Finite rank perturbations of contractions

Integr. equ. oper. theory 36 (2000) 253-268 0378-620X//030253-16 $ 1.50+0.20/0 �9 Birkh/iuser Verlag, Basel, 2000

I Integral Equations and Operator Theory

F I N I T E R A N K P E R T U R B A T I O N S OF C O N T R A C T I O N S

CHAFIQ BENHIDA and DAN TIMOTIN

We study finite rank perturbations of contractions of class C.0 with

finite defect indices. The completely nonunitary part of such a perturbation

is also of class C.0, while the unitary part is singular. When the defect indices

of the original contraction are not equal, it can be shown that almost always

(with respect to a suitable measure) the perturbation has no unitary part.

1. I N T R O D U C T I O N AND P R E L I M I N A R I E S

This paper continues the line of research initiated in [BT] by investigating the contractive

finite rank perturbations of a given C.0 contraction with finite defect indices. It is shown in section 3

that the completely nonunitary part of such a perturbation is also of class C.0, while the unitary

part is singular. When the defect indices of the original contraction are not equal, it turns out that

in "most cases" the perturbation has no unitary part. In particular, the results obtained extend

IN], which considers one-dimensional perturbations of isometries. Finally, it should be noted that

all the results can be trivially transposed (by passing to the adjoint) to the case of a Co. contraction.

A few words about the history of the subject are in order. The main class of perturbations

on which we focus appear for the first time explicitely in [C], where the defect spaces are one-

dimensional; although they had been implicitely used by De Branges in a series of papers (see

[deB] and the references therein). Clark's results have been extended by Fuhrmann (IF]) in the

case of finite dimensional defect spaces and Ball and Lubin ([BL]) in the general case; it should be

noted that formulas related to Proposition 2.1 below appear in these works. In particular, spectral

properties of the perturbation are investigated. From a different point of view, that of dilation

theory, contractions with finite dimensional defect spaces have been studied in [TW1] and [TW2].

In particular, Corollary 5.6 below has been proved in [TW1], in a different context; however, an

open question posed in this reference is solved in section 3 by using our methods.

Let ~/ be a separable, infinite dimensional, complex Hilbert space. We start by fixing

a contraction T C s with finite dimensional defect spaces ~)T and DT.. We will suppose that

254 Benhida, Timotin

T*nx ~ 0 for any vector x E 7-/; in the standard terminology of [SNF], T is of class C0. This

implies that d imDT _< dimDT.; suppose that d imDT = m, dimDT- = m + k. Our main aim is

to investigate all finite rank perturbations of T. We begin with a few simple remarks.

First, consider the decompositions 7-I = D r �9 DT r = DT* @ D~.. If the first is used

for the domain and the second for the range, then T is diagonalized: we have T(DT) C DT-,

T(i9~.) = D.~., and Ttz~ : DT ~ -+ D~.. is unitary.

Second, although T is completely nonunitary, it may happen that a finite rank contrac-

tive perturbat ion of T is no longer completely nonunitary; recall that an arbitrary contraction has

a decomposition as a direct sum between a unitary part and a completely nonunitary one.

Third, i f T - T ~ has finite rank, then there exists a subspace ~ C 7-I of finite codimension,

such that 5 C DT ~ V) 7:)Tl,, and TI.T = T'[iT. An important particular case happens when we may

take 9 v = D~,; that is, when T'ID ~. = T[D.~. Such finite rank perturbations of T will be called

standard. It is obvious that the standard perturbations of T are in one-to-one correspondence with

the contractions Z : DT --+ DT. ; the correspondence is obtained by the formula

z = T'IZ)r. (1.1)

We will denote by C(T) the set of all standard perturbations of T. It has an alternate

description, by means of fractional transforms. If W E s DT-) is a contraction, define Z =

CT(W) E s by the formula

Z = CT(W) = T "~ DT* ( I + W T * ) - I W D T . (1.2)

The inverse in the formula acts on D r . ; it exists since T* ]DT. is a strict contraction.

PROPOSITION 1.1. The map CT is a bijection from the unit ball of/Z(DT, DT-) to

itself; it maps strict contractions to strict contractions and isometries to isometries.

The proof (see Lemma 2.1 and Corollary 2.2 of [BT]) is based on the following formula,

which is a consequence of (1.2):

I - Z*Z = DT(I + W * T ) - I ( I - W * W ) ( I + T * W ) - I D T . (1.3)

We will have the occasion to consider different subsets of s DT*) (the class of all

isometries, the class of all operators whose range has a fixed dimension) which are algebraic (and

hence differential) submanifolds. Without entering into details, one can define the notion of subset

of measure zero of such a submanifold, by requiring that the image through each chart has Lebesgue

measure zero. Correspondingly, we know what 'almost everywhere' means. We denote by F v and

F~ the Grassmanian manifolds of the p-dimensional subspaces of DT and DT- respectively.

Benhida, Timotin 255

2. A F O R M U L A FOR THE C H A R A C T E R I S T I C F U N C T I O N

Remember that the characteristic function of T ([SNF]) is a contractive analytic function

@(A) : T)T --+ /)T*, defined by the formula

O(l) : -T + )~DT. (I -- )~T*)-IDT]~)T.

