0378-620X/89/020227-1451.50+0.20/0 (c) 1989 Birkh~user Verlag, Basel

Integral Equations and Operator Theory Vol. 12 (1989)

EXTENSIONS

OF I N T E R T W I N I N G

Vlastimil

RELATIONS

Pt~k and Pavia V r b o v ~

Let U 1 and U 2 be two isometries acting respectively on the Hilbert spaces H 1 and H 2 and let M 1 be a Ul-invariant subspace of H I . Let X : M I --+ H 2 be a bounded linear operator intertwining U 2 and the restriction to M 1 of U I , X( h I M r

= U2X

9

The authors give n e c e s s a r y and sufficient conditions existence of a bounded operator Y:H I --+ H 2 which extends X fies Y U1 = U2 y .

In the course authors

investigated,

of their

study

not long ago,

of generalized

intertwining

Hankel

relations

for the and satis-

operators

the

of the form

X T *1 = T 2 X

where H2

TI

and

T2

respectively.

to a relation

are given

contractions

The problem

considered

acting

on Hilbert

spaces

[3] was to lift the above

HI

and

identity

of the form

*

Y U1 = U2

y

the

U. being the minimal isometric dilations of the corresponding T. . I l It turns out that there is an essential difference between this lifting problem

and the classical

the form

X T1 = T2 X

considered

above

additional

condition

additional

requirement

to be satisfied

(where

by

one of Sarason-Nagy-Foia~.

may always I'* I

be lifted

is to be

dilated

has to be imposed assumes

to

Whereas

to a coisometry

for the lifting

the form of a stronger

X . The authors

it is easy to see that any operator

[4] called Y

relations

of

Y U 1 = U 2 Y , in the case

it

U1

to be possible. boundedness

R-boundedness:

intertwining

U~

and

This

condition indeed,

U2 ,

228

Pt~k and Vrbov~

YU~ = U2Y , satisfies the relation are the orthogonal projections

Y = P(R2)Y = YP(R I)

onto the subspaces

the unitary part in the Wold decomposition operator

Y

satisfying

Y = P(R2)YP(RI)

(Xhl,h2) =(Yhl,h2)

of

R.I

Ui .

where the such that

P(Ri) uiIR i

is

As a compression of an

the operator

X

clearly satisfies

= IYP(RI)hI,P(R2)h2)

9

It follows that I(Xhl,h2) l ~ IYI IP(RI)hl I IP(R2)h21

and this condition is stronger in general than ordinary boundedness. referee of [3] called the authors'

The

attention to the fact that an extension

problem for shift operators considered by L. B. Page [2] also requires an additional continuity condition in order to possess a solution, while the corresponding problem for coisometries precise:

given a coisometry

subspace

MC

striction of IT01

TO

H

S*IM

S*

on a Hilbert space

and an operator

T O @ B(M)

H , an

H

which commutes with

S*

and such that

is replaced by an isometry the analogous Indeed, Page considers a unilateral

Hilbert space

an invariant subspace for

H , M M

S*-invariant

which commutes with the re-

is false in general.

operator on

To be more

it is possible to show that there exists an extension

to the whole of If

S*

is always solvable.

commuting with

SIM .

S

and

T

ITI

extension result shift TO

S

on a

a bounded

He then shows that operators

TO

for

which an extension is possible satisfy an additional condition in the absence of which the result may fail: he exhibits an example of a two-dimensional shift and an operator S ,

TO

such that no extension of

(For a shift of multiplicity

TO

will commute with

one an extension is always possible.)

In the same paper Page formulated

the conjecture

that the necessary

condition is also sufficient for the existence of an extension. jecture was subsequently proved in the particular case when dimensional shift by C. F. Schubert

[5].

S

The conis a finite

The general case remained open.

In the present paper the authors consider

the more general case

of isometrics and replace the commutation relation by a relation intertwining two different

isometries.

This has another further advantage:

by separating

the domain and the range spaces of the given operator a deeper insight is

Pt~k and V r b o v a

obtained.

