HOUSTON
JOURNAL
OF
MATHEMATICS
Volume 21, No. 1, 1995
UNITARILY
EQUIVALENT
COMPOSITION
RANDALL
COMPACT
OPERATORS
K. CAMPBELL-WRIGHT
Communicated by Vern I. Paulsen.
ABSTRACT. If •b is an analytic function taking the unit disk D into itself
then the compositionoperatorC4 canbe definedonthe Hardy spaceHP(D)
for i _•p < co by C• (f) = f o•b.In thisworkit is shownthat if somepower
of C• is compactand q•hasa nonzeroderivativesat its uniquefixedpoint
insidethedisk,thenC4 is unitarilyequivalent
to Cq,onH 2(D) if andonly
if the equivalence
C4, - U*C•,U can be inducedby a unitary composition
operatorU = Ca with a(z) = eiøz.
1. Introduction.
If •b is an analytic function taking the unit disk D into itself then
the composition
operatorC4 can be definedon the Hardy SpaceHP(D)
for I _<p < oo by C•(f) = f o •b.An overviewof the propertiesof these
operatorscanbe foundin the recentbooksby Cowenand MacCluer[7]or
Shapiro[13].
Thiswork
explores
when
twocompact
orpower
compact
(thatis,C•/
compactforsomepositiveintegerN) composition
operators
C• andCe are
unitarilyequivalent(that is, C4 = U*C•U for someunitaryoperatorU)on
the HardyHilbertspaceH2(D). In the caseof compactor powercompact
compositionoperators,•band • haveuniquefixed pointsin D (denotedby a
andb respectively)
[5,page128].Shapiro[12]hascompletely
characterized
1991 Mathematics Subject Classification. Primary 47B38 Secondary 47B07.
Key words and phrases. Composition operator, unitary equivalence.
Research supported by a Dana Foundation faculty development grant from the
University of Tampa.
189
RANDALL
190
K.
CAMPBELL-WRIGHT
the compact compositionoperatorsin terms of the Nevanlinna counting
function.The resultof this workis that for C• compactor powercompact
with•(a) • 0 it turnsoutthat C• isunitarilyequivalent
to C• onH2(D)
if and only if the equivalence
C• - U*C•U can be inducedby a unitary
composition
operator
U - C•. This• is necessarily
oftheform•(z) - eiez
for somereal number•) [11,chapter2], sothat the operatorequivalence
is
thesameasthe functional
equation
•(z)The author[2], [3], [4]haspreviously
exploredothercomposition
operator equivalences
and foundthat the obviousequivalences
(inducedby
an invertiblecomposition
operator)arenot alwaysthe onlyones[3, section
4]. The presenttheoremshouldbe compared
with the followingresults,in
which the obviousequivalencesare the only ones.
Define the iterates of •bby •bl - •b, •bn- •bo •bn-1 for n a positive
integer.
Thenotation
•b© willbeusedforderivatives.
Theorem A. [3, corollary3.2] Suppose
qb,not a diskautomorphism,
has
its fixedpointa in D andsuppose
qbn
(0) is neverequalto a for n a positive
integer.ThenC• is unitarilyequivalent
to C• onH2(D) if andonlyif
q•(z)-- ei•b(e-i• z) for somerealnumber
O.
TheoremA coversmany of the sameoperatorsasthe theoremin this
article, but there are alsomany examplesof compactcompositionoperators
for which•bn(0) - a for somen. For instance,if
•5(z)
= O.5(1]a])z1-•'• -ka
with a in /]), then •b(a) = a and •b(O)= a. This C4 is compactbecause
I1•11•< 1 [11,page28].(Anevensimpler
example
is •(z) -- 0.5z.)
Theorem B. [3, Theorem3.5]. C4 is isometrically
isomorphic
to C• on
HP(D)for 1 •_p • c•, p • 2 if andonlyif qb(z)
- ei•b(e-•ez) for some
real number
O.
TheoremB makesno hypotheses
on C4 and C•, but there are very
few isometricisomorphisms
on HP(D) with p • 2 [9]. The Hardy Hilbert
spaceH2(D) hasmanyisometric
isomorphisms
(unitaryoperators).
UNITARILY EQUIVALENT COMPACT COMPOSITION OPERATORS
191
TheoremC. [•, Theorem
5]. Suppose
C• is compact
forsome
positive
integerN andqf(a) • O. ThenCq•is similarto C• onHP(D) for I •_p < c•
if andonlyif q)-- a o•boa -1 ,forsomediskautomorphism
a.
The proof of Theorem C is a direct calculationin the spirit of the proof
below,but is muchmoreinvolved
because
similarity(C4-- S-1C•S forS
invertible)is a moreflexiblepropertythan unitary equivalence.Whereas
unitary equivalencepreservesanglesand normsexactly,similarity only preservesthem approximately.
2. The
main
theorem.
TheoremD. Suppose
C• is compact
for some
positive
integer
N and
qf(a)• O. ThenCq•is unitarilyequivalent
to C• onH2(D) if andonlyif
qJ(z)- eiø•b(e
-iøz) for somerealnumber
O.
Guyker[10]hasdemonstrated
the effectiveness
of usingthe orthonormal basis
b•-1 -1a12)1/2
(n--O,
1,2,...)
- (1
az
iz-a
- az
for studyingthe teducingsubspaces
of compositionoperators. One reason
for the naturalness
of thisbasisisthat (b•} is an orthonormalization
of the
derivatives
kernels
K? ) = (•_uz)•+•
'•!z• ,so
named
because
If,K?))-
f('•)(a).
