Knibij

Since i^SV,- = 0, (RS(i,j))n

=

••• 0

= 0. This implies that 5(z,

j)£radA'(T).

0 Tk, then = radA'(Tk), k = 1,2, • • • , n.

The Similarity

Corollary 5.3.6 ®ij{J)- IfSijrji

Invariant

Let T =

of Cowen-Douglas

® Tk,

161

Operators

and Si:i€kerTT,

JGM(A'(T))

T./

= 0 for all r^eker-r^ T . / $ , j ( J ) , then Si:j = 0. n

Theorem 5.3.7 Let T = © Tk, then for each JeM(A'(T)),

there is a

fc=i

positive integer I j
By Corollary 5.3.3, $kk(J)

= A'(Tk)

or $kk(J)eM(A'(Tk)),k

= 1,2,-••

,n.

By Theorem 4.4.3, A'(Tk)/radA'(Tk) is commutative for k = 1,2, • • • ,n. Thus A'(Tk)/$kk(J)^C or A'(Tk)/$kk{J) = {0},fc = l , 2 , - - - ,n. Without loss of generality, we may assume that there exists an integer lj
= 1,2,-••

,lj,

and A\Tk)/$kk{J)

= {0}, k = lj +

l,---,n.

Thus, A'(T)/J

= {([Sy]j)nx„ :

S^ekerr^.

and [Skk]

=0,h
By (5.3.4), 0 = [S^R^j = [S^jSA'^/^iJ), where SijekerTT. T,, Rji£kerrT. Ti and / j - < i 1 and

A'(T)/J={

({Sij])ljxlj

0

°

0

I

OijKzK&TTrp.

rp_

Structure of Hilbert Space

162

{Claim 1}

For l
Operators

if

feerrTiiTi/$«(J)^{0} and kerTTjTJ$jk(J)^{0}, then kerTTiTk/$ik(J)

^ {0}.

Note that > l ' ( T i ) / $ « ( J ) S C and ^ ' ( ^ / ^ - ( J ^ C , l
If

fcerTTi>Ti/*o-(J)^{0}, by Corollary 5.3.6, there exist Sij&kerTTiTJ$ij(J) Sji€kerTT. Ti/$ji{J) such that SySji = [e^>(Ti)]j = 1Similarly, there exist Sjk£kerrT. T /$jk(J) and Skj&kerrT such that SjkSkj = [e^'(Tfc)]>7 = 1- Thus, SijSjk ^

and iT./$kj(J)

0 and

{Claim 2 } $U(J) ± kerrTiT, for l
^ {0},

l
and / c e r r T i | T i / * „ ( J ) = {0}, jo < j < / ^ . By Claim 1, we have kerT

Tj,Tj$ji(J)

= {0}- l<«<7'o,jo < j < / j - .

By Corollary 5.3.6, kerT^^./^jiiJ)

= {°}» 1<*<J0, Jo < J < / j .

Thus

•A'CH/J-j

^og(([5y]j) J0 xjo> ([sij}jhj-3oKh-Jo)) 0

This contradicts

° 0

'•

Sij€'ier"rTi,Tj

JGX(^'(T)).

{Claim 3 } M{T)/J^Mls{C). For l
163

The Similarity Invariant of Cowen-Douglas Operators

eij-eji = en,l
Since A'{Ti)/$u(J)=C,eiieu

&ij&ji ~ ei\BxjCj^e\i

= en. Assume that

— C^2'

and ji ij — ej\C\i€-i\&\j — Gjj.

Since for each Sij£kerTT, that

T./$ij(J)

ij

(l
==

there exists A^GC such

^ij^ij

and since

— X^ij&ji

— \ ij

"ij^iij^ij

^ij J^ii^ij

=

*-M

Sij = Xijeij. Thus the first part of the proof of Theorem 5.3.7 is now complete. lfTk~Tuk = 1,2, • • • ,n, then A'(Tk)/$kk(J)^C for all J e X CA'(T)). This completes the proof of the theorem. Theorem 5.3.7 implies the following properties. (5.3.9) If A'{TM$ii(J) JeM(A'(T)), then

?

{0} and A'^/^^J)

kerTTiiT. l*ijtf)^kerTTiiTi (5.3.10) lfA'(Ti)/$ii(J) then kerTT.iTJ$ik(J) Theorem 5.3.8

± {0} for

/^i(J)=C

= {0} for some i: l
= {0},fc= 1,2, • • • ,n.

LetT = ® Tfc, t/ien /or eoc/i Ji£M(A'(Ti)),

there is a

fc=i

unique JeM{A'{T)) such that $ U ( J ) = Jx. Proof We only prove the theorem when n = 2 and T = Ti®T%The general case can be proved similarly. By Lemma 5.3.1, there is an ideal J0 of A'(T) such that $u(J0) = Ji. Set J' = J0 + radA'{T).

164

Structure of Hilbert Space Operators

Then J' is still an ideal of A'(T) and $n(J') = Jx. By Corollary 5.3.6, radA' (T2)C.§22{J'')- Therefore we may assume that Jo = J ' , ^ ' ( T i ) / $ i 1 ( J 0 ) = C and A'{T2) / $22{JQ) is semisimple. By Theorem 4.4.3, A'{Tk)/radA'{Tk) is commutative, k = 1,2. Note that A'(T)/J0

={

Sll

S\2

Sij£kerTT,TJij(Jo),

S'21 'S'22

l
Denote ekk = [eA,{Th)]j,k = 1,2. Since A'^/^uiJo^C, {Case 1 } Assume that there are ei2€kerTTiiT2/$i2(J0),

e n = 1.

e2i€kerTT2iTJ$2i{Jo)

such that ei 2 e 2 i = 1. Set Qi = e 2 iei2 and Q2 = e22 — Qi, then Qi and Q2 are idempotents in A'(T2)/$22(J0) and Q1Q2 = Q2Q1 = 0. Let

A' = { [ | ;

2 Q^ 22

: Syekerr^/S^Jo),

l
and

.4" = {

0

0

:

0 Q2S22

S22£A'(T2)/$22(Jo)}-

{Claim 1} A'(T)/J0 = A'®A". It is obvious that for each S = 5*ii

(Sij)2x2&A'{T)/J0,

S'12

+

S21 Q1S22

0

0

0 Q2S22

where S n S12 S21 Q1S22

GA'

and 0

0

0 g2s22

eA".

For Sn

S12

S'21 Q1S22

eA'

The Similarity

Invariant

of Cowen-Douglas

Operators

165

and 0

0

o g 2 5 22

eA",

we have that tr =

0

0

S12Q2S22

0

rt =

0 Q2S22S12

0 0

To verify Claim 1, we need only to show that S12Q2 — Q2S21 — 0. For arbitrary S^e/cerr^ T 2 / $ i 2 ( J o ) and S2iekerTT2 Tl/$2i(Jo), we can find a A € C such that S ^ f ^ e n = Aei2e2i. By (5.3.4), we have (S12Q2 - Aei2)ei2 = 0 and
= e12e2i(5'i2<52) = ei 2 (e 2 i5 12 (52) = 0.

Similarly, we can show that Q2S12 = 0. Thus rt = tr = 0 and this proves Claim 1. {Claim 2} A'^M2(C). Let

A, = {Sn :

SneA'inyQuiJo)}

and A = {Ql5 2 2 : 5 2 2 e>4'(r2)/*22(Jo)}. Note that *4i=C. We define a map 0 : ^ 2 — y A i as follows: 0(6) = ei 2 6e 2 i

for all 6e.4 2 -

Clearly, 0 is a homomorphism. Since {Q\) = e n = 1,0 is surjective. If 0(6) = 0(6'), then e 2 i0(6)ei 2 = e2i(V)el2. Thus QxbQx = Qib'Qy and A2—A1=C. Similar to the proof of Theorem 5.3.7, we can get •4'=M 2 (C). Now we define a surjective map n : A'{T) —> A' as follows: 4(^)2X2) - [ ( [ 5 2 l ] U ([522]U

166

Structure of Hilbert Space

Then n is a homomorphism. Since A'=M2(C),

Operators

J = kern <E M{A'{T))

and

* n ( J ) = Ji. {Case 2 } If there is no ei2£kerrTi T2/$i2(Jo) and e2i£kerTT2Ti/$2i(J0) such that e 12 e 2 i = e n = 1. Since A'(Ti)/
errTir2/$12(Jo)

= {0}

and *erT^ i T l /*2i(Jo) = {0}. Thus A\T)/J0^A\Tl)/§u{JQ)®A'{T2)/<5>22{J0)^C@A'{T2)/<5>22{Jo). Similar to the proof in Case 1, we can find a JeM(A'(T))

such that

Now we prove the uniqueness. Suppose that there are J J'£M(A'(T)) such that $n(J) = $ n ( J ' ) = J i . Let J

= J + J'

Then J is an ideal of A'(T). Theorem 5.3.9

= {S + S' : SeJ,

and

S'GJ'}.

Since $ n (~J) = Jlt J = J' = ~J•

LetT = 0 Tk, then for each

SGA'(T),

fc=i

a(S)=

(J

^(S)),

jeM(T)

u//iere $ j - is a canonical map: A'(T)—>A'(T)/J,a($j(S)) is the spectrum of$j(S)inA'(T)/J. Proof We only prove the theorem when n = 2 and T = T\®T2. The general case can be proved similarly. If ^ ' ( T 1 ) / $ l x ( l 7 ) S C and A'(T2)/$22(J) = {0} for every JeM(A'(T)), then *J(S)

=

Aen 0 , 0 0

AeC

(see Theorem 5.3.8)

The Similarity

Invariant

of Cowen-Douglas

If Al{T1)/$n(J)^A'(T2)/*22(J)2iC,

Operators

167

then A l i e n Ai2ei2 A21C21 A22C22

where ei2e2i = en,e2iei2 = e22- This means that

|J

o-($j(S))ca(S).

J£M(T) If Xea(S), consider the maximal two-sided ideal J generated by - S) in A'(T), clearly \£a($j(S)) and this completes the proof of the theorem. (MH®H

2

Lemma 5.3.10 Let T = 0 Tfc, then the following are equivalent: fc=i

(i)

There exist a positive integer n and Xi€kerrT

T

, yi£kerrT

n

where i = 1,2, • • • , n , such that J2 Xit/i =

IA'(T)>

i=l

(ii) There exists an idempotent e£M„(A'(T2)) -f^'(r 1 )©°~a0®e

in

*

such that

Mn(A'(T2))\

Proof (i)=»(ii). Let 'y\

yixi [xi

••• xn]

•••

y„xi

&Mn{A'{T2)).

= _Vn^l

Vn

' ' ' yn%n _

By (i), 2/1

2/1 [xi

•••

[xi

xn]

.Vn

%n — e.

Vn.

Now set 0

0 [zi ••• xn] 0 0

0"

~yi'

0

(n+l)x(n+l) .Vn.

(n+l)x(n+l)

T

,

168

Structure of Hilbert Space

Operators

Then uv

*

Mn(A'(T2))

JU'(r,)©0 =

vu.

and

This implies that 7^ 1 ©0~ a 0©e in

(ii)=>(i). If

•4'(Zi)

*

*

M„M'(T2))

IAli

3e in '^'(Ti)

*

*

M„(^'(r2))

then by the basic properties of if-theory, we find u,v€

A'(T!) *

* M2(A'(T2))

such that /^'(T!)©0 = uv and 0©e = vu, and « = (^'(roMOQeJ.t; =

{0®e)v(IA.(Tl)).

n

Since

/A-(TI)©0

= uv, J2 xiVi = ^Aii=i

Proposition 5.3.11 Lei T = A^ and {Pi,--- , P m } be an (SI) decomposition, then m — l and Ai = A^\p-^(i)&Bn(fl). Proof We first show that m(s)(J) = where 5 = (Sij)lxieA'(A),JeM(A'(A)).

(
Pk = (Plkj)ixi,k = 1,2,

•••,m.

Then

1>(Pk)(J) =

MP&iJVixi-

The Similarity

Invariant

of Cowen-Douglas

Operators

169

I

Denote tr{ip{Pk){J))

:= £
nk>l. Note that £

m

p

k = / a n d PkPk> = Skk,Pk, thus £ tr(iP(Pk)(J))

=I

fc=i fc=i m

m

or £ tr(*l,(Pk){J))

= £ nfc = /. Therefore m
fc=i fc=i

Now we prove that yliGi?n(^)- Otherwise, assume that AiGBk(Q,), and k < n. Let 5 = ASAi. Since k < n, a simple calculation indicates that 0

kerr.

A2,Al

°

71

keTT

.

1*

GA'(S).

By the arguments similar to that used in the proof of Theorem 5.3.7, we can find Ji€M(A'(S)) such that A'(S)/J1^C. Set T1=A®T

=

AV+1).

By Theorem 5.3.7 A'(T)/j£*Ml+1(C) for all JeM(A'(Ti)). Note that Ti=A®Ai®---®Am and m
5.4

The Commutant of a Classes of Operators

FIR algebra. Let A be a Banach algebra, n is a representation of A on a Banach space X, dimX>l, if 7r is a nontrivial continuous homomorphism from A onto C{7i). If a linear subspace y of X satisfies ir(a)ycy for all a£A, y is said to be an invariant subspace of ir{A). A representation ir is said to be irreducible if a subspace y of X satisfying n(a)ycy for all a^A is either y = {0} or y = X. An ideal J'cA is called a Primitive ideal of A if J = kern, when 7r is an irreducible representation of A. Definition 5.4.1 A Banach algebra A is called an FIR algebra if for every irreducible representation ir : A—>£(X),TT(A) is finite dimensional, i.e., dimX < oo. A Banach algebra A is said to be n-homogeneous if there

170

Structure

of Hilbert Space

is a natural number n such that representation of A.

Operators

where n is a irreducible

TX(A)=M„(C),

Definition 5.4.2 Let A be a unital Banach algebra, A is said to be essentially commutative if A/radA is commutative. P r o p o s i t i o n 5.4.3 The com/mutant of every strongly irreducible CowenDouglas operator is essentially commutative. Definition 5.4.4 Let AC(7i). A is called typical strongly irreducible operator AG(SI) and A is essentially commutative. A is called a type1 operator if A has a finite (SI) decomposition (Pi,P2,--- ,Pn) and each A\PiH *s a typical (SI) operator. P r o p o s i t i o n 5.4.5

Every Cowen-Douglas operator is a type-1 operator.

By Gelfand Theorem, we know that an essentially commutative Banach algebra is 1-homogeneous. On the other hand, every 1-homogeneous algebra is essentially commutative. In fact, we have the following stronger result. P r o p o s i t i o n 5.4.6 Let A be a unital Banach algebra and let x(-4) be the set of all nonzero multiplicative linear functionals. For x£A, r(x) denotes the spectral radius of x. Then the following are equivalent: (i) A is essentially commutative; (ii) A is 1-homogeneous; (iii) a(x) = {(ii). It follows from Gelfand Theory. (ii)=^(iii). First we consider A = A/radA. For xGradA, (iv). For arbitrary x, y, z£A, a(xyz) = {p(xyz) : ipGx(A)} = {(i). Given x,y£A,

p£x(A)}

'•
cr(xy) = a(yx).

o(xzy).