229

We c o n s i d e r

subspace

two isometries

M I of H 1 and a bounded

operator

x(u 1 IMI) = h x The m a i n existence

result

of a b o u n d e d

operator

U1

zation

condition;

to the shift

parts

I.

and

around

Given we shall

by

of a H i l b e r t

from

onto

be no

doubt

B

adopt H

with

in the

the proofs,

.

vnp(Hi)V*nh

H For

.

for the

to the whole

of

H1

only refers

U2 .

A

by

operator

T

on

T-invariant

H

(resp.

subset AC H . If B is a A PB the orthogonal p r o j e c t i o n

and

we have UIE -L of

concerning

of the results A~L

we shall

, we have

centered

[I].

linear

the symbol

P(B)

whenever

to be denote

H = R ~

i

a unilateral U

so that

there

where

can

shift.

to

Pn ~

Denote

Since

B(H)

by

the space in

Z • = ~(L) Z .

the p r o j e c t i o n

, in other words,

(i - u n u * n ) h

and Z~t

K , used only

HC) R ; otherwise by

R

VIH = U . We shall

orthocomplements. as opposed

U n H = Ker U *n = H { ~ V n H •

h ~ H

theory.

is meant.

extension

convention

the book

the smallest

a given

we use

from d i l a t i o n

as well as for the theory

Red(A,T)

unitary

unitary

of facts

and a bounded

is an isometry

n ~ 0 ~

X

condition

that the condition

and

we denote

we shall u n d e r s t a n d

onto

1 - unu *n

A

statements

For each operator

and

space

UIR

the f o l l o w i n g

appears

U I I M 1 and U 2,

turns out to be a g e n e r a l i -

we suggest

containing

space

the m i n i m a l

results

H

. For brevity,

U e B(H)

are U - r e d u c i n g V ~ B(K)

H

as to w h i c h

If

U1

theorem

space

of

subspace A

for these

Inv(A,T)

subspace

extends

it is i n t e r e s t i n g

lifting

a Hilbert

denote

T-reducing)

which

sufficient

paper we use a n u m b e r

reference

the c o m m u t a n t

a Ul-invariant

and p r e l i m i n a r i e s

In the whole As a general

and

The c o n d i t i o n

of the isometries

Notation

U2EB(H2),

9

Y

U2 .

B(HI),

X : M I + H 2 intertwining

is a n e c e s s a r y

and i n t e r t w i n e s of Page's

U 1E

= h - vnp(H)V*nh

P

n

=

=

The r e l a t i o n P h = vnp(Hi)V*nh n

will play an i m p o r t a n t the equality

of norms

r$1e

in our further

work.

In particular,

it implies

230

Pt~k and Vrbov~

IPnhl = Ip(Hi)v*nhl for all

h~H

.

The following technical

lemma will play an important

r$1e in our

considerations. i.i

Let

Lemma.

MC

H

be invariant with respect to an isometry U .

A = (P(H •

Set

(M,V))-

Red

Then 1o

2~

Red ( M , U ) ~

the operator i.e.

I Inv(A,v)

relating subspaces of

greater generality. . Consider a

K

and

For simplicity, V-invariant Inv (QB,V)

This equality,

H

let us write

subspace Q

B C K

QB = P(Inv

Q Inv (QB,V) C Indeed, given a

to

A ,

is based on some iden-

which may be stated in a somewhat P

and

Q

for

P(H)

(QB,V))

.

of the inclusion

(i)

(QB)

y ~ Inv (QB,V)

Qy ~ (QB)-

and

and let us show that

in its turn, will be a consequence

follows from (i) that

V

9

The proof of the first part of the proposition

tities

P(H •

A ,

and its minimal isometric dilation may be

p(Hi)VIA = P(A)VIA V

G

coincides with the compression of

T = P(H•

identified with Proof.

R • = Inv (A,V)

Q

Hence

QB

we have

y • QB

IQyl 2 = (Qy,y) = 0

and it

so that

y = Py + Qy = Py . To prove inclusion and

QV = QVQ .