Computations involving the characteristic function in this section will imply operators acting either

on the whole Hilbert space or on different defect spaces. To kcep track of this distinction, we will

use the orthogonal projections P : 7/--+ 7:)T, P, : 7-/--+ T)T. and their adjoint embeddings J = P*,

J,=P**.

Suppose now that T ' is a standard perturbation of T which corresponds to a Z given

by formula (1.2). Our purpose is to obtain a formula for the characteristic function OT, of T ' (cf.

also IF], [T]).

PROPOSITION 2.1. The characteristic function OT, coincides (in the sense of [SNF])

with the contractive analytic function O'(A) : 7Pw --+ 7)w*, defined by

O'(A) = - W + D w . ( I - O(A)W*) - IO( t )Dw

Proof. Let ~ and ~ , be defined by:

g2DT, x = D w ( I + T*W)-IDTX,

~,DT, .X = Dw. (I + TW*)-IDT*x.

We will now refer to some results obtained in Section 2 of [BT]. It has been established therein

that g2 : 7)T, --+ 7)w and ~ , : 7)T,- ~ :DW. are unitary operators, as well as the following formula:

D w * ( I - E ) ( A ) W * ) - I D T . P , ( I - AT*) -1 = D w . ( I + T W * ) - I D T . P , ( I - AT'*) -1. (2.1)

Also, we will use the relation

OT(A)DT = DT* ( i - AT*) -1 (A - T), (2.2)

proved in [SNF], VIA, and its analogue

OT, (A)DT, -- DT,-(I -- AT'*)-I(A - T') . (2.3)

The proposition will be proved by establishing the identity

~ ,Or , (A) = [ - W + Dw. (I - O(A)W*)-lO(A)Dw].~ (2.4)


As

Let us app ly the left h a n d side of (2.4) to DT, X. Using (2.2), (2.1) and ( 2 . 3 ) , we o b t a i n

~2.0T, (A)Dy,x = Dw* (I + T W * ) - I D T. (I - AT'*) -1 (A - T ' ) x

= D w . ( I + T W * ) - I D T . P . ( I - AT'*) -1 (A - T' )x

: D w . ( I - 6 ) (A )W*) -XDT.P , ( I - A T * ) - I ( A - T ' ) x (2.5)

= D w - ( I - (-3(A)W*) - ] [O(A)DT - DT. P, (I - A T * ) - ~ J . D T . W ( I + T* W ) - I D T P ] X .

On the other hand, we have:

[ - W + D w . (I - | x

: - W D w ( I + T ' W ) - 1 D r x + D w . ( I - 6 ) ( A ) W * ) - I O ( A ) D ~ , ( I + T * W ) - I D T x

= - W D w ( I + T * W ) - ~ D T Z + D w . ( I - O ( A ) W * ) - ~ O ( A ) ( I + T * W ) - ~ D T Z

- D w - ( I - 8 ( A ) W * ) - 1 6 ) ( A ) W * W ( I + T*I/V)-aDTZ.

D w . ( I - O ( A ) W * ) - I O ( A ) W * W ( I + T * W ) - I D T x

= D w . ( I - O ( A ) W * ) - I ( @ ( A ) W * - I + I ) W ( I + T * W ) - I D T X

= D w . ( I - O ( A ) W * ) - I W ( I + T * W ) - I D T x - W D w ( I + T * W ) - I D T X ,

it follows that

[ - W + D w . ( I - O(A)W*)- IO(A)Dw]QDT, X

= D w . ( I - ( ~ ( A ) W * ) - I [ o ( A ) ( I + T * W ) - I D T X - W ( I + T * W ) - I D T x ] .

But we can write

O ( A ) ( I + T * W ) - I D T x - W ( I + T * W ) - I D T x

= 0 ( ) 0 ( I + T * W - T * W ) ( I + T * W ) - I D T X - W ( I + T * W ) - I D T Z

= O(A)DTX - (3(A)T*W(I + T * W ) - I D T Z - W ( I + T * W ) - I D T x

= e ( A ) D T x + ( - (9 (A)T* - I ) W ( I + T * W ) - I D T X .

Since - I - Q)T(~)T* = - I + TT* - ADT. (I - A T * ) - I D T T *

= -D~.. + D T . ( I - A T * ) - I ( - A T * + 1 - 1)DT.

= -D?r. + D 2. - DT. ( f -- AT*)-]DT-

: - -DT*(I - - A T * ) - I D T *,

we obtain

- W + Dw* (I - | X

= D w . ( I - e ( ~ ) W * ) - I [ O T ( A ) D T -- DT.]:).([ -- AT* ) - I J , D T * W ( I + T * W ) - I D T P J X . (2.6)


We may now conclude the proofi comparing (2.5) and (2.6) yields the desired for-

mula (2.4). !

COROLLARY 2.2. (i) The completely non-unitary part of T t is of class C.o.

(ii) Suppose that in formula (1.1) ][Z H < 1. Then T' E C.o.