The Similarity

Invariant

of Cowen-Douglas

171

Operators

Note that for each n > l and 0
= a((xy)n~k~1

xyxkyk)

= a(yk{xy)n-k-1xk+1y)

a{(xy)n-k-1xk+lyk+1).

=

This implies that cr((xy)n) = a(xnyn). r(x,y) = (r((xy)n))i

a{yk{xy)n~k~lxyxk)

=

Thus

= (r(xnyn))±

= ||* n ||*||y||*.

Let n—>oo, we have r(xy)
{xi, • • • , xn}eX.

Let SeC(X),

then 5 = (Sjj)„ x „), where

Sij£C(Xi,Xj).

Definition 5.4.7 Let X = Xi@ • • • ®Xn be the product Banach space of {Ai}™=1. An algebra AcC(X) is called a free matrix algebra of order n if S = (Sij)nxn£A implies that "0 •• • 0 • • • S(i,j)

0 •• •

:=

.0

••

&ij •

• 0 •

0"

• 0 6.4;

• o.

A is called a unital free matrix algebra of order n if I, the identity on X, is in A. Let AcB(X) be a free matrix algebra of order n. Denote 0 •••

•sxij

— l^ij

•

0

•••

0

0 • • • Si. 0

0

GA} •••

0

and Ai = An, then it has the following properties: (5.4.1) Ai is a Banach algebra and A is unital if and only if each Ai is unital, l
172

Structure

of Hilbert Space

Operators

(5.4.3) If SijGAij and Sjk&Ajk, then SijSjk&AikSij&Aij and Sji^Aji, then SijSjiGAi. Definition 5.4.8

The map 7Tjj : A —> Aij is called an entry map if Ttij \\&ij

for

Particularly, if

Jnxn)

==

^ij

{Sij)nxn£A.

In most cases, we take great care of the diagonal elements Ai's in a free matrix algebra. So we denote the free matrix algebra A of order n with the diagonal elements A\, • • • , An by A~diag(Ai, ••• , An)Example 5.4.9

Let Hi be a separable Hilbert space, l
and H =

© Hi. Let Ti£jO.(Hi) andT = © Ti&C(H). Then A = A'(T) is a typical free matrix algebra, where Ai = A'(Ti),Aij

= kerrT_ T ..

Definition 5.4.10 Let A be a unital Banach algebra. Two idempotents e, feA are called orthogonal, denoted by e_L/, if ef = fe — 0. A set of elements {e i }^ 1 c^4; n < +oo is called a finite decomposition of A if et±ej for l
he„ = 1.

Let {e\, • • • ,em}cA

be a finite decomposition of A. De-

M := {(Sij)nxn '• Sij = Xi = {eiSei : SeA}

eiSej,SeA}, \
Then Xi *s o, Banach space and Ai is a free matrix algebra of order n,Ai£C(X), acting on X = Xi® • ••©<¥„. Define a map : A—*Ai by (S) = (Sij)nxn,

SSA,

then 4> is a continuous isomorphism. Lemma 5.4.12 LetA~diag(A\,A2,---,An)andJ'cAbeanideal. Then J is a free matrix algebra and 3~diag{J\,Ji,-• ,Jn)- Furthermore, either Ji = Ai or Ji is an ideal of Ai, l
The Similarity

Invariant

of Cowen-Douglas

Operators

173

Since A is a free matrix algebra, a simple computation shows that either Ji = Ai or Ji is an ideal of Ai, l
^

IJit

V

Xi^yiGzA-i.

Thus Ai is an irreducible subalgebra of £(Xi),

l
Definition 5.4.14 Let A~diag(Ai,

A2, • • • , An)

and B~diag(B1,B2,-..,Bn) be two free matrix algebras of order n. The map <j> : A —> B is called a freely matrical morphism if (i) 4>(A)CB; (ii) (friaSx + j3S2) = a(Si) + /fy(S 2 ) for SUS2€A and a, /?eC; (hi) ^(5iS 2 ) = (S) = T, then 4>(S(i,j)) = T(i,j). If in addition 4>{IA) = LB, then <j> is called a freely matrical homomorphism. If 4> is a freely matrical morphism from A to B, then 4> induces a morphism from Aij to B^. In general, we will not distinguish with the morphism induced by it from Aij to B^. Lemma 5.4.15 Let A be a unital free matrix algebra of order n and JcA be a closed ideal of A. Then A/J is a unital free matrix algebra of order n. Furthermore, there is a freely matrical morphism (j>j from A to A/J. Proof Let j be the canonical map from A to A/J. Denote ei = I^.(i, i) and ft = 4>j{ei) for lj is a freely matrical morphism from A to Aj J.

174

Structure

of Hilbert Space

Operators

Theorem 5.4.16 Let A~diag(Ai, A2, • • • , An), then A is an FIR algebra if and only if each Ai is an FIR algebra. Proof "4=" Suppose that -K is a continuous irreducible representation of A acting on a Banach space X. Let Pi = n(ei), then n(A) is a unital free matrix algebra and -K(A)~diag(Tr(Ai),Tr(A2), • • • ,n(An)) acts on X = P\X® • • • ®PnX. By Lemma 5.4.13, every n(Ai) is a unital irreducible algebra, ir(Ai)c£(PiX) or PiX = 0, l
n = y^dimPjX x=i

< 00.

"=*>" Suppose that n is a continuous irreducible representation of Ai and J\ = kern. By Kaplansky [Bonsall, F.F. and Duncan, J. (1973)], there exists a unique primitive ideal JdA such that J\ — Jf\A\. Since A is FIR, A/J is a finite dimensional algebra and Ai/Ji is a subalgebra of A/J. Thus A\/J\ is finite dimensional and Ai is FIR. Similarly, we can prove that Ai is FIR for all i. By Example 5.4.11, we can restate Theorem 5.4.16 as follows. Theorem 5.4.16'

Let Abe a unital Banach algebra and {ei,e 2 ,---

,en}cA

be a decomposition of A. Denote Ai = eiAei, l
Then A is FIR if

An FIR algebra ,4 is said to be stable finite if Mn(A) natural number n.

is FIR for each

Corollary 5.4.17 Let A be a unital FIR algebra, then for each natural number, Mn(A) is FIR. Proof This is a straightforward corollary of Theorem 5.4.16'. Similar to the discussion in Section 5.3, we can get the following result n

about the type-1 operator T = 0 Tkfc=i

Theorem 5.4.18 Let Ji&M(A'(Ti)), then there exists a unique j£Ai(A'{T)) such that nn(J') = J\, where TTU is defined in Section 5.3. Theorem 5.4.19 Let T = T\®T2 be a type-1 operator. If Ji is a subalgebra generated by kerrT T and kerrT T , then

175

The Similarity Invariant of Cowen-Douglas Operators

(i) A'(T) is 1-homogeneous if and only if J\CradA'(Ti). Moreover, if A'(T) is semisimple, i.e., radA = {0}, then A'(T) is 1-homogeneous if and only if A'(T)*A'(T1)®A'(T2). (ii) A'(T) is 2-homogeneous if and only if there exist a positive integer n and Xi£kerrT T , yi£kerrT T ,l
Y^xiVi

=lA'(T1)

i=\

and n

Y^Vixi

= lA'(T2) + R,

i=l n

Furthermore, if A'(T) is semisimple, then Ylyixi

where R&adA'iT^)-

~

i=l

-U'(T2)Proof The first part of the theorem is obvious. We need only to show the second part. Suppose that A'(T) is 2-homogeneous. {Claim} There exist a positive integer n and XiGkerrT T , n

yi£kerrT Ti,l
Therefore, there exist Xi€kerrTiT2

and yi£kerrT2T

such that ^ xtyt =

n

-TA'(TI)-

Note that £2 Vixi ~ ^A'(T2) *s a nilpotent and idempotent, thus i=i

n

^yiXi

- IA^T2)£radA'(T2)

[Antonevich, A. and Krupmk, N. (2000)].

j=i

Conversely, if there exist a positive integer n and Xi£kerrT n

ytekerr

, l
T

,

n IA'^)

and £ yiX{ i=l

IA>(T2)

+

176

Structure of Hilbert Space Operators

R, R£radA'(T2). A'(T)/J^M2(C)

5.5

Similar to the Proof of Theorem 5.3.7, we can get for all JeM(A'(T)). Thus A'(T) is 2-homogeneous.

T h e (SI) Representation Theorem of Cowen-Douglas Operators

In this section, we always assume that T££(H) ator with index n.

is a Cowen-Douglas oper-

Lemma 5.5.1 Let TeB„(Q) and T = A@B, T = A+D be two decompositions ofT, then D~B. Proof We may find three idempotents PA,PB,PD&A'(T) such that -* \ranPA ~ -**> -*• \ranPs — &•> ± \ranPo

=

^ t

then PA + PB = I, PA + PD = I- Thus PD is can be regarded as an invertible operator from ranPg to ranPoSuppose AeBm(Q), then B,DeBn^m{n). Let (ef(A),--- ,e£(A)) be the holomorphic frame of ker(A — A) and (/^(A),--- ,/,f_TO(A)) be the is t n e holomorphic frame of ker(B - A). Then (PDff(X), • • • , Pofn-mW) X holomorphic frame of ker(D — A). Denote {PD\ranPB)~ = XD, then XDDPDf?(\)

= \XDPDf*{\)

= A/f(A) =

Bf?(\).

So B~D. Lemma 5.5.2 Let T = A(™l)®A2m2)®---®A{™k\ and Ai^Aj

for i

k

VM'(® A

))=N

(fc)

^

j , then

Aie(SI)nBni{£l), M {A'(T))^N^

i=

l,2,---,k

if and only if

, where {mi, • • • ,mk} and {ni, • • • ,nk} are two or-

i=l

dered sets of positive integers. Proof We need only prove the necessity of the lemma. By Theorem 4.2.1, \/(-4'(^))=N ( f c ) implies that 0 A\mmi)

has a unique finite {SI)

2= 1

decomposition up to similarity, where m = YLnni-

Set T\ = 0 A\" ,

i=l

then T[

n)

k

= ®^ i=l

nni)

i=l k

.

Since mmi>nnu

l
and 0 ^ i=l

mm,)

=

The Similarity

Ti ( n ) e © ^ m m , ~ n n , ) .

Invariant

of Cowen-Douglas

Operators

177

By Theorem 4.2.1 again, T ^ has a unique (SI)

decomposition up to similarity and

V(i'(®AK)))=N(fc». »=i

Remark 4.5.3 By Theorem 4.2.1 and Lemma 5.5.2, if Ai£(SI),

l
k

and Ai'/'Aj (l
tion up to similarity if and only if ( 0 Ai)^

has a unique (SI) decompo-

i=i

sition up to similarity. Lemma 5.5.4 Let A be a finite irreducible algebra and JQA be a closed ideal and 0 —> J -^ A —> A/J —> 0 fee an exact sequence. If \J{A)=N and \I_A\ = 1, t/ien the induced map nt : K0(A) -» tf0(-A/J) is injective. Proof Suppose that n is a natural number and p, q£Mn(A) are two idempotents. Since \J(A)=N, [p] = [er] and [q] = [es], where efc = diag(I^,--- ,1^, 0, •••) with k Ij^'s in the diagonal for k = r,s. If 7T*([p]) = 7r»([g]), then [7r(er)] = [7r(es)]. Since A is FIR, A/J is FIR. Moreover, ^4/v7 is stably finite by Corollary 5.4.17, thus r = s and [p] = [q]. This proves that ir» : /^o(»4) —> KQ(A/J) is injective. Lemma ideals of Proof Define 0

5.5.5 Lei .4 6e o unital FIR algebra and J\ ^ J2 be two maximal A. Denote J = Jif)J2, then A/J=A/Ji®A/J2. Suppose that fa be a quotient map from A to A/Ji, i = 1,2. : A—>A/Ji®A/J2 as follows: (a) =

fa(a)®fa(a),

aeA.

Then is a homomorphism. By Chinese Remainder Theorem, <> / is surjective. Since ker<j> = kerfankerfa = JiC\J2 = J, A/J=A/Ji®A/J2. Lemma 5.5.6 Let Ai (i = 1,2, • • • ,fc)6e strongly irreducible operators and Ai'/'Aj for l
178

Structure

integers and T = ® A\n

of Hilbert Space

Operators

®B be a Cowen-Douglas operator. Denote

j=i

51 = 4 ni) ©5,5 2 = 0 ^ H

>

i=2

i.e., T = S1®S2. If\/(A'{S2))^N(k-1\

then

n

J\ = {^XiVi,

Xi&kerTsus2,

yi&kerTs2,Si, ±
i=l

is a proper ideal of A'(Si). Proof If J\ = A'(Si), then there exist xi,x2,---

,xnekerTSus2,

2/i, 2/2, - - -

,yn^kerTs2,s1

such that zi2/i H

1- a;„j/n = /^'(s x )-

So yi

[ n - - - xn]eMn(A'(S2))

P = .?/«.

is an idempotent. By the similar argument used in the proof of Lemma 5.3.10, we have l ^ ( S l ) ® 0 ~ a 0 © P in A'iS^S^). Let S i € £ ( £ i ) , S^n)eC(fC2). Then by Lemma 4.2.4, A[n)®B = Si = (5i©5^" )|(/_4/(Si)©o)(AC1e/c2)~(5'i®5'2n )l(o©P)(/c1ffi/c2) = - V IPK:2 Since V(<4'(S'2))=iV(A:_1), a unique (SI) decomposition AI~AJ, where 2<j
it follows from Theorem 4.2.1 that S< n) has up to similarity. Since A\~ ®B~S2 \PK2, contradicts to our assumption that Ai / Aj proper ideal of A'(Si).

Lemma 5.5.7 Let A~diag{A\, • • • ,An} be a unital Banach algebra and J~diag{Ji, • • • ,Jn} be a closed ideal of A. Suppose that •K : A—>A/J, are quotient maps and A/J={Ai/J\,

7Ti :

Ai—>A\/J\

0, • • • , 0), then

The Similarity

Proof

Invariant

of Cowen-Douglas

Define a* : ^\*{KQ{A\))—*-K„{KQ{A))

Operators

179

as follows

a*(7ri*([e])) = 7r„([e©0©- • •©()]) for all

e&Mk(Ai).

We first show that a* is injective. If 7r*([e©0© • • • ©0]) = 0, then 7r»(e©0©---ffi0)~s0. Thus there exists

r

such that 7r,(e©0ffi- • • ©0)©r~ a 0©r. Then

7ri*(e)©°© • • • ©0©r~ a 0©r. Note that 0 © r ~ a r for each r. Let r' = 0© • • • ©0©r, then 7ri*(e)ffir'~a0©r'. This shows that 7Ti*(e)~s0. Therefore, [7i"i*(e)] = 5Ti*([e]) = 0. Remark For unital Banach algebra A, let p,q£Mcc{A) be two idempotents, we say p~3q if there exists an idempotent r in Moo (A.) such that p<$r~aq®r. Second, we will prove that a* is surjective. It follows from that for every (3£Ko(A), there is an e such that

A/J=A\/J\

*l.(H)=7T.([/3]). we

Similarly, for (Pij)nxn€Mn(Ko(A)), such that *"i*([eij])

have

\eij]nxn

= 7r

* ([/%])•

By the basic K-theory, we have [(7r*(e i ; ; -)®0---e0) n X „] = [7T,((ey)nxn)©0].