(i) observe

that

VHC

H

so that

VP = PVP

It follows that QV n = (QVQ) n = QVnQ

for every that

n .

QVnQBC

QV n = QVnQ

QB

To prove the desired for every

inclusion it obviously suffices

n ; this, however,

to show

follows from the identity

. Consider now the case

B = Red

(M,V) .

We have then

and Inv (A,V)

@

A = Inv (QB,V)

G

QB = P(Inv

(QB,V))

A = (QB)-

Pt~k and Vrbov~

231

Now let us prove PInv Since

pv*km = u*km

the inclusion (QB,V) C

for all PBC

Red (M,U)

m E M

Red

(M,U)

and

. we have

k ~ 0

.

Furthermore, pvkQB = pvk(I - P ) B C Red

(M,U) + U k Red

pvkB + p v k p B C

(M,U) ~

Red

PB + ukpB

(M,U)

.

We have thus proved PInv Since

(QB,V) C

and

Range Q • R

R

is

Red

(M,U)

V-reducing,

. we have

Inv (A,V) • R

and,

consequently, PInv the orthocomplement

of

It remains Since

first

Red

(A,V) C P ( K Q R ) R

in

H .

to prove the inclusion

(M,U) I~ R •

H

and

H • A

To prove

the inclusion

Red

R•

that

(M,U)~

lim VnQV*nh

for every

VnQV*nUPU*qm

,

m C M

P(R •

h ~ H .

Red

the inclusion

the identity

Red

QvnAc

A

a E A

and

n ~ 0 .

R•

Inv ( A , V ) Q A

Red (M,U)(~ R •

.

Inv (A,V) Since

to consider V*Q = QV*Q

P(R•

HC

= lira P h = n of the form

we obtain _ vPQV*qm

. and completes

QB~A

= p(A)vna

Furthermore,

A• 9

we observe

elements

_ Q)v*qm = vnQv*n-p+qm

and Qvna = p(A)QVna

for

(M,U)

(M,U)~

R•

QV n = QVnQ = (QV) n

QVnQB = Q V n B C so that

we have

Using the relation

E Inv (A,V) This proves

Red

(M,U)~

it suffices

VnQV*nuPu*q m = VnQV*n-P(1

Using

= R• ,

the proof of 1 ~ we deduce that

.

232

Pt~k and Vrbov~ (p(A)VP(A)) n = (p(A)v)np(A) = (QV)np(A) = Qvnp(A) = P(A)Vnp(A)

so that

V I Inv (A,V)

is the minimal isometric dilation of

P(A)V I A . N

The following proposition will he useful in the sequel. 1.2

Proposition.

a subspace of

B(HI)

U 1E

Let

H1

P2n E B(H 2)

and

B(HI)

and

U 2 E B(H 2)

invariant with respect to be the projections in

be two isometries ,

U 1 . For HI

n ~ 0 , let

and

H2

Pln ~

defined by the

relations

pl i n ,n n = - UIUI '

p2 I - U nU*n n = -2~2

X: M 1 -~ H 2

Suppose

x(h i 1)

satisfies the relation

(2)

= u2x

Then the following assertions are equivalent

l~

IP Xml for all

m ~ MI ,

2~

IPn ml

n >~ 0 .

IP(H~I ~k v2kXmkl--< ~IP(HI) k~ Vlkmkl

for all finite sequence

Proof.

ml,m 2 ..... m k ~ M I 9

Take a finite sequence

m I ..... m k ~ M 1 . Then

n n v , n - k .~mk = PIH~) k-1 _[ V2kXmk = P(H~) V ,2n k__LlV2 ,n (V2P(H ~ n V2

,n

:iv2n-k

2 n =

Suppose condition 1 ~ holds.

Then

n

n

n

alVIP(HI)VI k=~IV1 mkl

~,n~2= ~ .n-k = " 2 ~nAk~lUl mk "

= v,np2

n

n

Mi

Pt~k and Vrbov~

233

Now suppose

Ip2Xml

2 ~ satisfied.