Proof. Since T E C.0, it follows that OT is inner. In our case, (1.3) becomes

I - O'(A)*O'(A) = D w ( I - 0 ( A ) * W ) - I ( I - O(A)*O(A))(I - W*O(A))-XDw. (2.5)

The analytic function de t ( I - W*O(A)) is non-zero for A = 0 (since [](9(0)][ < 1); hence it has

nonzero radial limits almost everywhere on the unit circle. Take such a point (, and assume that

O(() is an isometry. If we let A tend radially to (, (2.5) implies that O'(~') is also an isometry.

Therefore the characteristic function of T t is inner, which implies (i).

As for (ii), if I]Z[] < 1, then ~T ' = :DT. I fNu is the space of the unitary part o f T ~, then

7-/u C T)~. But T'IZ~ = TI:D ~. Since T is completely nonunitary, it follows that 7-/u = 0. I

3. G E N E R A L P E R T U R B A T I O N S

We will use the following result from [Ca].

LEMMA 3.1. Let W1, W2 be isometries on a HiIbert space such that W1 - W2 has finite

rank. I f Wi = W~ @ W p is the Wold decomposition of Wi (i = 1, 2), then the absolutely continous

parts of W~ and W~ are unitarily equivalent.

The next lemma is an easy consequence for arbitrary contractions; the argument is

suggested by [TW2], Theorem 4.1.

LEMMA 3.2. Suppose T1,T~ C s are two contractions with I)TI and l)T2 finite

dimensional, such that T1 - T2 has finite rank. Let V1, V2 be their corresponding minimal isometric

dilations. Then the unitary parts of V1 and V2 have unitarily equivalent absolutely continous parts.

Proof. Define ~ = 7)T1 VT:)T2, and ~ = 74@D@/) . . . . Consider on K: the two isometries

0 0 . . .

W i = I 0 . .- , i = 1 , 2 . 0 I . . . : : ' . .

Then each Wi is the direct sum between the minimal isometric dilation of Ti and a shift. But

W1 - W2 has finite rank; by Lemma 3.1, the absolutely continous parts of their unitary parts are

unitarily equivalent. The statement of the Lemma follows. |


LEMMA 3.3. Suppose T E C.o and dim'~)T < oo. Let ~ C 7~ be a closed subspace of

finite codimcnsion. Then T P f E C.o.

Proof. Denote T1 = TPT. We will start with a particular case, assuming that $ C D~.

Then DT1 = U c, and DTI = P2•

If 7-s C ~ is a reducing subspace for 7"1 with T117r ~ unitary, then 7r ~ C D~. = O r .

Since T~ I) r = TI5 c, we obtain that TI74' is unitary. As T is completely nonunitary, we conclude

thatTIT~ ~ = 0 and therefore T1 is completely nonunitary.

On the other hand, T - T 1 has finite rank. Since T and T1 are both completely nonunitary,

their minimal isometric dilations V and V1 are absolutely continous. Applying Lemma 3.2, it follows

that the unitary parts of V and V1 are unitarily equivalent. But, since T E C.0, the unitary part

of V is {0}; the same is therefore true for V1, which means that T I E C.0.

We consider now a general subspace .Y" of finite codimension. If 7 = U (3 D r ~, then

77~ = TPy , is in C.0 by the first part of the proof. Also, its defect space is DT; = 7 • thus

D~; = or' = or M D.~ C or. Therefore

TllT)~; = TPyliP' = TPT, IJ:' = T~lYr';

it follows that T1 is a standard perturbation of T~. Also, llxlt = I[Tlxll = tlTPTz'II implies x E

5r M 79T ~, and thus x C 7 = DTI,; since DT; is finite dimensional, this implies IITllT~T;II < 1. We

may apply then Corollary 2.2 (ii) to conclude that T1 C C.0. |

The next theorem is the main result of this section; it characterizes the finite rank

contractive perturbations of a contraction of class C.o.

THEOREM 3.4. Suppose T E C.o, dimTPT < 0% and T' is a finite rank perturbation

of T. Then:

(i) the completely non-unitary part o f T I is of class C.o;

(ii) the unitary part of T ~ is singular, of multiplicity at most dim(DT V ker(T ~ - T) • -

dim iDT,.

Proof. We will denote by T I = U | T~ the decomposition of T ~ into unitary and com-

pletely nonunitary parts.

First, define $- = D r ~ N ker(T ~ - T). According to Lemma 3.3, T1 = T P ~ is also of class

C.0. But, since T~I~ = T [ ~ = Tl13 r, T ~ is a standard perturbation of T1; therefore its completely

nonunitary part Tc ~ is of class C.0 by Corollary 2.2 (i).

Now, the minimal isometric dilation of T' is the direct sum between U and the minimal

isometric dilation of T~. The last one is a pure isometry, and therefore the unitary part of the


minimal isometric di lat ion of T ~ is U. We may then apply Lemma 3.2 to T and T' ; since the

minimal isometric di lat ion of T has no uni tary part , it follows tha t U is singular.