Thus a» is surjective and so a , is an isomorphism. Lemma 5.5.8 Let T = Ax®A2, where A\ and Ai are strongly irreducible Cowen-Douglas operators and Ax^Ai- Suppose that n is a positive integer and

is another finite decomposition of T^n\ where mi,m2,m>0,Bi£(SI) Bi'/'Aj for \
and

Structure of Hilbert Space

180

{Claim 1} m,i + m n. Let B = £ i © S 2 ® • • • © S m , R = T{n^®A2, A'(R) =

A'{TW)

Operators

then

fcerrr(„,iAa

Suppose that J is a subalgebra generated by kerTT{„)A2 and kerrMTin). By Theorem 5.2.2, \/A'(A2)^N. By Lemma 5.5.6, J is a proper ideal of A'(T^). Let J\ be the closure of J, then J\ is a closed ideal of A'{T™). Jx kerTTi„)iA2 cA'(R), then J" is a closed ideal of SetJ = A'(A2) kerr A2,Ti") A'(R).

when rw = 4 n ) ®4 n ) , w e have: kerTAuA2 A;errT(„)>A2 =

kerTA1,A2 A'(A2) A'(A2)

:={

xn

2nxl

: a^e/cerr^

A2 ,

?/iG^'(yl 2 ),i = 1,2, ••• ,n}

2/1

J/n

and fcerT^2iTC„) = [kerTA2>Al, ••• , /ser-TA2,A,, ^ ' ( ^ 2 ) , ••• ,
,y'n) \x!i£kerTA

A

, y'i&A!{A2),i = 1,2, • • • , n } .

The Similarity

Invariant

of Cowen-Douglas

181

Operators

Thus kerrTln)

[kerTAuA2-kerTA2tAl}nxn *

tA2-kerTA2tT^)

* Mn(A'(A2)) (5.5.1)

where kerTAuA2-kerrA2,A1

= {xx' : xekerTAuA2,x' X\Xi

•• •

X\Xn

XnX-^

•••

XnXn

viy'i

•••

yiy'n

ekerrA^Al}, Xi€kerrAuA2 x^Gkerr^^

[kerTAuA2-kerrA2iAl}nxn

= <

yny\ •••

yny„

>.

yi,y'ieA,(A2) i = 1,2,--- ,n

j

Consider another decomposition of T^™':

T(")~4mi)©4m2)©51®- • -@Bm = 4 mi) ©4 m2) ©£~4 mi) ©£®4 m2) Similarly, we have that kerrT(n)

tM

-kerrA2

=

T(n)

diag([kerTAuA2-kerTA2:Al]miXmi,kerTB1,A2-kerTA2iBl, kerTBm,A2-kerTA2tBm,

• • •,

Mm2(A'{A2)))

(5.5.2)

Note that J can not be a maximal ideal of A'(R). Otherwise, by the construction of J, J\ is a maximal ideal of A'(T^ and A'(R)/J

A'{Ai(l)®A2n+1))/J^A!{T^/Jl^Mn{C).

=

But from (5.5.1) and (5.5.2), A'(R)/J=A'(AlTl)®B®A{p+1))/J^Mmi+m(C). This contradictsTOJ+ m > n. Now consider

A'(T™) = dm 5 (^(4 n) ).^'(4 n) )) and Ji~diag(J^,J{2).

182

Structure of Hilbert Space

Operators

Denote A = A'(T^)/Ji. By Theorem 5.2.2, \J(A'(A2))^N. Moreover, by Lemma 5.5.6, J{[ is a closed ideal of A'{AP), and J£2 = A'(A2n)) = Mn(A'(A2)). Therefore, A={A'{A((l))/J^)®0.

(5.5.3)

On the other hand, we consider A'iT^-diagiA'iA^),

A {By), • • • , A'(Bm),

A'(A{™3))).

Now, Jl = Jl~diag(Jn,

J22, ••• , Jm+2,m+2)-

By Theorem 5.2.2 and Lemma 5.5.6 again, Jn is a closed ideal of .4'(A^ m i ) ), 3u is a closed ideal of A\Bi—\), 2
= {A'{A[mi)®B)/Jl)®Q,

(5.5.4)

where J[ = J{ = diag(Jn,J22, ••• , Jm+i,m+i). Without loss of generality, we can assume m;, m2 > 0. Otherwise, we can consider that: T (2n) =

T ( " ) © T ( " ^ A [ " + m i ) © A ^ + m 2 ) © B 1 © - • -@Bm

and

r<2n> = 4 2 n ) e 4 2 n ) . By (5.5.4), there exists a surjective homomorphism
A'(A[mi)®B)-^A.

By Theorem 4.4.3, A'(Ai)/radA'(Ai) is commutative. Thus ^ ' ( A ^ 0 ) and A both are n-homogeneous. Therefore A is an FIR algebra. Furthermore, A = .4~cfoa3(.4i, A2, • • • , Am+i) = ^ ~ ^ a f f M ' ( 4 m i ) ) / ^ n , A'{Bi)/J22,---

,A'(Bm)/Jm+i,m+i).

Since each Jii is a proper ideal, Ai ^ 0, i = 1,2, • • • ,m + 1. Suppose J ^ is a maximal ideal of A\. By Kaplansky theorem [Bonsall, F.F. and

The Similarity

Invariant

of Cowen-Douglas

183

Operators

Duncan, J. (1973)], there exists a unique maximal ideal J2
= A/J2~diag(A1/J(1,

A2/$22(J2),

•••,

Ai+m/$i+m,i+m(J2))-

Since A is n-homogeneous, A/J2=Mn{C). Note that mi + m > n, by the arguments similar to that used in the proof of Theorem 5.3.7, we can see that there exist mi+m — n natural numbers {hi, k2, ••• , fcm+mi_„} in {1,2, • • • , m + 1} such that •Aj/$jj(J2)

= 0, je{ki,k2,

• •• , /em+mi_n}.

Without loss of generality, assume that when j = m + 1, Al+m,l+m/$l+m(J2)

= 0.

Suppose that J{+m 1 + m is a maximal ideal of Ai+m, using Kaplansky theorem again, we can find a unique maximal ideal J% of A such that ®l+m,l+m(J3)

=

Jl+m,l+m-

Thus J2 ^ J3. Since A is n-homogeneous, A/Jz=Mn{C). Ji^Jz- By Lemma 5.5.5, there exists an isomorphism

*i :

Denote J4 =

A-^A/J^Mn{C)®Mn{C)

such that $ i C U © 0 © • • • ©0) = (1©0© • • • ©0)©P, P ^ 0,

$1 (o© • • • ©o©/.A1+m) = o©(o© • • • ©o©i). Set $ = $!•(/>. Then $ is a surjective homomorphism from A'(Ai to Mn(C)®Mn(C) such that

^©5)

^ ^ ' c ^ r 1 ' ) ® 0 0 • • • ©°) = C1©0© • • • ©o)©^> P^O, $(0© • • • ®0®IA,{Bm))

= 0©(0© • • • ©0©1).

Since A'(R)/J = A'{T^)/JX®Q = A®0, there exists a closed ideal ED J such that A'{R)/E = A/Ji®0 = A'{A(™')®B)/ker§®0. Suppose that TT : A'{R)->A'(R)/J, 7r2 : A'(A{(ni)®B)—^Al{A{™l)®B)lker$ and

184

Structure

TTI : A'{Ai)^A'(Ai)/J"

of Hilbert Space

Operators

are canonical maps. Then by Lemma 5.5.7,

irlt(Ko(A'(A^))))^,(K0{A'(R)))^nit(K0{A'(A^ni)®B))). By Lemma 5.5.4, 7rl!#! is injective. So n24K0(A\A^l)®B)))^4K0(A\R)))^n14Ko(A'(A[n))))^Ko(A'(A^)))^Z. Furthermore, $ induces a homomorphism A'(A(Tl]®B)/ker^—+Mn(C)®Mn{C).

* : Therefore, *. =

*,-TT 2+

K0(A'(A[mi)(£B))—>K0(Mn(C)@Mn(C))^Z@Z.

:

Since \t* is an isomorphism,

$»(tfoU'(4 mi) ©5))) = **(^*(^o(^'(4 mi) ©5))))^Z.

(5.5.5)

Since $

( ^ ' ( A ( m i , ) ® ° ® • • • ©o) = (i©o© • • • ©o)©p

and $(0© • • • ®0®IA'{Bm)) = 0©(0© • • • ©0©1) we have $•{[!*

(Afi))®0®

• • • ©0]) = [l©0© • • • ©0]©[P] = 1©[P];

$*([0© • • • 0Oe/A'(B m) ]) = [0]©[0© • • • ©0©1] = 0©1. By (5.5.5) again, there exists n£Z such that * * ( [ ^ ( ^ i ) ) © 0 © • • • ©0]) = ™$*([0© • • •

®0®IA,{Bm)}),

i.e. 1©P = n(0©l) = 0(Bn£Z®Z. But this is impossible and so we verifies out claim that m4 + mn, and it suffices to prove that m\ + m>n. By Theorem 4.4.3, A'(Bi)/radA'(Bi), l
A = A'{T^)/J! =

A'{A^)/Jn

The Similarity

Invariant

of Cowen-Douglas

185

Operators

is n-homogeneous and since A = A^diag(A'(A^)/Jlu

A\B1)/J22,

••• , A'(Bm)/Ji+m,i+m,

0).

By the similar arguments used in the proof of Theorem 5.3.7, we have A/J'=Mi(C) for each J'£M.(A) and l<mi + n. Since A is n-homogeneous, m\ + m>n. Similarly, we may obtain that m2 + m>n. Therefore, m* + m = n for i = 1,2. Lemma 5.5.9 Let Ai,A2 be two strongly irreducible Cowen-Douglas operators. Assume that A\^f>A2 and T = A^'QA^2, where ni,n2 are two natural numbers. If A'(T)fradA'{T) is commutative, then\J(A'{T))^N^ and K0{A'(T))^ZW. Proof Since A'{T)/radA'{T) is commutative, A'(T) is 1-homogeneous and A\T)/radA\T)^(A\A[ni))/radA\A{ni)))®(A'(A2n2))/radA'{A{2n2)))

Thus,

K0(A'{T))^Z^.

Lemma 5.5.10 Let A\,A2 be two strongly irreducible Cowen-Douglas operators. Assume that A\^A2 and T = A± 1'®A2n2 , where n-\_,n2 are two natural numbers, then \J{A\T))^NM

andK0(A'{T))^Z^.

Proof By Remark 5.5.3, we may assume that T = Ai®A2. By theorem 4.2.1, we only to prove that for each natural number n, T^ has a unique (SI) decomposition up to similarity. Let T<"> = A^QA^ and assume that

T ^ W ^ W ^ 2 ' © . ^ © - • -®Bm is another finite decomposition of T^n\ where m i , m2, m>0, Bj£(SI), Bj'/'Ai for i = 1, 2 and l < j < m . By Lemma 5.5.9, mi + m = n, i.e., rrii = n — m for i = 1, 2. {Claim} m = 0, i.e. m, = n, i = 1,2. Since T = A1@A2, A'{T) = A'(T)~diag(A'(Ai), A'{A2)). By Theorem 4.4.3, A'(Ai)/radA'(Ai) is commutative for i = 1,2. From the proof of

186

Structure of Hilbert Space

Operators

Theorem 5.3.7, for every JeM(A'{T)) A'(T)/J^Mt(C), where I = 1 or 1 = 2. Without loss of generality, we can assume that there is a J&M.(A'(T)) satisfying A'(T)/J^M2(C). Then A'(T^)/Mn(J)=Mn(A'(T))/J)^M2n(C). SinceTO*= n — m, T (n)

„ A[ m i ) ©A( m 2 ) eBi©- • -®Bm

= 4 n _ m ) ® 4 " _ m ) © 5 i © • • • ®Bm = T("- m )eBi©- • -®Bm, Thus A'(TW) = A'(T™) = diag(A'(T^-m^),A'(B1),---,A'(Bm)). Therefore, for every J&M(A'{T^)), $n(j) = A'(T^-m^) or $ n ( J ) m is a maximal ideal of _4'(T("- )). This implies that A'(T^)/J^MS(C) for each JeM(A'(T^)), where s<2(n -m) + m. So 2n<2(n - m ) + m , i.e. 2m<m and m = 0. Repeating the arguments in the proofs of Lemma 5.5.8, Lemma 5.5.9 and Lemma 5.5.10, we have the following theorem. Theorem 5.5.11 Let A\,A2, • • • , Ah be strongly irreducible CowenDouglas operators. Assume that A^Aj for i ^ j , and T = Aj ®A22 ©• • •®A£lk , where (ni, • • • ,nfc) is a tuple of natural numbers, then \J(A'(T))^N^k\

K0(A'(T))*ZW.

From Theorem 5.5.11, we can get the following theorem. Theorem 5.5.12 Let T be a Cowen-Douglas operator, then for each natural number n, T^n' has a unique (SI) decomposition up to similarity. Theorem 5.5.13 Let A,B£t3n(Q,)

and assume that

A~4ni)e4"2)©---©4nt), where 0 ^ n ^ N , AiG(SI) for i = 1,2, • • • , k and A^Aj for i ^ j , then A~B if and only if the following two conditions are satisfied: (i) (K0(A'(A®B)), \/(A'(A(BB)), J)^(ZW,NW,1);

The Similarity

Invariant

of Cowen-Douglas

Operators

187

to N ( & ) satisfies

(ii) The isomorphism h from \J(A'{A®B)) h([I]) = 2n\e\ + 2n 2 e 2 H

h 2nfcefc,

where I is the identity of A'(A®B) and {ei}f=1 are the generators of N ^ ) . Proof "<$=": Since B is a Cowen-Douglas operator, we know

5 = 7^ Sl) ®5 2 /S2) ©---©74H where each Bi is strongly irreducible Cowen-Douglas operator for i = 1,2, • • • , m and B^Bj for i ^ j . Without loss of generality, we can assume that A _

A^)

B =

a, A^)

ffi

. .

.ffiA(nk)

B{1Sl)®B{2S2)®---®B^\

{Claim 1} VBi, i = 1,2, •• • ,m, there exist Aj,j = 1,2, •• • ,k such that Bi~Aj.

Otherwise, without loss of generality, we may assume that B\,- • • ,Bi, for l
(>i©5)~(yi[ ti) ©4 t2) ® • • • ©4 t f c ) ©£i s i ) © • • • ©5, (s,) ). By Theorem 5.5.11, \J{A\A®B))

= yiA'iA^^A^®

• • • ®A%k)®B{1Sl)® • • • ©£/ ( s , ) ))=iV ( f c + i ) .

This contradicts (i). {Claim 2 } m = k. It follows from Claim 1 that m
( A © £ ) ~ ( 4 n i + S l W 2 " 2 + S 2 ) © • • • ®Atm+Sm)®A{Z+V]®

• • • ®A{knk)).

By Theorem 4.2.1, the isomorphism h satisfies h([I}) = (ni + si)ei H

h (nm + sm)em + nm+i

By (ii),/i([7]) = 2niei+2n 2 e 2 H Ylnke-k- A contradiction. Thus m = k. Now we may assume that B\~A\, B2~^42, • • • , B^^Ak, i.e., {A®B)~(A
•••

®A^k+'k)).