We have then,

two examples

suggested

results

of this section as well as Theorems

Example

i. Let

UI

its restriction

be the multiplication

to the Hardy

the identity mapping

then obviously X

X(UIIM1)

space of

by the referee

MI

into

z

on

M I = H2~

H 2 , i.e. Pln= 0

H I = L 2, and let

i ~ of 1.2. This shows

(2) alone does not imply condition

i~

to a mapping

for which

only the zero operator the relation

L2

into

can intertwine

(n ~ O)

Example

2. Denote by

and the minimal lateral

HI = H2~

L2 by

X

cannot be

is contained

and let us define z

extension

L2(~ L 2

Ul-invariant

that

YU I = U2Y . Indeed,

and a unilateral Y

f ~ H2

shift:

in the inter-

is the following

on

and a bilateral

unitary

shifts on

following

a bilateral

for

the operator

and this is the zero snbspace.

sum of multiplications

sum of a unilateral

n a 0

Observe

YU~ = U~Y shows that the range of

/~U~ H 2

A little more challenging

direct

H2

U 2 be

that the intertwining

extended

of

the

L 2 . If we denote

X : f--+ f

for all

relation

Y

illustrate

3.2 and 3.4.

by

H 2 = H 2 . Set

= U2X . Since

does not satisfy condition

section

n ~ 0 ,

= a Iplml

The following

X

m 6 M1 ,

n _t V2,n Xm I = Ip(H~)V2nxm I <= a ~1,,n = IV2P(H2) ~n vl m[ =

~ IVlP [H• i)~i m

by

for

H2

shift. V1

subspace

Then

of

defined by

and

UI

VI:

U I ~ B(HI)

L 2 ; thus R 1 = (0) O

is the direct

U1

as the is a direct

L2, A~I = H 2 0

sum of two bi-

(f,g) --+ (zf,zg) . Consider

M 1 = {(f,f)

(0)

the

: f 6 H 2} . It is easy to show

that

Inv (~,U~) Red

= {(h,f)

( ~ , U I) / ~ I

: feL 2

=~I

and

=H2~

C0) ,

P(HI) Red ( ~ , V l ) = H ! ~ (0) and Here, H2

and

if

U 2 6 B(L 2)

as usual,

H 2 = L2~H

denotes

Red ( M I , V I ) =

Inv ~(H~)

{(f,f) : f E L 2] ,

Red (MI,VI)) = L 2 ~

the orthogonal

the identity

is the operator

is defined

Red (M1,U1) = H 1 ,

projection

of

L2

(0). onto

2 .

This illustrates

--+ e 2

P+

P+f = h ] ,

by

1~

in Lemma

of multiplication

X : (f,f) --+ f

then

by

i.i. Furthermore, z , and if

X(UIIM I) = U2X

and

X :M 1 IxI = i / ~

.

234

Pt~k and Vrbov~

It is easy to show that the set of all extensions intertwine

U1 _

--+ ~f + (i tension of

and

~)g X

U2

consists

with some

of all operators g ~

~ ~

. The choice

whose norm does not exceed

2.

Operators

intertwining

of two coisometries.

from the commutant

2.1

S~ppose

U 1 ~ B(HI)

operator

MIC

H1

and

be given such that

H1

which Y~ : (f,g)

yields

an ex-

X .

of coisometries operators

lifting

intertwining

of extensions

theorem.

are isometries and let

U 2 C B(H 2)

M 1 9 Suppose that the

U~MIC

satisfies

X E B(MI,H2)

xIu I MII then there exists a

~ ~ 1/2

of extending

directly

Theorem.

to

In this case the existence

may be deduced

a subspace

X

of the form

that of

parts

We deal first with the problem restrictions

of

, which extends

Y ~ B(HI,H2)

x

and intertwines

u 91

and

U2 , Y I MI ~ Y

Proof.