In order to obta in the est imate on the multiplicity, define s = 79T V ker(T ~ - T) • O 7:)7-,

and 3.4 ~ T'ks = Vk=o Consider the contract ion T" C L(7-/), defined as

T , = { O T o n e on E •

T" 79• = T " l f = TI$- = Tl[79.fil. Then T " is a Remark tha t T" = TI + T'Pz) r, and 7-1

s t andard pe r t u rba t i on of TI. Furthermore, IlT"179T~ II = HT'[79T ' II < 1, since the dimension of :DT,

is finite (note t ha t 79T' C ~ • By Corollary 2.2, (ii) it follows t h a t T" E C.0.

If T ' - T" = F , then kerF = E • and F(7-t) = F ( ~ ) = T ' (E) . In part icular , F (7 / ) C 2td

and kerF* = F(7 / ) • DAd z. Therefore M • is invariant for T ~* and T'*IAd • = T"*13A •

Take now x E ~ ; suppose x = xt +x2, x l E 3,t, x2 E .hd • Then Tf*nx2 = TtJ*nx2 -+ 0

(since T" C C.0); therefore

Hx[I = lira IlT'*r~xll = lira IIT'*nxl + T'*'~x21I

= limllT'*~xll l _< Ilxll].

Consequently x2 = 0, and thus 7/~ C 34. Therefore, the mult ipl ici ty of U is at most

equal to d i m e = dim(T?7- V ker(T ' - T) • - dimT?7-,. !

In part icular , if T r is a s tandard pe r tu rba t ion of T which means t ha t T'I~)T • -= TIT)T •

or equivalently t ha t ker(T ~ - T) • C 79T, then the multiplicity of the singular par t of T ~ is at most

d im 797- - d im 79T'.

Theorem 3.1 of [TW1] (which in tu rn is a generalization of [N]), s ta tes tha t , if T is a

cont rac t ion which can be wr i t ten as T = S + F , S a pure isometry and F a finite rank operator ,

then T = U ~ T1, with U a singular uni tary and Tl E C.0. An a l ternate proof can be ob ta ined by

combining Corollary 2.2 with Theorem 3.4. Moreover, in [TW1] it is asked as an open quest ion

whe ther the inequali ty #(U) + dim79T, _< r a n k F is true. Theorem 3.4 above implies a positive

answer to this conjecture.

4. I S O M E T R I E S ON A HILBERT SPACE

Up to now we have supposed only t ha t dim79T _< dim79T-. For the rest of the paper, we

will suppose t ha t we have str ict inequality; tha t is, according to the nota t ions in the in t roduct ion,

t h a t k >_ 1; this assumpt ion will allow more refined results. We will s ta r t by establ ishing in this

section some prel iminary facts about isometries, whose one-dimensional analogues appear in IN].


If 1C is a Hilbert space, s C K is a closed subspace, and V E s is an isometry, define

the analytic function

G E ( A ) = P E ( I + A V * ) ( I - A V * ) - l t $ , A E D . (4.1)

Since Ge(A) is the compression to s of the Cayley transform of the strict contraction AV, it is n accretive--that is, ~GE(A) > 0. Denote also 5 r = Vn=0 V g.

Let us consider now gP C K ~ and V ~ E s with corresponding function G~, and

invariant subspace Y spanned by g~.

LEMMA 4.1. If there exists a unitary co : g ~ $' such that wGE(A) = G~,(A)w, then

VI.T is unitarily equivalent to V~[~ ~.

Proof. We use the fact that 5 r and 7 are spanned by the vectors ( I - AV)-I~, ~ E

g, A E D a n d ( I - A V ' ) - I ~ ', ~ C g ' , A C D r e s p e c t i v e l y . I f ~ , ~ l C g a n d A , # E D , then

(I + ~ v ) ( I - Av) -1 + (I + ~ y * ) ( i - ~v*) -~

~- ( I -- # V * ) - 1 [ ( I - # Y * ) ( f -~- ,~V) -~- ( I -b ~ t V * ) ( I - / ~ V ) ] ( I - /~V) - 1

= 2 ( ~ - p ~ ) ( I - p y * ) - ~ ( 1 - ; ~ v ) - ~ .

Therefore

<( I - A V ) - ~ , ( I - p V ) - ~ > = < ( I - ~ V * ) - 1 ( I - A V ) - ~ , ~ > 1

- 2(1 - p A ) ( < a e ( # ) ~ , 7 / > + <GE(A) *~ , rl>

1 - 2(1 - #A) (<co* O', (#)co~, ,~> + <co* O~, (;~)*co~, ~>

1 i * - 2(1 - ~A~ (<a~,(p)co~,co~> + <az,(~) co~,~,~>

= ( ( I - A v ' ) - % L ( I - ~ v ' ) - ~ > .

I t follows that the map ~ : . r -~ 2 ' defined by ~q((I - AV) - I~ = ( I - AV~)-Iw~ is unitary. I t is

simple to show that it interwines V and V ~, and the theorem is proved. !