188

Structure

of Hilbert Space

Operators

Repeating the proof of Claim 2, we have Si = rii, i = 1,2, • • • , k. "=»" Since A~B, B = fl{ni)®fl
4 2 " l) )-

The remainder of the proof of the theorem is a consequence of Theorem 4.2.1. Theorem 5.5.13 is the Jordan canonical theorem for Cowen-Douglas operators. By the proofs of Theorem 5.5.11, Theorem 5.5.12 and Theorem 5.5.13, we have the following result. Theorem 5.5.14

and T i ~ ® Afi],

Let Ti,T 2 e£(ft)

where At is a

i=l

strongly irreducible Cowen-Douglas operator for every i, andTi is not similar to Tj if i j£ j . Then T\~T2 if and only if: (i) (^oM'(Ti©T 2 )), VM'CTieTa)), J)S(Z(">, N<">, 1); (ii) The isomorphism h from \J (A'(Ti®T2)) to N' fc ) satisfies h([I}) = 2kiei + 2k2e2 + ••• + 2knen, where I is the identity of A'(Ti®T2)

and {ej}™=1 are the generators

ofN^nK

In the following we will consider a more general form. Let T = ® T> , where T^S^^'iT^/radA'iTi) is commutative, \J{A'{Ti))^N,i = 1,2, •• • , n, and Ti is not similar to Tj if i ^ j . By the arguments used in the proofs Theorem 5.5.11, Theorem 5.5.12 and Theorem 5.5.13, we have the following theorem. Theorem 5.5.15 V ( - 4 ' ( T ) ) - N ( n ) andKQ{A'{T))^ftn\ FuHhermore, T has a unique (SI) decomposition up to similarity and T~G€:£(H) if and only if the following are satisfied: (i) (K0(A'(T®G)), \J(A'(T®G)), /)s*(Z<">, N<">, 1); (ii) The isomorphism h from \J(A'(T®G)) to N^ satisfies h([I]) = 2fciei + 2k2e2 + • • • + 2knen.

Corollary 5.5.16

Let T = ($ A\ k=i

Cowen-Douglas operators, Ak,Bj£(SI)

k)

@®

B)'',

where Ak's,

Bj"s are

j=i

and A^AK,

Bj^Bji

if k

^

The Similarity

k'J^f.

Invariant

of Cowen-Douglas

Operators

189

Then \f{A'(T))^N^mi+m^

K0{A'(T))^Z{mi+m2).

and

This corollary implies such T has a unique (SI) decomposition up to similarity.

5.6

Maximal Ideals of The Commutant of Cowen-Douglas Operators

In this section we will use the i^o-group to characterize the commutant of Cowen-Douglas operators. The main theorem of this section is as follows. Theorem 5.6.1 Let Ai,A? be strongly irreducible Cowen-Douglas operators. Assume that Ai^A2 and T = A^'QA^12, where ni,ri2 are natural numbers. Then for each JGA4(A'(T)), J must be one of the following two forms.

J,i i

kerr

(»l) -1

J= kerr<„„,

4

("2)

1^*1

("2h i] A\A^ )


(ii) l("-i). A'{AF>)

kerr,ni)

(na)

J kerr

("2>

4

22 J'.

("l)

where Ja is a maximal ideal of A'(A\n''),i = 1,2. Proof First, we assume that T = A\®A2Since A\,A2 are strongly irreducible Cowen-Douglas operators, by Theorem 4.4.3, A'(Ai)/radA'(Ai) and A'(A2)/radA1 {A2) are commutative. Assume that J has neither form (i) nor form (ii). Then

J=

Jl\

Jll

190

Structure of Hilbert Space Operators

According to the discussion in Section 5.4, Jii£M(A'{Ai)),{i Then A'(T)/' J=M2(C) O^J

= 1,2), Ji2 % kerTAuA2!J2i

C

kerTMtM.

and we have the following exact sequence: - U A'{T) ^ U

A'(T)/J-^0,

which induces the following six-term cyclic exact sequence:

K0(A'(T)/J) di K!(A'{T)/J) «-- KM'(T)) <— K,{J) K0{J)

-^> K0(A'(T)) - ^

d\

Since A'(T)/J^M2(C),K0(A'{T)/J)^Z, rem 5.5.11, K0(A'(T))^ZM. Note that n*([diag(IA,{Al),0)])

Ki(A'(T)/J)*0.

By Theo-

= 1, TT*([diag(0,IA,{A2))}) = l.

Thus 7T» is a surjective map from Z©Z to Z, and we get a split exact sequence: 0-^K0(J)

-±> Z 0 Z £± Z ^ O .

Since 7 r»( J fr 0 (^ / (T)))SZ, K0{J)=Z. Consider the following split exact sequence: 0—^J—>J+1^(J+1)/J—>0, Using the six-term cyclic exact sequence again and by the fact that d : KQ((J+1)/J)—>Ki(J) is a zero map, we have the following split exact sequence: 0—+Ko(J)-^Ko(J+l)^Ko((J+l)/J)—>0. Since ( j 4 - l ) / J ' S C © C f # o ( ( J + l ) A 7 ) = Z ( 2 ) . K0(J)=Z, we have K0{J+1)=Z©Z©Z. Note that •TA'(^I) °

0

0

and P2 =

0 0

0 IA'(A2)

Combining it

with

The Similarity

Invariant

of Cowen-Douglas

191

Operators

are two minimal idempotents of Moa(A'(T)) and P i / Q P 2 in Moo(A'(T)). It is obvious that Pi,P2 are also two minimal idempotents of M 00 ( l 7-j-l) and P i / a P 2 in M 0 0 (J-i-l). Since K0{J+1)=Z®Z@Z, there exists a minimal idempotent P in M o o ( J + l ) such that P^aP\, P^ap2 in M o ^ J + l ) . Without loss of generality, we assume that P£A'(T). { C l a i m } I - P~aP in M^J+l). Otherwise, / - P^aP in M^iJ+l). Then / - P ~ Q P i ox I - P~aP2 in MooiJ+1). Thus P~aP2 or P ~ a P i in MooU+1). This contradicts the choice of P. Since M 0 0 ( J + l ) c M 0 0 ( ^ ' ( T ) ) , we have P ~ a P i and (I - P ) ~ a P i in Moo(.4'(T)) or P~ Q P2 and (J - P ) ~ a P 2 in MooM'(T)). By Lemma 4.2.4, there exists a natural number n such that T(n'\ranPm

=

T\ranP~T\ra,nP1

(oi T\ranP2) = A\ (pi A2)

and r ( n ' | r O n ( / - P ) e 0 = ^ | r o n ( / - P ) ~ y | r o n P 1 (or T| ron f> 2 ) = A\

(or^)-

This implies that T = AX®A2 or T = A2®AX. By Theorem 5.5.13 T has a unique (SI) decomposition up to similarity. Thus A\~A2. The contradiction indicates that J must have the form (i) or (ii). Now we consider the general case, i.e., T = A" ®A2n 2 • It follows from the proof of Theorem 5.3.7 that A'(T)/ JS*Mk(C) for every JeM(A'(T)), where A; = n\ or k = n 2 or k = n\ + n 2 . Thus we need only to show that k^ m + n 2 . Otherwise, there exists J^M.(A'{T)) such that A'(T)/J^Mni+n2(C). Observing A\T)

=

A\T)~(diag(A'(A1®A2)),diag(A'(A{ni-1)(BA2n2-1)))),

J =

J~{diag{irn(J),ir22(J))).

Then fm(J)£M(A'(Ai@Aa)) *u(J)

and =

J\\ Jl\

J\2 J22

192

Structure of Hilbert Space

Operators

where Jn£M(A'(A1)),J22eM(A'(A2)),Ji2 £ kerrAuM, J2i$kerTA2iAl. This contradicts the conclusion got in the first part. Thus, for each JeM(A'(T)), A'(T)/J is not isomorphic to M„ 1 + „ 2 (C). By the arguments similar to that in the proof of Theorem 5.6.1, we have the following theorem. Theorem 5.6.2 Let Ai,A2,---,Ak be strongly irreducible CowenDouglas operators. If Ai is not similar to Aj if i ^ j and T = Ay11'Q)A2 '®- • -®Aknk', where n\,--- ,rik are positive integers, then for each JeM(A'(T)), J\\ J =

J\k J2k

J\2

J21 J21

.Jk\

Jkl

Jkk

where Jij satisfies the following properties. (i) Jij = kerrAint) A(nj), i ^ j ; (ii) There exists a unique i, \
such that Ju€M(A'(A\ni)))("i)-> and

Corollary 5.6.3 Let T = 4 n i ) © 4 " 2 V • -®Aknk), where AUA2, ••• ,Ak are strongly irreducible Cowen-Douglas operators and Ai is not similar to Aj if i ^ j . Then for each JeM(A'{T)), A'(T)/J^Mi(C), where le{ni,n2,--,nk}. 5.7

Some Approximation Theorem

Let fi be an analytic Cauchy domain, T = dft. Denote L2(T) the Hilbert space of all square integrable complex functions with respect to the arc length measure on I\ Denote M\(T) the multiplication by A operator acting on L2(T). Denote H2(T) the subspace of L2(T) generated by the rational functions with poles outside fi. Then H2(T)GLatM\(T) and MX(T) =

M+(T) Z 0 M_(r)

H2(T) L2(T)eH2(T).

It is easy to see that M£(r)e#i(fi*) and a{M+(T)) = Q.. If Q is the unit disk D, M+{T) = Tz. Furthermore, A'{M+{T)) = H°°(n) [Conway, J.B. (1990)].

The Similarity

Invariant

of Cowen-Douglas

Operators

193

Proposition 5.7.1 Let 0, be a connected analytic Cauchy domain, fii,Ti2 be two nontrivial invariant subspace of M+(T). Then Hir\H2 ^ {0}. Proof Given AeO. Since (M+(T) - \)H2(T) is closed subspace with codimension 1, (M+(T) - \)H\ is closed. Clearly, (M+(T) \)HieLatM+(r)£Hi. Thus (M+(r)-A)Wi^Wi. In fact, if (M+(r) - \)Hi = Hi, then for arbitrary f£Ki,f(z) = (z >^)n9n(z),gn€H1. This implies that /< n) (A) = 0, (n = 1,2, • • •). So / = 0, a contradiction. Given a nonzero function 4>€HiQ[(M+(T) - X)Hi]. For each heR(T), 0 = < (M + (T) - A ) A ( M + ( r ) ) ^ > = y (z - X)h{z)\cj>\2d^ where R(T) is the set of rational functions with poles outside fi. This implies that ^L{g^ReR{T) : g{\) = 0}{cReL2(T)). By [Fisher, S.D. (1983)], the orthogonal complement of ReR(T) in ReL2(T)) is the linear combinations of finite functions, denoted by Qi, • • • , Qm in C°°. Clearly, constant functions is orthogonal to {g£ReR(T) : g(X) = 0}. Thus \(j)\2 is a linear combination of Qi, • • • , Qm and 1 and sup |<£(z)| < oo. Therefore zer h = h(M+(T))4> <= Hi for each heR(T). Since R(T) is dense in H2(T) and since cj> is bounded, c/>H2(T)cHi. Similarly, we can find a nonzero bounded function 4'€'H2 such that

^F2(r)cw2. This means that

ninn2D^H2(T) # {o}. Lemma 5.7.2 Let Q be a connected analytic Cauchy domain, then lranTM+m,M+m}nlkerTM+(r)iM+m}

= {0},

i.e., if there exists an X&C(H2(T)) such that XM+(T)=M+(r)X and M+(r)Y-YM+(T)

=X

194

Structure of Hilbert Space

Operators

for some Y&H%{T), then X = 0. Proof

Since A'(M+(r))

= H°°(Q,), there exists g€i?°°(n) such that X = Mg{Y),

i.e., M 9 ( r ) / = ff/ for all feH2{T).

Thus

M + ( r ) y / = YM+(T)f =
Y(1)GH2(T).

X2h-2Xg.

Ar(A"- x ) - Y(Xn) = Xn~lg

Y(Xn) = Xnh-nXn-1g

(n = 1.2,---)-

Without loss of generality, we can assume that QcD. from lAI^IAI 2 *"- 1 ) (A <= fl,n = 1,2,- • •) that w 2 / r |A| 3 < w - 1 >dm / r \X\2ndm

•oo,

Thus it follows

(n — • oo).

HnA"

This implies " i|A"|i ""2(r)—>oo, (n —• oo). Therefore, if g ^ 0, then Y is unbounded, a contradiction. This means that g = 0 and X = 0. For each natural number n, define

M+(r) e£((tf2(r))(">).

M„(r) = 0

/ M+(r)J

Theorem 5.7.3 Lei M n ( r ) AeA'(Mn(T)),

A =

be defined as above.

Mh Mr.

Mh Mh

Mh.

Then for each

The Similarity

Invariant

of Cowen-Douglas

Operators

195

where fi£H°°(T) (i = 1,2, • • • , n). Moreover, A'(Mn(T)) is commutative; (ii) Mn(T)e(SI);_ (Hi) a(Mn{T)) = Cl,
•••

Aln

eA'{Mn{T)),

A = _-^*nl " ' " Ann i.e.

An

•••

•™-nl ' ' *

Ain

M+(T) I

M+(T)

-Ann

i M+(T) I 0

M+(r)

A n ••• A i n

M+(T)

A , ... A -"^nl **-nn

/ M+(T)_

At the (1, n), (1, n — 1) entries, we have AlnM+(T)

=

M+(T)Ain

and Ai, n _iM+(r) + Ai„ = M + ( r ) A i , n _ i . By Lemma 5.7.2, Ain = 0 and Ai
(l
Compare the (i + l,i) entries ( l < i < n — 1), we get A i + M M + ( r ) + A i + i , i + i = Ai:i + M + ( r ) ^ i + i , i . By Lemma 5.7.2 again, A i + i , i + i = Aiti and

Ai+itieA'(M+(T)).

Structure of Hilbert Space

196

Operators

At the (i + 2, i) entries (l
M+(r)Ai+2,i.

Thus Ai+2,i+i = -4i+i,j and Ai+2,i&A'(M+(T)). Using this argument and Lemma 5.7.2 respectively, we get the general form of A. (ii) Let P be an idempotent commuting with Mn(T). By (i), Mfl P = Mh

Mh

Since P2 = P, f\ = fa. Since fi is connected, / i = l or / i = 0 . In either case, fa = h = ••• = / „ = 0, i.e., P = / or P = 0 and M n ( r ) e ( S / ) . (iii), (iv) and (v) are obvious. Lemma 5.7.4 Suppose that 0

Qll

Ql2

• ••

Qlm

A2 Q21 Qii---

Q 2m

R = ci

0 C2

with respect to the orthogonal direct sum n

m

« = (®Wi)©(0«i) j=l

where (i) {a(Aj)}^=1 and {a(ci)}^l1 pact sets such that n

a(R) =

are two families of pairwise disjoint comm

[\Ja(Aj)M\Ja(ci)} j=l

and cr(R) is connected; (ii) Aj,Ci£(SI) and kerr

(n,m
i=\

A.

i=l

= {0} (l<j
l
The Similarity

Invariant

of Cowen-Douglas

Operators

197

(Hi) Either o-(Aj)nj > i>l or n + m>j > i>n + 1), i.e., P has an upper triangular matrix representation with respect to the given decomposition of the space. Moreover, the same arguments indicate that Pji = 0 (l<j < i
or n + l<j < i
Since Pjj is an idempotent and Aj, Cj£(S7), Pjj = 0 or 1, (j = 1,2, • • • , n+ m). Without loss of generality, we can suppose that P\\ = 0. Since cr(R) is connected, there must exist an integer i (l
==

-^l-* l,n-H i S;H-*n+i,n-f-i

or

Since Qu£ranTAi ,Pn+itn+i = 0. If a(Aj)Cia(ci) ^ 0, by the same argument, Pjj = 0. Since cr(R) is connected, after a finite number of steps Pjj = 0 (l<j
198

Proof

Structure of Hilbert Space

Operators

Denote Bi I 0 B2

B

H

where 5 i = A/+(r*), B2 = M+(T), T* = dCl* and T = d£l. Then B satisfies (i), (ii) and (iii). Suppose that P is an idempotent commuting with B. By Lemma 3.2.3, P =

P1P12

0 P2

n n'

Since B1,B2£(SI),P1 = 0 or / , P2 = 0 or / . If Pi =I,P2 = 0, it follows from P S = £ P that / + P12P2 = P1P12, or l£ranTBiB2. But this is impossible. Thus Be(SI). Note that A'(Bi) commutes with A'(B2) and for each AeA'(B), A = where Ai&A'(B{),A2&A'(B2).