Since

P(M1)UIX*

X(U~

Y UI

= U2

IY1

= Ixl

I M 1) = U~X

the contraction To see that,

it suffices

I = Inv (MI,UI) Z @ B(H2,/)

X*: H 2 -~ M 1

to see that the minimal may be identified

to observe

P(MI)UIP(MI) According

such that

Now define of

, its adjoint

P(MI)U 1 I M 1

and, consequently,

element

,

satisfies

= X*U 2 . It is not difficult

a

,

*y

UIM~C

with MS

to the commutant

UIZ = eu 2 ,

dilation (MI,UI)

Iz[ = Ix*[ of

theorem

and

and

PIM1Z = X*

Z

considered

IYI = IZ*l = IZl = IX*l

-

set

there exists

We have then

YU~ = U~Y

of

P(MI)UIP(M~)

For brevity,

lifting

as the adjoint

U I I Inv whence

= (P(MI)UIP(MI))n

Y: H I --+ H 2

B(H2,H 1) .

that

isometric

= [XI

as an

= 0

235

Pt~k and Vrbov~

For

hI ~ M 1

and

k2 6 H2

(Yhl,k2) It follows

that

3.

we have

= (hl,Zk2)

Y

= (h1,P~iZk 2) = (hl,X*k2)

is an extension

The general

of

siderations.

3.1

Theorem.

Let

X .

isometries

We begin by considering

a different

9

case of two isometries

The case of two arbitrary

requires

= (Xhl,k2)

requires more delicate

two particular

con-

cases - each of them

treatment.

H1

isometries acting on

and

H1

U]-invariant subspace

H2

and

be two Hilbert spaces and

H2

M lC H 1

respectively.

U 1 , U2

two

Suppose we are given a

and a bounded linear operator

X: M 1 --+ H 2

such that X(U 1

r M I) o u2x

Then the following assertions are equivalent there exists a bounded linear operator

1~

such that

Y: H l -~ H 2

Y U 1 = U2 Y ,

YJM I = P(R )x , Iyl ~ a

the operator

2~

X

,

satisfies the inequality

aIPmJ for all

m C M1

n

YU 1 = U2Y = Y*(H 2

e

= ply,p2 n>

n ~ 0 .

It is easy to see that 1 ~ implies

Proof.

n

and all

2 ~ (see also

[2]).

if

then YUIH I_C U2H 2 for n > 0 . This implies that Y* Range p2 n y,p2 U2H 2) C (H 1 • UIH I) = Range pl in other words n

, or equivalently,

p 2 y p 1 = p2y n n n

n

.

~

9

This

implies,

0 , p2Xmn = P~P(R~)Xm Ip2Xml

for all

Indeed,

n

m E M1

~ IP2YI and

= p2ymn = P2ypImn n " IPnlml _< IYl

n ~ 0 .

We have thus

Iplml

This proves

2~.

for

n

m ~ M1

and

236

Pt~k and Vrbov~ Conversely,

operator

assume that 2 ~ is satisfied.

Z 0 : [P(H~) Red (MI,VI))- --+ H •2 ZoP(Hi)V~km

for all

m C MI

and all

guaranteed by assumption

Z0

the

defined by the relation

= P(H~)v~kxm

k ~ 0 .

The existence and continuity of

2 ~ and Proposition

Let us introduce

Denote by

1.2; in particular,

two contractions

T1

and

T2

Z0

is

IZ01 ~ ~ .

defined as follows

T I = P(AI)V IIA I = P(Hi)V 11A1 where

A 1 = P(Hi) Red (MI,V I)

and

T 2 = e(H~)V21H i and let us show that of the form

Z0T 1 = T2Z 0 .

P(Ht)V~km

with

It suffices to prove this for elements

m ~ M1

and

k ~ 0 .