For ~ ~ K, define .q~(A) = ( ( I + A V * ) ( I - ) , V * ) - ~ , ~ ) ; thus, i f~ ~ g, then 9r =

(G~(A)~,~). Obviously, the condition in the hypothesis of Lemma 1 may be stated as g~(A) =

9~(~)(A)a~ for any ~ ~ g. Also, 9~(A) is an analytic function with positive real part, and thus the

positive harmonic function ~g~ is thc Poisson transform of a positive measure #~:

1 f~r e it q- A " it ~g~(A) = ~ j_ ~ d # r ) (4.2)

i r

Since g((0) = 1, we have I]#~]l = [l~]l 2, and the radial limits of Ng~ are equal to the Radon-Nikodym

derivative of #( with respect to Lebesgue measure.


The next lemma gives the computation of the function 9~ for an important case.

LEMMA 4.2. For a positive measure # on T, consider the space H2(C k, #) (the closure

of the analytic polynomials in L2(ck ,# ) ) ; let S be multiplication with e it. Then, if f C H2(C k, it),

1 f ~ e it + ~ . e i t . = IIc~ dit(e ). gy(;~) ~ J -~ e_77Z_f_~ll] ( ) 2 ~t

Proof. Since multiplication with e - i t on L2(C k, #) is a dilation of S*, we have

f --it 1 ~ 1 + Ae it it ( ( I + , ~ S * ) ( I - s f ) = ~ ~ l = ~ - - ~ ( f ( e ) , f ( e )}ckdit(e it)

1 / f f e it + ,~ (f(eit) -- 27r 7r ~ . . . . . f(eit)}ck d#(eit)"

(4.3)

The lemma is proved, !

COROLLARY 4.3. Suppose V C s is an isometry, ~ E 1C. If jz~ is the invariant

space spanned by ~, then VllP ~ is unitarily equivalent to multiplication with e it on H2(it~).

Proof. In order to apply Lemma 4.1, define /C' = H2(#~), V' multiplication with e~t;

then $, g ' are the one-dimensional subspaces spanned by ~ and the constant function 1 respectively,

while w(~) = 1. By Lemma 4.2, g~(A) = g~(~); since 1 is cyclic for V', VI$" is unitarily equivalent

to V ~. |

5 . I S O M E T R I C S T A N D A R D P E R T U R B A T I O N S

We investigate in this section the isometric standard perturbations of T; they correspond

to Z being isometric in formula (1.1). If dim 7:)7 = dim DT-, all these perturbations are obviously

uni tary (and singular, according to Theorem 3.4). However, we prove in this section that, if

d imDT < d imDT. , then in "most cases" the resulting isometry (which cannot be unitary) is in

fact pure.

The first remark is a consequence of Corollary 2.2. Among all contractions in C(T) there

is a distinguished one T1, which corresponds to Z = 0 in formula (1.1). We have T1 = TPw~ (which

has already been used in the proof of Theorem 3.4). Corollary 2.2 (ii) implies, in particular, that

T C C.0 if and only if T1 E C.0; on the other hand, it is obvious that C(T) = C(T1). Therefore, in

order to simplify computations, we will assmne in this section that T = T1; that is, that TIeD T = O,

or, equivalently, | = 0.

We remind the following well-known lemma.


LEMMA 5.1. S u p p o s e A E s B E s I f I n ~ - A B E s is

invertible, then I7~1 - B A E s is also invertible, and

(I7~1 - B A ) -~ -- In~ + B ( I u : - A B ) - I A .

Let us fix a s t andard isometric pe r tu rba t ion V of T, corresponding to an isometry

Z : ~D T - + ~D T - in formula (1.1). We will use the nota t ion G(A) for the f lmction defined by formula

(4.1), wi th s = l~)T-

LEMMA 5.2. With the above notat ions , / fd(A) = Z*O(A), then

G(A) = ( I + d ( l ) ) ( I - d ( l ) ) - i .

Proof. Let P and P, be the orthogonal projections onto "D T and DT, respectively. The

formula for the characteris t ic function of T becomes in this case

| = AP , ( I - AT*) - iP ,

and we have the relations P V * = V ' P , = P V * P , , T = V ( I - P ) , T* = V* - P V * P , . Therefore

and

I - AV* = I - AT* - APV*P, = (I - AT*) ( I - A(I - A T * ) - i p v * P , )

( I - AV*) - i : ( I - A ( I - A T * ) - i p v * P , ) - I ( I - AT*) -1,

To compute the first inverse on the right, we will use lemma 5.1, with A : 7-/--+ DT, B : :DT -+ ~t,

A = P V * P , , B = A(I - A T * ) - i P . Then

( I - A(I - A T * ) - I P V * P , ) - i = I + A(I - A T * ) - i P [I - P V ' P , A ( I - A T * ) - I P ] - 1 P V * P ,

-- I + A(I - A T * ) - I p ( I - d ( A ) ) - i p y * P , .