Ai 0

An A2

Thus A'(B)/radA'(B)

is commutative.

Lemma 5.7.6 Let T be in C(H) with connected spectrum cr(T). Ifaire{T) is the closure of an analytic Cauchy domain (I. Then there exists an (SI) operator A such that the spectral picture A(A) of A is the same as the spectral picture A(T) of T and min-ind(A - X)k <min-ind(T

- X)k, (fc>l, A €

ps-p(A)).

Proof Let (Cli,ki),(Q2,k2),• • ,(Cln,kn) be the finitely many components of a(T)\U, where ki = ind(T - A), A <E Qi (i = 1,2, • • • , n). Thus $7* are pairwise disjoint and each fii is a connected Cauchy domain. If \ki\ < 00, by Theorem 5.7.3 and Lemma 5.7.5, there is an (SI) operator Ai — A(Q.i,ki)eC(H) such that a(Ai) — H,, ind(Ai - A) = ki (A € fii) and either min-ind(Ai — A) = 0 (if k{ ^ 0) or min-ind(Ai — A) = i (if ki = 0) and A'(Ai)/radA'(Ai) is commutative. If \ki\ = 00, by Proposition 3.13 of [Jiang, C.L. and Wang, Z.Y. (1998)], we can find an operator Ai = A(Qi, ki)££(H) such that a(Ai) = fij, ind(Ai — A) = kit min-ind(Ai — A) = 0 (A € fij),

A'(Ai)/radA'(Ai) is commutative and V ker(Ai - X) —H.

The Similarity

Invariant

of Cowen-Douglas

199

Operators

Let $ i , $ 2 , - " !$m be all the components of Cl. By Theorem 3.7 of [Jiang, C.L. and Wang, Z.Y. (1998)], we can find (57) operators ci, C2, • • • , cm such that a(ci) = aire(c,) = $£ and A'{ci) is commutative. Set M

Qll M

A

Ql2

•••

Qlm

Q21 Q22 • • • Q2m

An ~~

Ujnl V n 2 ' ' ' Sinm C2

€£(H{n+m)),

0

0 where {Qij : l < i < n , l < j < m } are defined as in Lemma 5.7.4. By Lemma 5.7.4, AG(SI) and satisfies all the requirements of the lemma. Lemma 5.7.7 Let A be given in Lemma 5.7.6, then A'(A)/radA'(A) commutative. n

Proof

n

Since A'{@Ai)/radA'(0 i=l

i=l

commutative and since kerr „ i=l

m

Ai) and A'(0 m

m

Cj)/radA'(0

j=l

= {0}, A1(A) IradA!(A)

is

Cj) are j=X

is commu-

J= l

tative. T h e o r e m 5.7.8 Let Te£(W) with connected spectrum a(T), then there exists a sequence of (SI) operators {Tn}^L1 satisfying (i) A'(Tn)/radA'(Tn) is commutative for all n; (ii) lim \\Tn - T\\ = 0. n—*oo

Proof By Theorem 1.27 of [Jiang, C.L. and Wang, Z.Y. (1998)], we can find a sequence {^n}^.]^ of operators such that for each n (i) °~ire(An) is the closure of an analytic Cauchy domain; (ii) cr(An) is connected; and (iii) lim \\An - T\\ = 0. n—>oo

By Lemma 5.7.6 and Lemma 5.7.7, for each An, there exists a Bne(SI) such that A(B„) = A(An) and min-ind(Bn — \)k<min-ind(An - X)k, (k>l, A e PS-F{A)). Moreover, A'(Bn)/radA'(Bn) is commutative. It follows from Similarity Orbit Theorem that AneS(Bn)~•

Thus we

200

Structure of Hilbert Space

can find a sequence {Tn}^Ll of the theorem.

Operators

of (SI) operators satisfying the requirements

Applying Theorem 5.7.3 and repeating the arguments above, we have the following theorem. T h e o r e m 5.7.9

Given A£Bn(Cl), there exists a sequence of operators {Ak}cBn(£l)n(SI)

such that (i) A'(Ak) is commutative for each k; (ii) lim \\Ak - A\\ = 0. k—>oo

Definition 5.7.10 Let fl\,0,2 be two bounded connected open subsets of C and let n and rn be two natural numbers. An operator TG£(H) is said in the operator class Bn,m(fli,Q,2) if (i) QiCpF(T)ruT(T), (i = l,2); (ii) dimker(T - A) = n for A s fli and dimker(T — fi)* = m for /i € 1^2; (hi) V {fcer(T-A), ker(T-^)*} = H. [M.J. Cowen and P.R. Douglas [I]]Proposition 5.7.11 Given TeSi,i(fii,fi 2 ), A'(T)/radA'(T) is commutative. Proof Denote Hr{T) = \J ker(T - \),Hi(T) = \J ker(T - / * ) * . By Apostol's triangular representation, Hr(T)±.Hi(T) Tr T\2 0 T[

and

Hr{T) Hi(T)'

where Tr = T|^ r (j) and T; = (r*|-^ ( (^))*. A simple computation shows that T r €Bi(fii) and T^eB^fl^). Thus A'(Tr) and A'(T{) are commutative. This implies A'(T)/radA'{T) is commutative. Proposition 5.7.12 Given T£Bm,m(Q,i,Q,2), for each e > 0, there exists a compact operator K with \\K\\ < e such that A'(T + K)/radA'(T + K) is commutative.

The Similarity

Proof

Invariant

of Cowen-Douglas

Operators

201

By the argument of Proposition 5.7.11, Tr T\2 0 Ti

T

nr{T)

where Tr£Bm(Qi) and T^eB^il^)By Theorem 3.2.1, we can find compact operators K\ and K2 such that maa;{||ii'i||, H-ft^H) < § and Tr + J Ff 1 eB m (fi 1 )n(57), (T, +

K2)*GBn(n*)n(SI).

Set K

Kx 0 0 K2

By Theorem 4.4.3, T + K satisfies the requirement of the proposition. In this section we proved that for "almost every" strongly irreducible operator T, A'(T)/radA'(T) is commutative. For the matter of that, we conjecture that every (SI) operator T has the property, i.e., A'(T)/radA'(T) is commutative. 5.8

Remark

The results in Sections 5.1 and 5.2 are contributed by [Jiang, C.L. (1994)]. The work in Section 5.3 are due to [Fang, J.S.(2003)] and [Jiang, C.L. (1994)]. The work in Section 5.4 belongs to [Fang, J.S.(2003)]. The results in Section 5.5 were proved by [Fang, J.S.(2003)], [Jiang, C.L., Guo, X.Z. and Ji, K.], [He, H. and Ji, K.]. The work in Section 5.6 are due to [Jiang, C.L., Guo, X.Z. and Ji, K.]. The work in Section 5.7 are given by [Jin, Y.F. and Wang, Z.Y.(l)], [Jiang, C.L. (1994)] except that Proposition 5.7.1 is due to [Fong, C.K. and Jiang, C.L. (1993)]. Here we must point out that the work of classification of Cowen-Douglas operators was inspired by the work of [Elliott, G. and Gong, G. (1996)], [Elliott, G., Gong, G. and Li, L.], [Dadarlat, M. and Gong, G. (1997)].

5.9 1.

Open Problem Let Ts0„ iTn (f2i,172)) given a necessary and sufficient condition for

Te(SI). 2. Let T&B n , m (fii,fi 2 )n(S'J). Is A'(T)/radA'(T)

commutative?

202

Structure

of Hilbert Space

Operators

3. If TeBn,m(Qi,n2)ri(SI), is K0{A'{T)) isomorphic to Z? 4. Let T~{wfc} be an injective unilateral weighted shift. If for each M^LatT,T|x~T, then is T similar to aTz for some positive a ? 5. Let Tf be an analytic Toeplitz operator. Is the following statement true? Tfe(SI) if and only if A'(Tf)/radA'{Tf) is commutative. 6. Let fl be an analytic Cauchy domain. Does there exist an (57) operator A satisfying the following conditions? (i) A e B i . i ^ n ) ; (ii) A'(A) is commutative. 7. Let Q be an analytic Cauchy domain. What is Ki(H°°(fl))7 8. Given TGC(H) with connected cr(T), does there exist a sequence {Tn} of (57) operators satisfying (i) lim \\Tn - T\\ = 0; n—*oo

(ii) Tn — T is compact; (iii) A'(Tn)/radA'(Tn) is commutative. 9. Let Tf be an analytic Toeplitz operator. Is the following statement true? Tf£(SI) is equivalent to Tfe{RI). 10. If A is a unital subalgebra of 77°°, is Ko(A) isomorphic to the integer group Z?

Chapter 6

Some Other Results About Operator Structure

6.1

i^o-Group of Some Banach Algebra

Theorem 6.1.1 Let fi be a bounded connected open subset of C with (f2)° = Q,. Let H°°(£l) be the unital Banach algebra consisting of all bounded analytic functions on Q. Then K0(H°°(fi))^Z and V(#°°(ft))=N, where (il)° denotes the interior of closure Cl of Q. To prove Theorem 6.1.1, we need some lemmas. For a bounded connected open set fi, if (fi)° = fi, then there exists a probability measure JJ, such that support /x = T := dfl satisfying f(z) = f fdfi

for all / analytic on H

[Herrero, D.A. (1974)].

Denote N(T) :=the "multiplication by A" on L2(T,fi) and H2(T) := "the subspace generated by all analytic functions on fi". Then H2(T)£LatN(T) and N(T) =

N+(T)

o

Z

w_(r)

H2(T) L2(r»aH"2(r) •

Lemma 6.1.2 (i) N(T) is normal, and N+(T),N-(T) normal; (ii) ^ ; ( r ) 6 B i ( f i ) ; (in) A!(N+{T))^H°°{Q). [Conway, J.B. (1978)]

are essentially

Lemma 6.1.3 Let G = Z and (G,G+) be an ordered group. Then there exists an isomorphism <j>: G—»Z such that (/>(G + )cN. Proof By the definition of ordered group, we can assume that there is an n, 0 < n£G+. If 0 > raeG+, then {-m)GeG+ and mneG+. 203

204

Structure of Hilbert Space

Operators

Note that {—m)n + mn = 0, by the definition of ordered group again, we have {—m)n = 0 and mn = 0. Thus G + c N . Suppose that (f> is the identity isomorphism from G onto Z, then (G+)cN. Now we are in a position to prove Theorem 6.1.1. Proof of Theorem 6.1.1 For the given Q, by Theorem 5.1.6 K0(A'(N+(r)))S£Z. It follows from Lemma 6.1.2 that K0(H°°(Q,))^Z. By Lemma 6.1.3,

\/(A'(N+(r))) = \J(H°°(n))cN. We need only to prove that \J(A'(N+(T)))^N. Set P = diag{I,0,0,---)€Moo{A'{N+(T))) and r = [P}£\f(A'(N+(T))). Then r is a positive integer, since A'(N+(T)) is stably finite. Suppose that q£Mn(A'(N+(T))) is a nonzero idempotent, then 0^[q] =

ss\/(A'(N+(T))).

Suppose that B = (N+{T))^\pnn, then A'(B) is ^-homogeneous and k>l. Note that rs = r[q] = s\p], there exists n'>n such that Q = diag(q,q!,--in Mn(A'(N+(T))).

,qr,0,---

,0)~adiag(P,Pi,

• • • ,P S ,0, ••• ,0) = P

By Lemma 4.2.4, p>{r) _

A(n')\

,,^4(n')|

, „ — 4(s)

Thus ^ ' ( B ( r ) ) ^ ^ ' ( A ( s ) ) , i.e., M r ( > t ' ( B ) ) S M , ( / ( y l ) ) . Since M r (^t'(B)) is rfc-homogeneous and M.,(.4'(.<4)) is s-homogeneous, s — rk. Therefore, \/(A'(A)) Since (KQ(A'(A)),\/(A'(A)))

= {kr: k = 0,1,2,- • • }. is an ordered group, r

=

1.

Thus

V(.4'(;v+(r)))=N. In Theorem 5.1.6, we have directly proved that \/(A'(N+{T)))^N. In the proof of Theorem we give a new proof by using a different method. Let 0 be an analytic connected Cauchy domain and W22(Q) denote the Sobolev space W22(ST) = {f&L2(Cl,dm) : the distributional partial derivatives of first and second order of / in L2(Q,dm)}, where dm us the planar Lebesgue measure.

Some Other Results About Operator

205

Structure

Set feW22(Q)},

W(Q) = {Mf :

where Mf is "multiplication by / " on W22{Q). Denote R(£l) the subspace of W 22 (fi) generated by the set of all rational functions with poles outside U. Note that R(Q)£LatM\. Denote M\(Q) = MX\R(Q.). By Proposition 4.5.1, ^'(M A (Q))=i?(fi). Theorem 6.1.4 K0(R{n))^Z and \/(R{Q))^N. Proof Since M^EBi(Q*), by asimilar argument of the proof Lemma 6.1.2 we can prove Theorem 6.1.4. For an analytic connected Cauchy domain fi, denote A(fi) = {/ : / is analytic in ft and /€C(fi). Then A(£l) is a unital Banach algebra with the norm ||/|| = max|/(.z)|. zen A(D), when Cl = D, is called disk algebra. Given an f£A(Q,), there exists a sequence rn of rational functions with poles outside Q, such that lkn-/|U(fi)—»0 Theorem 6.1.5 Proof

K0(A{D))^Z

and

(n—• oo). \/(A{D))^N.

We need only to show that for each idempotent P~adiag(l,l,---

in M 0 0 (A(D)), where k>n. Since

, l fc ,0, •••) PtMn{A(D)),

'fn(z)

fuiz)---

Jnl{z)

fnl{z)

fm(z)-

P= •••

fnn(z).