Indeed, ZoTIP(H~)v~km

= ZoP(H~)VIP(Hf)v~km

: p (Hi~V 2 " - 2*k~' Avlm : P(H~)v~ku2xm = P(HI)V2P(H~)v~kxm

= ZoP(H~)VIV~km

: P (H~)V2(P(H2)

= T2ZoP(H~)v~km

According to the commutant there exists an operator

= Z~P(H• *k"u l m u " i'-i

+ P(H~))V~ kXm

9

lifting theorem and Lemma i.i

Z : Inv(Ai,Vl) --+ Inv(H~,V2)

such that

ZV 1 ] Inv(Ai,V I) = V2Z ,

z0=P(s )z I A 1

,

Z(Inv(AI,V 1) ~ A I) C Inv(H~,V 2) @ H i and

Set

Y = ZP(D 1) I H I

Lemma i.I we have and that

where

D I = Inv(Ai,V I) ~ J

Inv(H~,V2) @ H E = H 2 ~ R 2 = B~

D I = Red (M1,UI)(~ R~

so that

DI

so that

is reducing for

I .

According

to

Range Y C H 2 UI .

It follows

237

Pt~k and Vrbov*

YU I = ZP(DI)U I = ZUIP(DI) To complete P(R~)X

.

Since

= ZVIP(DI)

the proof,

Y = P(R~)Y

m ~ MI

and each positive

already

that the relation It follows

let us show that

n ,

YU 1 = U2Y for

9

is an extension

to prove that,

p2ymn = P2Xm implies

= V2Y = U2Y

Y

it will be sufficient

integer

that,

= V2ZP(DI)

"

of

for each

We have observed

p2y = p2yp1nn

for all

n .

m E M1 ,

p2ym = Pn2yplm ~ V2 P (H2) • V2,n ZP (D1) Pnlm " n Since

plmn ~ Red(MI'UI) (~ Rt ~ D1 n

•

~n

the last expression

equals

i

V2P(H2)V ~ ZPnm = V2P(H2)V n s ,n n • ,n 2 ZVIP(HI)V 1 m . E A1

Observing

that

P(Hi)vlnm

we may simpl$fy

the above expres-

V2(P(H~)Z)

i ,n n • ,n n • ,n P(HI)V 2 m = V2ZoP(HI)V 1 m = V2P(H2)V 2 Xm -- p2Xm n

sion to

Let us state explicitly theorem.

If

R~ = H 2 3.2

and

every UI .

is a shift operator

so that,

Theorem.

H1

U2

S2

in the preceding

Let

H1

and

~2

a shift operator on

x C H 2 ). Let

two particular

Let

M1

X : M 1 -~ H 2

then

theorem,

U2 Y

cases of the preceding is an isometry

(thus

be a subspace of

S~S 2 = 1 H1

for which

will be an extension

be two Hilbert spaces, H2

"

UI and

of

X .

an isometry on s~nx -~ 0

invariant with respect

for to

be a bounded linear operator such that

x ( h I 11

hx 9

Then the following assertions are equivalent i~

there exists a bounded Zinear operator and satisfies YU 1 = S2Y , IYI ~ a

2~

the operator

for

a~

X

m ~ M~

.

satisfies

and

all

the inequality

n > 0

.

Y : H I -~ H 2

which extends

X

238

Pt~k and Vrbov~

Specializing conjecture 3.3

stated

Theorem.

further we are able to establish

in the 1971 paper

Let

H

defined on

M

S

of the

[2] of L. B. Page.

be a Hilbert space and

respect to a unilateral shift

the validity

on

which commutes with

M

Let

H .

a subspace invariant with X

be a bounded operator

S 9

Then these are equivalent 1~

there exists an extension con~nutes with

S

2~

and

Y

of

X

to the whole of

H

which also

IYl ~ ~ ,

IPnXm I ~ aIPnm ] for all

m s M

and all

n ~ 0

(here

P

= 1 - sns *n ). n

3.4

Theorem.

H 1 , U2

Let

HI

and

H2

be two Hilbert spaces,

a unitary operator acting on

-invariant subspace

MIC

HI

H2 .

U1

an isometry on

Suppose we are given a

and a bounded operator

x : M 1 --+ H 2

U lsuch

that

xIu I I MII = u2x Then there exists a bounded operator YU 1 = U2Y

Y : H I -+ H 2

such that

,

YIM 1 = X ,

IYl : Ixl Proof. U1 .