Therefore

( [ - A Y * ) - i P = ( I - A T * ) - i P + A(I - A T * ) - Z P ( I - d ( A ) ) - i p y * P , ( I - A T * ) - i P

= (I - A T * ) - i P + ( I - A T * ) - i p ( I - d ( A ) ) - l d ( A ) P

= ( I - A T * ) - i p ( I + ( I - d ( A ) ) - i d ( A ) ) P

= ( I - A T * ) - i p ( I - d ( A ) ) - i P

and P A V * ( I - A V * ) - i P = P V * P , A ( I - A V * ) - I P

= P V * P , ( I - A T * ) - i p ( I - d ( A ) ) - i P

= P d ( A ) ( I - d(A)) - IP .


Then P ( I + A V * ) ( I - AV*) - IP = P ( I + 2 A V * ( I - A V * ) - I ) P

= P ( I + 2d(A)(I - d ( ) Q ) - I ) P

= P ( I + d(A))(I - d ( A ) ) - l P

The lemma is proved. I

It follows from Lemma 5.2 that, if ~ E 7)T, then

~gr = [ [ ( I - d(A)*d(A))U2(I - d(A))-l~[[ 2.

The radial limits of ~g~(A) are almost everywhere 1 [ ( I - d ( e i t ) * d ( e i t ) ) l / 2 ( I - d(eit))-l~][ 2 (note

that O and consequently d being pure, the determinant of I - d(A) does not vanish idcntically~ and

thus ( I - d(eit)) -1 is invertible almost everywhere). Therefore

II(I - d(eit)*d(eit))U2(I - d(eit))-l~[]2 dt <_ II#~ll ---- I]~'[] 2. (5.1) 7T

Since dim:DT = m, dimT)T. = m + k, we may fix an isometry j : C k --4 7)T* 0 Z(:DT).

If b()0 = frO(A), then b is a contractive analytic function from ~T to C k, and

tlb(eit)~[]2 + [id(eit)~]12 = ][~[]2 (5.2)

for any ~ E Z)T, almost everywhere on T. Then formulas (5.1) and (5.2) say that b(I - d ) - l ~ is a

flmction in H2(Ck), and

J[b(I- d)-l~[l~2 _< I1~17. (5.3)

We will call ~ E 7)T pure if the measure tt~ is absolutely continous with respect to

Lebesgue measure. This is equivalent to equality in formula (5.1) or (5.3).

PROPOSITION 5.3. I f a dense set of vectors ~ E 7)T are pure, then V is a pure

isometry.

Proof. Denote ~" = Vn~__o vnT)T; Y is invariant to V and .P• is invariant to V*, and

V, iS-• = T*l y •

If there exists an invariant subspace 7/~ which reduces V to a unitary operator, take

x C 7/u, and suppose x : xl + x2, x1 E 5 r , x2 C 9 r • Then V*nx2 = T*~x2 ~ 0, whence

]}x]] = lim []Y*~xl] : lim ][V*~xl + V*~x2[[

= lim ][Y*~x~[[ _< ]]x~[].

It follows that x2 = 0, and thus 7-/u C ~ .

Let us define ~(~) = b(I - d ) - l~ . Formula (5.3) shows that ~ : 7)T --+ H 2 ( C 2) is a

contraction. If ~ is pure, then ][~(~)t[ = []~][; since such vectors are dense, ~ is actually an isometry.


Denote g = t(:DT), and G~(A) = GE(A) the analytic function associated, as in section 1, to the

shift opera tor on H 2 ( C k) and the subspace s

Applying formula (4.3) to f = g = t(~), we obtain, using (4.2),

1 ~ e it ~- A �9 = d(e") lI2 e t

= I J ( • - d(~)*d(~))~/2( I - d ( :~ ) ) -~ l l 2 = ~ < a ( : % ~>.

Since (G(0)~,~) = (G'(0)t~, t~) = II~H 2, it follows tha t G(A) = t*G(A)t. We may apply Lemma 4.1

wi th g~ = D r , w = t*; we obta in a uni tary equivalence between V[.T and the restr ic t ion of S to an

invariant subspace. Since the last operator is a pure isometry, V I ~ is pure. But we have shown

tha t 7-/~ C 5", and thus V is pure. !

We want to show in the sequel tha t "almost all" V-s satisfy the hypothesis of Proposi t ion

5.3. Note first t h a t the set of isometric s tandard per tu rba t ions is in one-to-one correspondence

wi th the set Z of all isometries from ~DT to :DT*. If F ~ is the Grassmanian manifold of all m-

dimensional subspaces of :DT ~, then there is a na tura l map p : Z --+ Fm, mapping each isometry

to its range space. The fiber over each F C F*~ is isomorphic to the group N(m) of all unitaries

on the m-dimensional space :Dr, which is natural ly endowed wi th the Haar measure.

We will use the following lemma, whose proof is left to the reader.

LEMMA 5.4. I f F C :DT i8 a proper subspace, and b( C l.t(m) is defined by bF = {u C

bt ] u has an eigenvector in F}, then bF has measure zero.

The next theorem is the main result of this section.

T H E O R E M 5.5. For almost all Z C Z the corresponding isometry V is pure.

Proof. or(t) = O(eit)(:DT) is an m-dimensional subspace of :DT'. Since it can take only

a countable set of values on a set of str ict ly positive measure on T, it follows tha t for almost all

F E F~ , F # or(t) almost everywhere on T.