P£Mn(A(D)),

Structure of Hilbert Space

206

Operators

Let y = ^ + z and define

P'

7 n (£ + * ) / « ( £ + *)• • fin(m + z) ./nl(£ + *)/»2(£+*)-- fnn(rn + z) 7 n ( y ) / i 2 ( y ) ••• / i n ( 2 / ) " Jnl(y)

fn2(y)

•••

fnn(y)

Since ||P'—P\\ can be arbitrary small when m is big enough, P' is homotopic to P , denoted by P'~hP in M n (A(D)). Note that P'eMn(A(D)). It follows from \/(R(D))^N and the tftheory that P'~hdiag(l,l,--in

,lk,0,---

,0)

Min{A{D)). Let fti(t) : [0,1]—>Mn(A(D))

be a continuous map satisfying

h1(0) = P, /ii(l) = P ' and h,2(t) : [0,1]—>Min{A{D)) be a continuous map satisfying h2(0) = P', h2(l) = diag(l, 1, • • • , lfc, 0, • • • , 0). Denote hi{t) = (h(£)©0(3">), then /n(t):[0,l]—M 4 „(i4(2?)) is a continuous map and /H(0) = P©0( 3 "\ /n(l) = P'©0 ( 3 "). Thus /i 3 (t) = (fe2o/ii)(*) : [0,1]—>M4n(A{D))

satisfies

h3{0) = P, h3{l) = diag{l, 1, • • • , lfc, 0, • • • ,0). Therefore, \/(A{D))^N 6.2

and

K0(A(D))^Z.

Similarity and Quasisimilarity

We have the following question: Is A~B if A^n^B^n\ number.

where n is a natural

Some Other Results About Operator

207

Structure

Theorem 6.2.1 Let A,BeMk(C). Then A~B if and only Proof By matrix theory, we can easily obtain the theorem. Theorem 6.2.2

Let A,B££(H)n(SI).

ifAln)~BW.

If

V(A'(A))S\/(A'(B))SN, A'(A)/radA'(A) and A'(B)/radA'(B) are commutative, then A^B if and only if A^~B(n^ for each natural number n. Proof By Theorem 5.5.15, A~B if and only if K0(A'(A@B))^Z and by Theorem 5.5.15 again, \f(A'(A®B))^ where I = { J J

A

f^

\/{A'(A®B)(n))^NV\

. Therefore, A~B if and only if

A^^B^.

^ 0 it 7l~X3

T h e o r e m 6.2.3 Suppose that A = 0 A<-ni),B = 0 J9J mj) , where {Ai} t=i

i=i

and {Bj} are two families of (SI) and pairwise not similar operators. If foreachi andj, (l
Suppose that AeBn(tt)n(SI) W(In

for each XGC(H),

+ A) + (In +

A)X^In

then T =

with a(A) = T), and

In + A In l^SI) 0 -In - A

208

Structure of Hilbert Space

Operators

and ifker(IH + A) = {0}, then T^q.s.(In + A)®(-In - A). Proof In + AeBn(D(l, 1)), -(IH + A)&Bn(D(-l, 1)), where D(l, 1) = {zeC : \z - 1| < 1}, D(-l,

1) = {z£C : \z + 1| < 1}.

Thus I>(1, l)Cap(In Since £>(1, l)r\D(-l,

+ A), D(-l,

l)c<rp(-(In

+ A)).

1) = 0 and

\J ker(Iu + A - z) = \J ker{-In ze£>(i,i) zeD(-i,i)

- A •z) = n,

by Lemma 3.2.3,

Let P be a nontrivial idempotent in A'(T) and P n P12 P21 P22

Since (I+A)P2i+P2i(I+A) = 0 and P2\ = 0, P n and P22 are idempotents in A'(A). Since Ae(SI), Pn = 0 and P 2 2 = / or Pn = I and P 2 2 = 0. Assume that P n = 0 (otherwise consider I — P). If follows from PT = TP that P 1 2 ( / + ^ ) + (7 + A)P 1 2 = - P 2 2 . By the assumption of the proposition P 2 2 = 0. This shows that Set X

2(1 +A) I 0 I

n n

Y =

I I 0-2(1 +A)

H

Te(SI).

and

Then XT = ((I + A)®(-I - A))X,TY implies that T~ g . s .(/ + A)®(-(I + A)).

w

= Y((I + A)®(-I

- A)). This

Example 6.2.6 Let S be the injective unilateral shift on H = I2. Then S*£Bi(D). Let { e n } ~ = 1 be the ONB ofH and Sen = en+1 (n = 1,2, • • •).

Some Other Results About Operator

If for some Xe£{H),

209

Structure

X(I + S*) + {I + S*)X = I, then XS* + S*X + 2X = I. n

A simple computation indicates that Xen

= \ ^2, ( — l) f c _ l e n+i-fc ( n

=

fc=i

1,2,- ••) and\\Xen\\ = ^ —>oo (n—»oo). Thus X(I+S*) for all XeC{H). By Proposition 6.2.5,

+ (I+S*)X

^ I

I + S* I 0 -(I + S*) H

n

and Y =

I I 0 - 2 ( 7 + A)

ne{SI)-

Theorem 6.2.7 Suppose that AeBn(n)r\(SI) and B~q.s.A, then B<=(SI). Proof Without loss of generality, we assume that the minimal index of A is n. Since B~qs.A, there exist operators X and Y, with trivial kernels and dense ranges, such that AX = XB and YA = BY. A simple computation indicates that Q, £ crp(B) and dimker(B-z) = n for all zeQ.. Note that AXTY = XBTY = XTYA for each TeA'(B). This implies that XTY&A'(A). Since n is the minimal index of A, by Theorem 4.4.3, cr(XTY\ker(A_z)) is connected for each z€Q. Since YA = BY,Y(z) is a linear transformation from ker(A — z) to ker(B — z), where Y(z) = y\ker(A-z)-

Similarly, X(z) = X\ker(B-z) to ker(A — z). Since

*s a linear transformation from

AXY = XBY

=

ker(B-z)

XYA,

XYGA'(A)

and T{XY){z)

=

X(z)Y(z)

for all z£tl. Similarly, YXeA'{B) and T{YX)(z) = Y(z)X(z) for all zeQ. By Theorem 4.4.3, a(X(z)Y(z)) = {X(z)},X(z) ^ 0. Since X,Y are injective, Y{z)X(z) is invertible. Thus a(Y(z)X(z)) = {X(z)}.

210

Structure of Hilbert Space

For arbitrary TeA'(B),

since

T(XTY)(z)

XTY&A'(A),

=

and a(X(z)(T\ker(B-z))Y(z))

Operators

X(z)(T\ker(B_z))Y(z)

is connected. Assume that

a(X(z)(T\ker(B_z))Y(z))

= {»(z)},

then a(Y(z)X(z)(T\ker(B_z)))

= {/x(2)}U{0}.

{Claim 1} If n(z) = 0 for some zed, then cr(T\ker(B_z)) = 0. The claim can be proved as follows. Since Y(z) and X(z) are invertible, 0£a(T\ker(B-z))If 0 ^ is n o t Xea(T\ker(B-z)), then XIker(B-z) -T\ker(B-z) invertible. Repeating the argument above, we have a(X(z)(\Iker{B_z)

-T\ker{B_z))Y(z))

= {0}.

tr(X(z)(XIker{B_z)

-T\ker(B_z))Y{z))

= {0}.

Thus

But
T\ker{B_z))Y(z))

= tr(AX (z)y(z) - x(z)(r| fcer(B _,))y( z )) =

\tr(X(z)Y(z))-tr(X(z)(T\ker(B„z})Y(z))

= n\X(z) ^ 0 A contradiction. {Claim 2 } If there exists z£Q, such that /J,(Z) ^ 0, then <^{T\kerrB_z\) is connected. Suppose that /3(z)£a(T\ker{B-z))> then P(z)Iker(B-z)

~

T\ker(B-z)

is not invertible. Repeating the proof of Claim 1, we have a(X(z)((3(z)Iker{B_z)

-T\ker(B_z))Y{z))

= {0}.

Some Other Results About Operator

By Claim 1, (r{T\ker{B_z)) U that

211

Structure

= {(3{z)}. It follows from V

V ker(T - /3(z))n = H.

ker B

(

~ z) =

z£D

This implies that a(T) is connected.

z€D

Therefore <x(T) is connected for each TeA'(B). {Claim 3 } B£{SI). Otherwise, there exist M,NeLatB such that MnAf = {0} and M + M = H. Denote Bx = T\M,B2 = T\#, then B = By+Bi. Set T = IM+2I//, then TeA'(B), but a{T) is disconnected. This contradiction implies that Be (SI). Proposition 6.2.8

Suppose that AeBn(Q)n(SI)

and B~q.s.A,

then

A'{B)/radA'{B) is commutative. Proof Without loss of generality, we still assume that the minimal index of A is n. Then Qcap(B), dimker(B — z) = n (z€fi) and \J ker(B — z) = H. For X,Y€A'(B), {XY

— YX)\ker(B_z)

= X\ker(B-z)Y\ker(B-z)

—

Y\ker(B-z)X\ker(B-z)

for all zefl. It follows from Claim 2 in the proof of Theorem 6.2.7 that a((XY-YX)\ker{B-z)) for all zeO. A'(B)/radA'(B)

Thus ((AT - YX)\ker{B_z))n is commutative.

= {0} = 0 for all zed

and

Next we will discuss the similarity of irreducible operators. Since irreducibility is unitarily invariant, some important behavior of C*-algebras and irreducible operators can be described in terms of irreducible C*algebras and irreducible operators. F. Gilfeather proved that if Ne£(H) is a normal operator with empty point spectrum, then N is similar to a irreducible operator. [Jiang, Z.J. and Sun, S.L. (1992)] proved that each self-adjoint operator with infinitely many point spectrum is similar to a irreducible operator. D.A. Herrero posed the following conjecture in [Herrero, D.A. (1979)]: An operator QeCCH) is similar to an irreducible operator if and only if (i) X — Q is not finite rank for each complex number A;

212

Structure

of Hilbert Space

Operators

(ii) Q does not satisfy any quadratic equation ax2 + bx + c = 0, where |a| + |6| + | c | ^ 0 ; (iii) A — Q is not a direct sum of a finite rank operator and an operator satisfying a quadratic equation for A G C. In the following we will answer Herrero's conjecture in the case of nilpotent operators, normal operators and Cowen-Douglas operators. Denote Afk = {TeC{H) : Tk = 0 and T ^ 1 ^ 0}. For TeAfk(H), T admits a kxk operator matrix representation of the following form: 0 T 1 2 T13 • • Tifc-i 0 T230

T =

' • Tak-i

0

kerT © kerT0 kerT2 © kerT1 kerT3 © kerT2

Tifc

Tzk

0

Tk-ik 0

kerT''-1 0 kerTk kerTk © kerTk

where fcerT0 = {0},kerTk = H, kerTj-i j = {0},j = 1,2, ••• ,fc [Davidson, K.R. and Herrero, D.A. (1990)]. Using Lemma 7.8 of [Herrero, D.A. (1990)], we can prove that TGAfkCH) (k>3) is similar to an operator of the form 0 Ti Ti2 • • • Ti fc_2 Ti k-i 0 T2 • • • T2 fc_2 T2 fc_i W2 0

T3 k-i T-j, k-\

0

where fcerT, = {0} (z = 1, 2, Or

0

(6.2.1)

Tik-i Hk

, k — 1) and ranT\,ranTi

0 Ti T 12 0 T2 T~0Hl(

Tk-x 0

W3

are dense.

Ti fc_2 Ti fc_i ?2 fc_2 T 2 fc_l T3 k-2 T3 fc_i

0

(6.2.2)

Some Other Results About Operator

Structure

213

where 0-^ is the zero operator acting on H\,dimrii = n, (l
(6.2.3)

and .F has finite rank.

Theorem 6.2.9 Given TGAfkCH), T is similar to an irreducible operator if and only if the following conditions are satisfied: (i) T is not finite rank; (ii) T 2 # 0; (Hi) T ^ T\®F, where TiGAfk{H) and F is finite rank. Before we prove Theorem 6.2.9, we need several lemmas. Lemma 6.2.10 [Kato, T. (1984)] Suppose that A,BeC(H). If A is positive with kerA = {0}, then there exists a positive number S such that ker(A + XB) = {0} for |A| < S. Lemma 6.2.11 Given A, BGC(H). If A is positive with kerA = {0}, then there exists a positive number A such that ran(XA + B) is dense in "H. Proof Since ran[XA + B]~ = [ker(XA + B*)]-L, we need only to show that ker(XA + B*) — {0} for some positive number A. Note that XA + B* = X(A + iJB*). Thus Lemma 6.2.11 follows from Lemma 6.2.10. Lemma 6.2.12 Suppose that A,B&C(H), A is a positive operator with kerA = {0} and cr(A) is uncountable. Then there exists an operator E&C{H) such that the C*-algebra generated by A,AE + B and I is irreducible. Proof Since cr(A) is uncountable, there exist disjoint uncountable Borel uncountable setsCTi,0"2such that cr(A) = aiL)a2 and 0 ^ 2 - Suppose that E is the spectral measure of A satisfying E{a{)E{a2) = 0. Then dim(E(ai)H)

= oo, and dim(E(a2)H)

= oo.

Denote A =

Ai 0 0 A2

E{a1)H E{a2)H'

where A\ = A\E(ai)H> ^2 = ^|£(a2)W- Then A\,A2 are positive operators and kerA\ = kerA2 = {0}. Since 0 ^ 2 ) A2 is invertible.

214

Structure of Hilbert Space

Operators

Assume that B =

Bn B12 B21 B22

E(a2)H'

By Lemma 6.2.11 ran(XAi + B12) is dense in E(ai)Ti for some A > 0. Set E =

0 -A2-1B2i-A2-1(B22-(d

A + V))

where d > \\Bu || + 1 , V is the Volterra operator. Thus V is irreducible and a(d+V) = {d}. AE + B =

0 XAi -B21 d + V-B22_

+

Bn B\2 B21 B22_

?n A A i + 5 1 2 " 0 d+V We are now to prove that the C*-algebra generated by / , A and AE + B is irreducible. Suppose that P is a projection commuting with A and AE + B, and P =

P11 Pia P21 P22

Since PA = AP, P21A1 = A2P21 and A1P12 — P12A2. It follows from E{ai)E{
Bn XAi + B\2 0 d+ V

Bn XAi + B12 0 d+V

PiO 0 0

Thus Pi(A^4i + P12) = 0. Since the range of XAi + B12 is dense, P x = 0 and P = 0. Therefore the C*-algebra generated by A, AE + B and / is irreducible. Lemma 6.2.13 Suppose that A,BG£(H), kerA = {0} and a(A) is uncountable.

A is a positive operator with Then there exists an operator

Some Other Results About Operator Structure

215

E££(H) such that the C*-algebra generated by EA + B,A and I is irreducible. Proof The lemma follows from Lemma 6.2.12. Lemma 6.2.14 Let Ai,A2,B£C(7i). If Ai is a positive diagonal selfadjoint operator, A\ is a positive operator and kerA\ = kerA^ = {0}. Then there exist X, E££(H), X is invertible, such that: (i) A2X is a positive diagonal self-adjoint operator with trivial kernel ker(A2X)

= {0};

(ii) The C*-algebra generated by AiE+BX, A2X and A\ is irreducible. Proof Since A2 is a diagonal operator, A2en = Xnen for some ONB {enj^Li- Since kerA2 = {0}, A n ^0 for all n > l . Assume that A\ admits the following matrix representation with respect to the ONB {e„}J° =1) an ai2 0.21

a

0-2r>

ii

Ai = Onl 0-n2

-

'

and each row vector (aji,Oj2> • * • ,Qjn, • • • )^0. Otherwise, if (a.,1,0,2, • • • , a.jn, • • •) = 0. Since A\ = A\, (o-ij,

&2j, • • • , a,nj , • • • ) = 0 .