Let

V 1 ~ B(KI)

Denote by

W

V I* I Red(M I,V 1)

be the minimal

the compression

is the minimal

to

unitary

extension

M1

U *1

isometric

of

of the isometry

and let us show that

dilation of the coisomtry

W

Indeed,

P('l)V~ k I M i = (vk I Mi)* : (ui k I Mi)* = (U I I MI )k*

IMi)*k: W k

: (U 1 for all

k ~ 0 . Since

the minimal

kYOV~ kMl

isometric

x According

: Red(MI,V I)

dilation :

of

.~x(~ I L MI)

to Lemma

W . :

it follows that Since

~x.*

U2

* I Red(MI'VI)

VI

is unitary we have

.

2.8 of [4] there exists a unique

operator

is

239

Pt~k and Vrbov~

X 0 : Red(M1,Vl)

--+ H 2

such that

X 0 = U~X(V~

I Red(MI'VI))*

= U~XV 1 I

Red(MI,VI)

'

X0 I M 1 = X ,

lXol NOW set to

Y = XoP(RedfMI,VI) ) ] H 1 .

R , we have then,

for each

YUIh = XoRUlh At the same time,

Ixl.

=

for

Abbreviating

P(Red(MI,VI) )

h C H1 ,

= XoRVIh

= XoVIRh

= U2(U~XoV1)Rh

= U2XoRh = U2Yh

9

m 6 MI ,

Ym = XoRm = X0m = Xm . The proof is complete. Theorems general 3.5

3.1 and 3.4 make

it possible

to describe

fully the

situation.

Theorem.

Let

H1

isometries acting on

and

HI

H2

and

space invariant with respect to

U1

be two Hilbert spaces,

H2

respectively.

U1 .

Let

Let

MICH

X : M 1 --+ H 2

and 1

u2

two

be a sub-

be a bounded

linear operator such that

x(u1 i MI) = u2x Then

YU 1 = U 2 Y

for all

X

may be extended to an operator

i f and only i f there exists a positive

m ~ M1

and

a

these are equivalent

there exists an operator

y : H 1 -+ H 2

such that

YU I = U2Y , YIM I = X , IP(R2)Y I ~ IP(R2)X I 2~

for every

m E MI

and every

and

n ~ 0 ,

IP(R~)YI

H2

such that

n ~ 0 .

More precisely: 1~

Y : H1 ~

$ a ,

satisfying

240

Pt~k and Vrbov~

Proof.

Suppose first condition 2 ~ satisfied.

X = P(R2)X + P(R~)X

Y1

of

on the whole of P(R~)Y = Y2

in the form

Then apply Theorems 3.1 and 3.4; we obtain an

P(R2)X

IYI[ = [P(R2)X [ and

and

X

and verify that each summand satisfies the same

intertwining relation. extension

Write

and an extension

Y2

of

P(R~)X

such that

[Y21 $ m , both fulfilling the intertwining relation

H I . The operator

Y = Y1 + Y2

satisfies

P(E2)Y = YI '

and satisfies thus the requirements of the theorem.

The authors acknowledge a debt of gratitude to the referee for a number of comments which have contributed to the improvement of the presentation.

References

[i]

B. Sz.-Nagy and C. Foia~, Harmonic analysis of operators on Hilbert space, Akad~miai Kiad~-Budapest, North-Holland Publishing Comp.-Amsterdam-London, 1970.

[2]

L. B. Page, Operators that commute with a unilateral shift on an invariant subspace, Pac. J. Math. 36 (1971), 787-794.

[3]

V. Pt~k and P. Vrbov~, Lifting intertwining relations, Intt Eq. Operator Theory ii (1988), 129-147.

[4]

V. Pt~k and P. Vrbov~, Operators of Toeplitz and Hankel type, Acta Sci. Szeged (in print).

[5]

C. F. Schubert, On a conjecture of L. B. Page, Pac. J. Math. 42 (1972), 733-737.

Institute of Mathematics, Czechoslovak Academy of Sciences Zitng 25 115 67 Praha i, Czechoslovakia

Submitted; Revised:

A p r i l i0, 1 9 8 8 J u l y 15, 1 9 8 8