Fix now a space F with this property, and an isometry Z0 with image equal to F.

Consider then the measurable space E = T x / ~ (with the product of the Haar measures) . Define

the measurable subset ca C E by

ca = {(e i t , ~t) ] uZ~PFO(e it) : �9 ~ DT has an eigenvalue of modulus 1 }.

Now, if e it is fixed, any eigenveetor corresponding to an eigenvalue of modulus 1 of u Z ~ . P F O ( e it)

has to belong to l;t = O ( e i t ) - l ( a ( t ) M F) , which is a proper subspaee of :Dr. Fix a un i ta ry

opera tor Ut : DT --+ :DT, which coincides with Z~PFO(e it) on ~t; any eigenvector as above is also


an eigenvector of uUt belonging to ))t. But, according to Lemma 5.4, the set of all unitaries in

b/ having eigenvectors in Vt is of #-measure 0. Since multiplication with Ut* preserves the Haar

measure, it follows that the set of all u-s for which (e it, u) E w has #-measure 0.

We can then apply Fubini 's theorem to deduce that for ahnost all u E U the set of

e it E T for which (e i t ,u) E w has Lebesgue measure 0. Consequently, for ahnost all Z E Z , d(e it)

has no eigenvalues of modulus 1 for almost all t.

Fix then such a Z E Z. The last property is obviously preserved if we replace Z by

aZ , and correspondingly d(e it) by ad(ei t) , with a E C, ]a I = 1. Denote by Ga the corresponding

analytic function (see Lemma 5.2); fix also ( C DT. The measures #~ appearing in (5.1) depend

on a = eiS; we will denote them by #~ (since ~ is fixed). According to (5.1), we have

if IIb(eit)( I - c~d(eit))-l~l[2dt <- [I#~ll = 1, (5.4) 7r

(note that I - ad(e it) is invertible for all a and almost all t).

Consider now the double integral

2"-~ ~ ~ ~ Hb(eit)(I - eiSd(eit))- l~l[2dtds

i f i f = - - ]lb(eit)(I - eiSd(eit))-l~[12dsdt" 27C zr z:

We have

f if 1_ [ib(eit)(i _ eiSd(eit))_l~ll2d s = ~ 1[(/_ d(e i t ) ,d (e i t ) ) l /2 ( I _ eiSd(eit))_l~ll2ds 2 7 r 7r ~

If ~" E C m and A is an m x m matrix which has no eigenvalues of modulus 1, then the

analytic function r = D A ( I - A A ) - I ~ is in H2(cm) . Indeed, since IIAPH --+ 0, it is equal to oo ~]p=o ABDAAP~, its radial limit is ~-~,~=o eiPSDAAP~, and thus

11r _ IIDAAP([[2 = '}-'~([[Ap~[[2 _ [[AV+I(IIZ) = 1](1]2. p = 0 p = 0

Therefore

f f __1 i[b(eit)(i _ eiSd(eit))_l~l[2d s = 1 = 2~r ~ ~ ~ l lDd(e">(I-- ei~d(e~t) ) - l~ l l2ds = II~ll 2 1 .

It follows then that .

I f ~ I f ~ l f_ ~ ~r ~ ~ ][b(eit)(I - e i~d(ei t ) ) - l~N;dsdt = ~ 7r l d t = 1.

But (5.4) implies that we must have actually [ [ b ( e i t ) ( I - ad(e i t ) ) - l ( [ l~ = 1 almost

everywhere in a; therefore { is pure for almost all a E T. Applying this argument to a countable


dense set of vectors ~ E ~)T, Proposition 5.3 implies that a V is a pure isometry for almost all

c~ E T. Since the last fact is true for almost all Z E Z, the assertion of the Theorem follows. !

COROLLARY 5.6. I f T E C.o has defect indices m, re+k, then T is unitarily equivalent

to a finite rank perturbation of dimension m of the k-dimensional shift.

Proof. Suppose T = V + F, F finite rank. The proof of Theorem 5.5 shows that V is

obtained by perturbing T on 7PT, which has dimension m; thus rank F _< m. On the other hand,

7PT C (kerF)• thus m = dimZIT < rankF. !

As noted in the introduction, corollary 5.6 is proved, with different methods, in [TW1].

6. S T A N D A R D C O N T R A C T I V E P E R T U R B A T I O N S

The next result is a rephrasing of Proposition 3.2 of [BT] (it also follows easily from

Lemma 3.3 and Corollary 2.2 (ii) above).

PROPOSITION 6.1. I f a standard isometric perturbation V of a contraction T is pure,

then T E C.o.

We may then use Theorem 5.5 to obtain results about standard non-isometric pertur-

bations. Let us define, for 0 _< p _< m, the set 79p as the collection of standard perturbations T I of

T with dim 79w, = p. Obviously 7)0 is the set of standard isometric perturbations of T, while 79m

is characterized by the property that ]lZII < 1; these cases have been dealt with in the preceding

two sections.