This implies that 0£ap(Ai) and contradicts kerA\ = {0}. Without loss of generality, assume that aij1^0. Set

u

ei

h

0

0

tn

X=

where tj£[l, 2] satisfying Xit^Xjtj Clearly, X is bounded.

{i^j).

e2

216

Structure of Hilbert Space

Operators

Assume that ~bu

&12 •

&21 &22

-

•

b i

•

b

n

• ••'

2 n

•••

BX = bnl

bn2 • ' ®nn •

:

Set e n e i 2 • • e i „ •• e2i e22 • • e2« • •

e n i e„2 • • ^nn

where ejin (j,k)^(j\,n),

= y°+ny, an € (0,1] such that ahejin + 6ln^0. If then e^ = 0. It is easy to see that E is bounded. Since «i

a2

0

0

a„

AoX =

where an = Xntn and {a„} are pairwise distinct. Thus ker(AzX) and A2X is a diagonal self-adjoint operator. Assume that ^11 ^ 1 2 •

• Wn •

h\ h2 •

• hn •

-

' 'un *

= {0}

^iE + SX = Inl ln2

where lXn = ai^e^n + bln^0 n = 1,2, • • •. We now prove that the C*-algebra generated by Ai,A2X is irreducible.

and

A\E+BX

Some Other Results About Operator

Structure

Suppose that P is a projection commuting with A2X, A% and Since P commutates with A%X,

217

AiE+BX.

Pi

Vi

0

0

pn

Since P2 = P,Pi = 0 or / , i = 1,2, • • •. Without loss of generality , we assume that pi = 0 (Otherwise, consider I — P). From P commutates with AiE + BX, i.e., Pi P2

0

j? n

hi

hi •

hn

hi

I22 •

hn

hi hi

' l l ^12 '

• 'in

- -

'21 ^22 •

• hn

• •

hi 'n2 '

' Inn ' '

Pi •••

*nn ' '

P2

0

Pn

We have pi/12 = '12^2Since p\ — 0 and '127^0, we get pi = 0. Similarly, since pihn = hnPn and Zi„^0, pn = 0 (n = 1,2, • • •). Thus P = 0. This implies that the C*-algebra generated by A\,A2X and ylii? + BX is irreducible. Lemma 6.2.15

[0 0 0 = kerA2

Assume that T =

Ai 5 ] w 0 A2 H, where Ai and Ai are 0 0 . H = {0}. Then T is similar to an

positive operators and kerA\ irreducible operator. Proof (1) If o~{A\) is uncountable, by Lemma 6.2.12 there exists an operator Ee£(H) such that the C*-algebra generated by Ai, A\E + B and / is irreducible.

218

Structure of Hilbert Space

Operators

Define 10 0 0 1 -E 00 I

X then

70 0 01 E 00 I_

xDenote 1

Ti = XTX-

=

0 Ai AiE + B 0 0 A2 0 0 0

We are now to prove that T\ is irreducible. Suppose that P is a projection commuting with Ti and P n P12

Pis

P21 P22 P23 •P3I A 2 P33

It is obvious that kerT^LatP,

kerTi = W©080, fcerTj2 = ft©W©0. Thus

Pn = P31 = P32 = 0. Since P is a projection, Pi 2 = P13 = P23 = 0. Therefore we may assume P =

Pi 0 0 0 P2 0 0 0 P3

Since PTi = T i P , P u 4 i = AiP 2 . So A : P i = P2Ax. Thus Pi A? = P i ^ i ^ l i = A1P2A1

= A\PX.

By functional calculus, P ^ i = A\Pi. Thus AiPi = A ^ , i.e., A\{Pi P2) = 0. Since A\ is positive and kerA\ = {0}, Pi — P 2 = 0. Thus pi = p 2 . Similarly, we have Pi = P 2 = P3 and Pi 0 0 P = 0 Pi 0 _ 0 0 Pi.

Some Other Results About Operator

Structure

219

Since the C*-algebra generated by A\, A\ E + B and 7 is irreducible, Pi = 0 or 7, and therefore P = 0 or 7. Thus T is similar to the irreducible operator T\. (2) If
Gi =

70 0 07 0 0 0X"1

G;1

70 0 07 0 0QX

then

Thus, GxTGl - l

'QAi 0 0 0 0

BX A2X 0

Set G2 =

70 0 0 7-£ 00 7

G?

70 0 07£ 00 7

G2

OAi AXE + BX 0 0 A2X 0 0 0

then

Therefore,

G2G{TGX

Clearly, G 2 GiTGf ^ J ducible operator.

1

is irreducible. Thus T is similar to an irre-

220

Lemma 6.2.16

Structure

of Hilbert Space

Operators

Let 0 Ax A12 A13 • • Aln 0 A2 A23 • • A2n 0 A3 • • A3n

T = 0

' • An 0

If Ax,A2 are positive operators and kerAk = {0} (A; = 1, 2, • •-,n). is similar to an irreducible operator. Proof Set

ThenT

OAx Ax2 0 0 A2 0 0 0

Tx

By Lemma 6.2.14 and Lemma 6.2.15,there exists an invertible operator Xx such that

XxTxX^

=

0 Ax M2 0 0 ~A2 0 0 0

e(Sl),

where Ax and A2 are positive injective operators. Therefore, there exists an invertible operator X such that 0 Ax Ap Ay3 0 A2 A23 XTX

-l

0

A3

••

Mn

••

A2n

'••

A3n

An 0 where Ax and A2 are positive and kerAk = {0}, k = 1,2, • • • , n. Now we prove that XTX~l£{RI). Suppose that P is a projection commuting with XTX-1. By the argument similar to that in the proof of

Some Other Results About Operator

Structure

221

Lemma 6.2.15, we can prove that Pi Pi

0 P3

0

p„+1 Since X T i X f 1 £ ( # / ) , Px = P2 = P 3 = 0 or / . Without loss of generality, assume that Pi = P 2 = P 3 =_0. Since PXTX~l = XTX~XP, P3A3 = I3P4 and A3P4 = 0. Thus kerA3 = {0}. This implies that P 4 = 0. Similarly, Pi = P2 = P3 = P4 = • • • = Pn+1 = 0. Therefore T is similar to an irreducible operator. Lemma 6.2.17 Suppose that TeJ\fk{H) is of the form (6.2.1), then T is similar to an irreducible operator. Proof Without loss of generality, assume that OTi 0 T2 * 0 T3 T = '••Tk

0 where kerTj = {0} (j = 1,2, •• • , k) and ranT\ and ranT2 are dense. By the Polar Decomposition Theorem, T\ = A\Ui and U\T2 — A2U2, where Ai is positive, kerAi = {0} and Ui is a unitary operator (i = 1,2). Set

C/j

0 U2

G =

I 0

222

Structure

of Hilbert Space

Operators

then OAi 0 0 A2 * 0 0 0 T3 GTG-1

=

oooo'-. II I '•• '•• Tfc_i 0 0 0 0 ••• 0

By Lemma 6.2.16, T is similar to an irreducible operator. Lemma 6.2.18

Suppose thatT is similar to 0-H^TI,

\QA1 B~\ 0 0 A2 0 .

.° °

where dirriHi = oo.

n n, n

where A\ and A2 are positive operators, and kerA\ = kerA2 T is similar to an irreducible operator. P r o o f Without loss of generality, we can assume that 00 0 0 T = 0Hl®Ti

=

OOAx

Set

E

I -7 0 0 0 7 00 -7 0 / 0 0 0 07

Then

E~

B

0 0 0 A2 00 0 0

7700 0700 7770 0007

{0}, then

223

Some Other Results About Operator Structure

and -Ai Ai 0 0

ETE-1 =

-Ax -Ax Ax Ax 0 0 0 0

-B B A2 0

Suppose P be a projection commuting with ETE

1

and

Pn P\i Pn Pu Pi\ P22 P23 P24 P31 P32 P33 P34 P41 P42 P43 -P44

It follows from PETE'1

= ETE~lP

-PxxAx + P12Ax -PxxAx -P31A1 + P22AX -P21Ax -P3lAx + P32AX -PsxAx -PAXAX + P42AX -PixAx S

that

+ Px2Ax -PxxAx + P22AX -P2xAx + P32Ax -PzxAx + P42AX -PAXAX

+ P12Ai -PnB + P22A1 -P21B + P32A1 -P31B + P 4 2 ^ i -P41B

+ P12B + P13A2+ P22B + P23A2 + P 3 2 P + P33A2 + P 4 2 P + P43A2.

T (6.2.4)

W

V

Where S

=

T =

-AxPxx - AxP2x - AxP3x - 5 P 4 i -AxP12 - AxP22 - AxP32 - BP42 AxPxx + AxP2x + AxP3x + BP41 AxP12 + AxP22 + AxP32 + BP42 -AxPxs - AxP23 - AxP33 - -BP43 A1P13 + A i P 2 3 + ^ i P 3 3 + BP43

W =

A2P41 0

V =

A2P43 A2P44

0

- AxP2i - AxPM - PP44 AxPx4 + AxP24 + AxPzi + BP44

-AXPXA

A2P42 0

0

Compare the (4, 1) entry of (6.2.4), we have -PixAx

+ P42AX = 0.

224

Structure of Hilbert Space

It follows from [ran^i]

Operators

= Ti that P41 = P 4 2 .

(6.2.5)

Compare the (4, 4) entry of (6.2.4), we have -P41B+P42B + P43A2 = 0. By (6.2.5), P43A2 = 0. Since [ranA2}" = H, P43 = 0.

(6.2.6)

Compare the (3, 3) entry of (6.2.4), we have -P31A1+P32A1 = A2P43 = 0. Thus P31 = P32.

(6.2.7)

This indicates that

w=\00[ '0 0 J ' A2P41

= A2P42

= 0 and

P 4 1 = P 4 2 = 0.

(6.2.8)

Note that the sum of the (1, 3) entry and (2, 3) entry of the right side of (6.2.4) is 0. Thus -(P11 + P12 ~ P21 + P22)B + P13A2 + P23A2 = 0 and P13A2 + P23A2 = 0, i.e., {P13 + P23)A2 = 0 or P13 + P23 = 0. By (6.2.7) and A* = A,P13 = Pai.fta = P32 = P31. Thus P13 = P23 = 0.

Some Other Results About Operator

Structure

225

Therefore the (6.2.4) can be written as follows:

-PuAi + P12A1 -PuAi + P12Ai - P n A i + P12Ai -PuB + P12B -P21A1 + P22A1 -P21A1 + P 2 2 Ai - P 2 i A ! + P22A1 - P 2 1 P + P22B 0 0 0 P33A2 0 0 0 0 -AxPxi - A1P21 -AxPi2 - AXP22 -A1P33 -PP44 A1P11 + A1P21 A1P12 + A1P22 AxP33 BP44 0 0 0 A2P44 0 0 0 0

:= (/?«)

Since a 2 i = #21, -P21A1 + P22A1 = A i P n + A1P21 or AiP 2 i + P2iAi = P22AX - A i P u .

(6.2.9)

It follows from a 12 = /?i2 that -P11A1 + Pi 2 Ai = -A1P12 -

AXP22.

Taking the adjoint of both sides, we get —AiPu + A\P2\ = —P2\A\ — P22A\ or A1P21 + P21A1 = -P22A1 + A i P u . Prom (6.2.9) and (6.2.10) we get P 2 2 Ai = A i P u . Repeating the proof of Lemma 6.2.15, we can prove that P\\= P 22 Ai = AXP22. Since A2+/S22 = 0, a? 12 +a 2 2 = 0 or ~PnAi 0, i.e., P12A1 - P21A1 = 0. Thus P21 = P 12 Since a 2 2 = (322, -P2lAx

(6.2.10)

P22.TI1US (6.2.11)

+P12A1-P21A1 +P22At

=

(6.2.12)

+ P22AX = A1P12 + A1P22. By (6,2,11), P21A1+A1P12=0.

By (6.2.12), A1P12 + P12A1 = 0.

(6.2.13)

226

Structure of Hilbert Space

Operators

Note that P* = P. Thus P{2 = P 2 i and P i 2 is self-adjoint. Thus P?2 is a positive operator. From (6.2.13) we know Pi2Ai = —P12A1P12 = AiP^2. Thus P12AX = AiP12. From (6.2.13), PuAi = 0. Thus P12 = 0 and P 2 i = 0. The (6.2.4) can be written as - P i i i 4 i -PuAi

-PuAi

-PuB

P22A1

P22A1

P22A1

P22B

0

0

0

P33A2

0

0

0

0

-AtPu AiPn 0 0

-AXP22 AXP22 0 0

-A1P33 -BP44 AXPSA BP44 0 A2P44 0 0

By a proof similar to the proof of Lemma 6.2.15, Pi 1 and XTX~1G(RI) for some invertible X. Lemma 6.2.19

P22 = Pi33

44

Suppose that 0 i 4 i A12' 0 0 A2 0 0 0 H

n n,

T = (W where A\ and A2 are positive T is similar to an irreducible Proof If dimHi = °°, then If dimHi < 00, say dimHi

operators, and kerA\ = kerA2 = {0}. Then operator. by Lemma 6.2.18 the conclusion is true. = n. Denote 0 Ai A12 0 0 i2 0 0 0

Then by Lemma 6.2.15 there is an invertible operator X\ such that

A = X1AX11

=

0 Ai An 0 0 32 0 0 0

e(Ri),

where Ai and A2 are positive operators, and kerAi = kerA2 = {0}.

Some Other Results About Operator Structure

227

Thus "0 0 0 0 " 0 0 Ai An H = = Ti 0 0 0 A2 H .0 0 0 0 H

m

l

XTX~

for some invertible X. Let be an ONB of Hu / i , / 3 , • • • Jn&kerA EAh

0 0 0A

w<3>

ei,e2)-

QkerA,EeL{n{3),Hi)

such that

x

= eu EAf2 = e 2 , • • • , Elfn

= e n and £([ V 3 / * ] ) = 0. fc=i

Denote

Y =

IE 0 /

Hi W<3)

and 0E_A 0 3

T2 = Y T j y - 1

WW

Now we are to prove that T2£(RI). Suppose that P is a projection commuting with T 2 . Denote M = (0®kerA)f)PkerT2,

Af = {0®kerA)n{I -

M' = PkerT2eM,

N' = (I -

P)kerT2,

P)kerT2QN.

It is easy to see that dimM.1 = h
V x£M',

n

x = ^caei

+ zx, zx£M,

V y€Af', y = ^faei

i=l

+ z2, z2GAf.

i=l

Thus, for ei£Hi, et = x + y, x£M',

y&A/"', where

n

n

x = ^ a , e i + z\, zxeM,

y = ^ f t e j + z2,

i=l

i=\

Since A4JJV, we obtain x = 0 or y = 0. This implies that ei£PkerT2 or eiG(7 - P)kerT2

(i = 1,2, •• • , n).

z2eN.

228

Structure of Hilbert Space

Operators

Without loss of generality, we assume that e

i , e 2 , • • • ,ek€PkerT2,

€(I-P)kerT2.