THEOREM 6.2. For any O < p < m, almost every T' E 79p is in C.o.

Proof. First, if we denote by 79Ip the set of partial isometries in 79p, we may identify

79p : 79Ip N: •1, where Bl is the open unit ball in L(C p, Ck+P). (Note that the range of an element

in 79Ip has dimension rn - p . ) For any T' E 79p there exists a T~ which is a partial isometry and

coincides with T ~ on 7P~.,. According to Corollary 2.2, T ~ E C.0 if and only if' 5r~ E C.o. It is then

enough to show that almost all f~ E 79Ip is in C.0.

Now, suppose we fix $ E F,~_p, that is, a subspace of ~)T of dimension m - p. Any

isometry V E 79o can be decomposed as a direct sum V = VPc + V ( I - PE), and VP~ E 79Zp,

V ( I - Pe) E 79Im-p. Conversely, if V ~ E 79Ip is given, the choice in completing it to an isometry

in 79o is in an arbitrary isometry with domain 7?T O $ and range 7?T* Q V~C. These last subspaces

have dimensions p and p + k respectively; thus, if 1)(p, p + k) denotes the set of isometrics from C p

into CP-bk~ we may write locally

79o • Fm-p = 79Ip x V(p,p + k). (6.1)


If PP C P0 is the subset of pure isometries, then, by Theorem 5.5, Po p • Fm-p has full

measure in P0 • Fro-p; the same happens therefore with its image under the identification in (6.1),

which, therefore, has nonvoid intersection with ahnost all ~ E Pip. It follows that almost all

~t E P~p has a standard perturbation that is pure. By Proposition 6.1, any such Ft is pure. The

Theorem is proved. I

[BL]

[BT]

[Ca]

[c]

[deB]

[F]

7. N O N - S T A N D A R D P E R T U R B A T I O N S

As noted in the preliminaries, if T - T I has finite rank, then there exists a subspace

5 ~ C ~ of finite codimension, such that 5 ~ C :D~ M :D~,, and TI$- = T']iP. We may now obtain

results about this general situation. Fix Y, and denote by P(5 r) the set of all these Tr-s.

THEOREM 7.1. If T E C.o, then almost every T ~ C ~ ( ~ ) is in C.o.

Proof. If we consider T1 = TPj:, then we may apply Proposition 6.1 to any pure

isometry in C(T) (which exists by Corollary 5.6), and obtain that 7"1 E C.0. Now 7)(~ ") = C(T1),

and we end the proof by applying Theorems 2.1, 5.5 and 6.2. I

A particular case happens when T is a pure isometry. In this case we obtain the following

more precise result, generalizing [N].

THEOREM 7.2. If T is a pure isometry, almost every isometry in P(.~) is pure.

Proof. The proof is similar to that of Theorem 7.1: To E C.0, and consequently, by

Theorem 5.5, almost all its isometric standard perturbations are pure. But these are exactly the

isometries in P(Sr). I

R E F E R E N C E S

J.A. BALL and A. LUBIN, On a class of contractive perturbations of restricted shifts, Pacific J. Math 63 (1976), 309-324.

C. BENHIDA and D. TIMOTIN, Functional models and finite dimensional perturbations of the shift, Integr. Equat. Oper. Th. 29 (1997), 187-196.

R.W. CAREY, Trace class perturbations of isometries and unitary dilations, Proc. Amer. Math. Soc., 45 (1974), 229-234.

D.N. CLARK, One dimensional perturbation of restricted shifts, J. Analyse Math. 25 (1972), 169-191.

L. DE BRANGES, Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood Cliffs, N J, 1968.

P.A. FUHRMANN, On a class of finite dimensional contractive perturbations of restricted shifts of finite u~ultiplicity, Israel J. Math 16 (1973), 162-176.


[N]

[SNF]

[TWl]

[TW2]

IT]

Y. NAKAMURA, One-dimensional perturbations of the shift, Integr. Equat. Oper. Th. 17 (1993), 373-403.

B. SZ.-NAGY and C. FOIAS, Harmonic Analysis of Operators on HiIbert space, North- Holland, Amsterdam, 1970.

K. TAKAHASHI and P.Y. WU, Dilation to unilateral shifts, Integr. Equat. Oper. Th. 32 (1998), 101-113.

K. TAKAHASHI and P.Y. WU, Singular unitary dilations, Integr. Equat. Oper. Th. 33 (1999), 231-247.

D. TIMOTIN, Redheffer products and characteristic functions, J. Math. Anal. Appl. 196 (1995), 823-840.

Chafiq Benhida UFR de Math~matiques - URA CNRS 751, B~t. M2 Universit~ des Sciences et Technologies de Lille F-59655 Villeneuve D'Ascq Cedex France e-mail: b enhida@gat, univ-lille 1.fr

Dan Timotin Institute of Mathematics of the Romanian Academy PO-Box 1-764 Bucharest 70700 Romania e-mail: [email protected]

Mathematics Subject Classification: 47A45, 47A55.

Submitted: February 25, 1999 Revised: August 3, 1999

Finite rank perturbations of contractions

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