By a routine proof, P admits a block upper triangular matrix representation with respect to the decomposition H\®H®'H®'H. Since P is a projection, Pi 0 P = 0 .0

0 P2 0 0

0 0 P3 0

0 0 0 PA

Note that A e ( P J ) , thus P 2 = P 3 = P 4 = 0 or I. Without loss of generality, assume that P2 = P3 = P4 = 0. Since PT2 = T 2 P, P\EA = 0. But .EM is surjective, thus P\ = 0 and P = 0. So T is similar to an irreducible operator. The proof of next lemma is similar to the proof of Lemma 6.2.17. Lemma 6.2.20 Suppose that TeNk(H) and T = 0 W l ©A is of the form (6.2.2). Then T is similar to an irreducible operator. So far we have proved the sufficiency of Theorem 6.2.9. As for the necessity, we need only to show that if T2 = 0 or ranT is finite rank, then

TftBI). (1) If T2 = 0. We may assume that Q

0A H 00 H

and dirriH = oo. By the proof of Lemma 6.2.15, Q=

QA 0 0

where A is a positive operator. Suppose that Pi is a nontrivial projection commuting with A. Set P = Pi©P2, then P commutes with Q. (2) If T is finite rank, then T* is finite rank. Denote M = y{ranT, ranT*}, then M. is a reducing subspace of T. Thus the proof of the necessity of theorem 6.2.9 is now complete, Next, we are going to discuss normal operators.

Some Other Results About Operator

Structure

229

Theorem 6.2.21 Suppose that N€.C(H) is a normal operator. Then N is similar to an irreducible operator if and only if the following conditions are satisfied. (i) X — N is not finite rank for all X £ C; (ii) N does not satisfy any quadratic equation ax2 + bx 4- c = 0, where \a\ + \b\ +

\c\?0.

In order to prove Theorem 6.2.21, we need several lemmas. Lemma 6.2.22 Suppose that AE£(H) and each operator in the similarity orbit S(A) of A is irreducible, then each operator in S(A"(A)) is also irreducible, where S{A"(A)) denotes the similarity orbit of A"(A). Proof Assume that Be A" (A) and X is invertible. Since XAX~X <£ (RI), XAX~l commutes with some nontrivial projection P, i.e., P&A'{XAX~l). l Thus X^PXeA'iA) and therefore BX~ PX = X~lPXB or 1 1 XBX^P = PXBX- . Namely XBX' 0 {RI). Lemma 6.2.23 Suppose that Ai€C(H) (i = 1,2,3) and A = Ai®A2®A3 satisfying kerrA A. = {0} (i ^ j). Then A is similar to an irreducible operator. Proof By Lemma 6.2.22, it is sufficient to show that there is some operator in A"(A) that is similar to an irreducible operator. Since kerrA_ A, = {0} (i •£ j), calculations indicate that A'{A) = {Bi©B 2 ©-B 3 : Bi&£(H),i

= 1,2,3}.

Therefore, it is sufficient to prove that B = ai©a2©«3 is similar to an irreducible operator, where {«i}f=1 are pairwise distinct complex numbers. Define T=

"<*i 1 D' 0 a2 1 0 0 a3_

where D is an irreducible operator. Suppose that P&A'(T) is a projection and Pll

Pl2

Pl3

P21 P22 P23 P31 P32 P33

230

Structure of Hilbert Space

Operators

Since {ai}:f=1 are pairwise distinct, Ptj = 0 (i ^ j) and P n = P22 = P33. It follows from PUD = DPn and De(RI) that P n = P 2 2 = P33 = 0 or / . Thus TG(RI). It is easy to see that T^a1®a2®as. Lemma 6.2.24 Suppose that A i € £ ( C m ) (m < 00), A 2 , ^ S ^ W ) satas/2/m# kerrA. A. = {0} (i ^ j ) , t/ien A = Ai®A2®A% is similar to an irreducible operator. Proof By Lemma 6.2.22, it is sufficient to prove that B = a\@a.2@a3 is similar to an irreducible operator, where {&i}f-i are pairwise distinct complex numbers. Without loss of generality, we assume that % = L2(0,1). Define ai F 0 0 a2 M 0 0 a3 where M is "multiplication by the idempotent variable", i.e., (Mf)(t) tf(t), F is a surjective operator from L2(0,1) to C m given by

=

Ff = ([ tf(t)dt, f t2f(t)dt,-.- , / tmf(t)dt), /eL 2 (0,l). Jo

Jo

Jo

It is not difficult to see that T ~ P . It is sufficient to prove that T£(RI). Suppose that PGA'(T) is a projection. Since on 7^ aj (i ^ j), P = Pi©p2©P3, where each P is a projection and P2M = MP3,PiF = FP2. Thus P2M2 = MP3M = M{MP3)* = M(P2M)*

= M2P2.

Therefore P2M2k = M2kP2

for fc>l.

Using functional calculus we have P2M = MP2 and P 2 = P3. The spectral theory of self-adjoint operators asserts that there exists a Borel subset Ec[0,1] such that P2f = XB! for all f£H, where \E denotes the characteristic function of E. If the Lebesgue measure of E, 11(E) = 1, then P 2 = P3 = 1. Since P\F = FP2 and since F is surjective, P\ = 1, i.e., P = 1. If n(E) < 1, let Ai,A 2 ,--- ,A m be nonzero, pairwise distinct Lebesgue points of F = [0,1]\ (AeP is a Lebegue point of F if limn{Fn[\ - e, A + s})/2e = 1).

Some Other Results About Operator

231

Structure

Since the matrix Ai A2

• • Am

Aj A2

1

• A_, A"

is invertible, there is a 8 > 0 such that the matrix (aij)mXm is invertible provided that | a y — AM < 5 (i, j = 1,2, • • • , m. Choose a sufficiently small £ > 0 such that '2e

fdt-XU

/

<6, i,j=

1,2, •

,m.

[\i-e,\i+£]nF Set

fj = i^XlXi-e.K+e^F-

Since EnF = 0,

P1Ffj = FP2fj = FxEfj

= 0

(j = 1,2, • • • , m).

On the other hand, if {efc}jj=1 is an ONB of C m , we have 0 = P1Ffj = P ^ tfjdt, fi t2fjdt, ...,fi

fc=l

E(i

[Ai-e,Ai+e]nF

/

k=l

tmfjdt)

tkdt)Pxek.

[\i-e,Xi+e]nF

Define Oti

2e

I

tldt,

{\i-E,Xi+e]r\F

then (ay) m xm is invertible. Thus Piek = 0 (A; = 1,2, •••,m), i.e., P x = 0. Since F P 2 = P\F,FP2 = 0. Note that FP2e = (^tXEdt,J^t2XEdt,--,^tmXEdt), where e&H2,e(t) = 1. Therefore fi{E) = 0, i.e., P 2 = -P3 = 0 and P = 0. Thus T€(PJ"). Now we are in a position to prove Theorem 6.2.21. Proof of Theorem 6.2.21 By the arguments similar to that used in the proof of Theorem 6.2.9 we can prove the necessary condition of the theorem. Now we prove the sufficient condition. If (i) and (ii) of the theorem are satisfied, then there are three possibilities:

232

Structure

of Hilbert Space

Operators

(a) a(N) consists of infinitely many points; (b) ae(N) consists of at least three points; (c) cr(N) is a finite set, cre(N) consists of two points and a(N)\cre(N)

^

0. In cases (a) or (b), there exist pairwise disjoint Borel sets <Ji,(T2 and 03 such that a(N) = 0-1U0-2U0-3 and dimE(ai) = 00

for

i = 1,2,3,

where E(-) is the spectral measure of N. Set Nj = N\E(ai)H (* = li 2,3). Then kerrN^N, = {0} (i ^ j). By Lemma 6.2.23, N is similar to an irreducible operator. In Case (c), assume that cre(N) = {Ai, A 2 }. Thus dimE({\i})H

= 00 (i = 1,2)

and 0 < dimE{a(E)\ae(N))

< 00.

By Lemma 6.2.24 N is similar to an irreducible operator. Proposition 6.2.25 Given T££(H) such that A'{T) is abelian, T is similar to an irreducible operator. Proof (i) Suppose that there exists a projection P&A'(T) such that dimPTi = 00 and A =

T\PHe(RI).

Fix a \<jLa{A). Then \®Ae£({PH)s-®PH)£A'(T). Since A'{T) is abelian, A'{T) = A"(T).

Thus

X®AGA"{T).

Some Other Results About Operator

Choose

D£JC(PH,

(PH)~L)

Structure

233

such that D is surjective, then

By Lemma 6.2.22 it suffices to prove that P =

{PH)L PH '

XD 0 A

\®A~B

Pll Pl2

BE(RI).

Suppose that

€A'(B)

P21 P22

is a projection. Since A ^ CT(J4),P 2 I = P12 = 0- Since A is irreducible, P22 = 0 or I. It follows from DP22 = PnD and D is surjective that P 2 2 =Pn=0oi I and Be(RI). (ii) If there is not any projection PGA'(T) with infinite rank such that T\PT-[£(RI), then we can find projections Qi, Q2 and Qz&A'(T) such that dimranQi = 00 (i = 1,2,3). Set Ti = T\ranQi (i = 1,2,3), then T = T1®T2@T^. Thus 0 ' ranQi ranQ 2 e„4'(T) = ,4"(T) A2 _ 0 As. ranQz "Ai

where {A4}?=1 are pairwise distinct numbers. By Lemma 6.2.23, F is similar to an irreducible operator and by Lemma 6.2.22, T is similar to an irreducible operator. Proposition 6.2.26 reducible operator.

Every Cowen-Douglas operator is similar to an ir-

In order to prove Proposition 6.2.26, we need some lemmas. Lemma 6.2.27

Given B&Bn(fl)

and A 6 fl. Denote

Hk = ker(B - X)kQker(B

- A)*" 1 (fc = 1,2, • • •),

then H =

@Hk fc=i

234

Structure of Hilbert Space Operators

and A B12 B13 * A S23

B = 0 where dirnHk = n and each Bkk+i is invertible. Furthermore, there is a unitary operator U such that XB12 * 0 A £23 UBU* =

H2

(6.2.14)

0 where Bkk+i is a positive invertible operator for each i. Proof Without loss of generality, we assume that A = Oefl. It is easily seen that dimHn = n and 0 B12 B13

*

0 B23 B 0

Let ko = min{k

'••

: kerBkk+i

7^ {0},&>1}. Then there exists a vector fco Xk0+i 7^ 0 such that Bk0k0+ixk0+i — 0- Note that M = 0 Hk = kerBko and dimM = &on. But y = (0,••• ,0,Xfc o+ i,0,• • -)&kerBk° and y $. M. This contradiction indicates that each Bkk+i is invertible. Thus B12 — UiB[2, where U\ is a unitary operator and B[2 is a positive invertible operator. Similarly, U1B23 = ^2-B 2 3 , U2B34 = UzBM,

• • • , UkBk+i fc+2 = Uk+iB'k+1

fc+2,

and each Uk is a unitary and each B'k k+1 is positive and invertible.

•

Some Other Results About Operator Structure

235

Define U = ® Uk, where U0 = I, then U is a unitary operator and fc=0

0 Si2 * 0 0 B23

UBU* =

0 0

where Bkk+i = ^kB'kk+lUk

(k>l) is positive and invertible.

Lemma 6.2.28 Given B£Bn(Q) with the representation (6.2.14). If £ l f c o e(i?J) for some k0>2, then BG(RI). Proof Suppose that PGA'(B) is a projection, then P admits the following oo

expression with respect to the decomposition H = © Hk, k=l

Pi Pl2 * P 2 P23

P = 0

Since P* = P, P = Pi©P 2 © • • • . Since PkBk fc+i — -Bfcfc+i-Pfc+i and since Bkk+i is positive and invertible, Pk = Pi for k>2. Thus Blk0Pko = PlP>lfc0Since Blkoe(RI),Pk fore Be(RI).

= Pi = 0 or I (k = 2,3, • • •), i.e., P = 0 or / . There-

Proof of Proposition 6.2.26 without loss of generality, assume that Oefi. Because of Lemma 6.2.27, we may assume that

0 5i2 0 B =

* " Hi B23

o '••

n2 W3'

0 where each Bkk+i is positive and invertible. Set B\ = {B12, B13, • • • )£C{H1

, Tii)

236

Structure of Hilbert Space Operators

and 0B23

*

0

£34

B2

£C{Hi,Hi).

0 '••

Let Jn be the nxn Jordan nilpotent. Then Jn£C(Cn)t~\(SI). E — B\2 (-^13

—

Jn)

and 1 £0 * 1 0 X1

1 '••

&C{Hi,Ht)

1 0 OXi

€C(H).

Set X-Then X is invertible and XBX-1

=

"1 0 ' '0 Bi oil 0B2 B1X11

'0

OXtl

h xr>\

0 B12 Jn 0

*

B23

0 '••

By Lemma 6.2.28,

XBX^^RI).

"1 0 OXf1

Set

Some Other Results About Operator

6.3

237

Structure

Application of Operator Structure Theorem

Theorem 6.3.1 Let SI be a connected open subset ofC and (Sl)° be finitely connected. Assume that {Pk(z)}™=i,z^ *s a class of Mn(C)-valued holomorphic functions such that: (i) Pk(z)eMn(H°°(Sl)), l
(Hi)

Y.Pk{z)~In-

fc=l

Then for fixed ZQ^SI, there exists an Mn(C)-valued ible function X(z)€Mn(H°°(Sl)) such that X{z)Pk{z)X-\z)

=

holomorphic invert-

Pk(z0),

where l
PiPj = 5ijPi,l
and £ Pfc = JW(»).

Denote Hk = PkH{n) (l
then

Denote Tk = T\uk- Since T^ has a finite (57) decomposition up to similarity (Theorem 5.5.11), there exist invertible Xi such that XiTiXr1 Set X = X : -i- • • • +Xm, then XeA'(T) XPkX~l

=

A(ni\ and

= 0 Wl © • • • ®QHk-M-Hk®Qnk+l®

Denote Y(z) = X| f c e r ( T _ 2 ) , then Y(z)&M„(H°°(Sl)) Y(z)Pk(z)Y(z)-1

• • • ©0 Wm .

and

= 0 f c e r ( T l _ z ) © • • • ®Iker(Tk-z)®Oker(Tk+1-z)®-

Set X(z) = Y~1(z0)Y(z),

then X(z)eMn(H°°(Sl))

X(z)Pk(z)X-1(z) and X(zo) = / „ , l
= Pk(z0)

and

•"©0feer(Tm_z).

238

Structure of Hilbert Space

Operators

In the following, we will discuss the winding number of some analytic functions. Let D be the unit disk and / be an analytic function on D. For a&D, the winding number of / ( e l t ) with respect to / ( a ) is given by 1

1 Z" W(f,f(a)) = — /

u

f ^f'(e >) dt.

Lemma 6.3.2 Given feH°°. If there is an aeD such that the inner function inn(f — f(a)) of f — / ( a ) is a finite Blachke product, then there exists a natural number n such that Tf~Tg , where g&H°° and Tg€(SI). Proof Denote h = inn(f — / ( a ) ) , which is a finite Blachke product. By Corollary 2.1 of [Cowen, C.C. (1978)], there exists a finite Blachke product