It followsfromGuyker'sdescription
[10,page369]of b,•that
k--O
Onecancalculate
C•K•
© byusing
FakdiBruno's
formula
[8],[1,
RANDALL K. CAMPBELL-WRIGHT
192
pages
823]forthederivative
ofacomposition
offunctions.
,,-,*
u'(,•)
<f,•o•
>-<cof'K? >-- (f o½5)(")
m=O
• j a• =n
• aj •m
={f,
• K•
•) • n](•(a))•(•"(a))•2'"(•(
•)
m=O
• j a• =n
• aj=m
(using
thefactthat•(a) = a)
Thus,
n!(½5,(a))•(½5"(a))
•' . . . (½5('•)
(a))
m=O
• j a• =n
(1!)•al!(2!)•'a2
!.--
Note
thatforn>_1,C;K•
© isalinear
combination
ofderivative
kernels
K•© withm > 1.Thisfactwillcome
uplater.
Thematrix
forC0withrespect
tothebasis
{b,}(orthederivative
dernels
{Ka(n)})is
lower
triangular
with
diagonal
[1,½5'(a),½5'(a)2,
...] [10,
page
369]
sotheadjoint
C;isupper
triangular.
The
adjoint
C;iscompact
andhasone-dimensional
eigenspaces
forA: ½5'(a)
• [6,page
93],sothe
.k- ½5'(a)
• eigenvectors
ofC•arelinear
combinations
ofbo,
b•,b2,
..., b,,
(orofK•,K'•,...,
K•('0)
ß
Proofoftheorem
D. <=. Obvious.
• . Several
restrictions
canbemade
on½5
and•bwithout
loss
of
generality.
IfCoisunitarily
equivalent
toC•then
C•isalso
compact
and
½5'(a)
=•b'(b)
[4,Lemma
1].Since
Cff=C0t•
one
can
assume
that
N= 1.
One
canalso
assume
thata = b,since
[a[= lb[[3,corollary
3.3]andone
could
conjugate
byarotation
tomake
thefixed
points
of½5
and
•bthesame.
Thecases
a = 0anda •: 0require
substantially
different
approaches
and
will be dealt with separately.
UNITARILY
EQUIVALENT
COMPACT
COMPOSITION
OPERATORS
193
Case 1. a: b: 0, ½5'(0): 0'(0) •= 0, both C0 and C• are compact.
The short proof is that C0 unitarily equivalentto C• implies that
•5= a o0 oa-• where• is a diskautomorphism
taking0 to 0 [4, Theorem
5 - quotedasTheoremC in thispaper].The onlysuchdiskautomorphism
is a rotation•(z) = e•øz.
However, it should not be necessaryto quote a similarity theorem
(with a long combinatorialproof!) in orderto provean analogous
result
about unitary equivalence.Unitary equivalenceshouldbe easy enoughto
calculate with directly. The followingdirect unitary equivalencecalculation
is givenin hopesthat it will be easierto followthan the proofof [4], thus
makingthe presentpaperindependent
of [4]andmaking[4]moreaccessible.
The followinglemma providesa meansfor linking •band 0. It is easily
provedby equating Maclaurin series.
Lemma A. 65(z)= eiøO(e-'øz
) for somerealnumber0 if andonlyif
O(•)(o)= •-(•-•)iøO(•)(o)/orall,• _>o.
For a = 0, Guyker's basis reducesto the standard orthonormal basis
b,•= z'• (n = 0,1,2,... ) andthederivative
kernels
areK?) = n!z'•. It
followsfrom Formula(2) that
(O,(o))•(O"(o))a'---(0(•)(o))•
m-'-O
• j aj -'-n
NowC; andC• areuppertriangular
with respect
to the standard
orthonormal basis and both have one-dimensionaleigenspacescorrespond-
ing to the diagonalentriesA = •b'(a)'• [6, page93]. Thus the subspace
ofHa(D)spanned
by{1,z,z2,... ,z'•} isinvariant
under
C; orC•. Since
C•,= U*C•U andC• = UC•U*, it follows
that U andU* arebothupper
triangular- and hencediagonal.Thus U(z'•) = c,•z'• wherethe diagonal
entries c,• of U have absolute value 1.
RANDALL
194
K.
CAMPBELL-WRIGHT
ApplyUC; andC•U to z• to getUC;(z'•) = C•U(z'•),yielding
• m!Cm
zm
(3)
m=0
• j aj =n
• ay--m
(0'(o))"'(0"(o))",..- (0(•)(o)
(l])"xal 1(2!)"2a21.--
(½,(o))-,(½,,(o))-•-... (½(•)(o))-•
m=0
• j aj =n
(l!)"xal !(2!)"2a2!---
Equatingthe coefficients
of z"• in Formula(3) for m = n - 1 yields
(n-1)!c,•_•
[q•'(O)]'•-•'
•"(0)
_(n-1)!c,•
[•'(0)]'•-•'
•"(0)
(,•-•)!
•
(,•-•)!
•
Thisimplies
•b"(0)= c•C---•_•
%b"(0).
Equating
thecoefficients
ofzminFormula
(3) for m = n- 2 yields
'(O)]n--3C•m(O)
[(•'(0)]
n--4
[•)t/(0)]2
}
(n-3)!3! + [¾f
(n-4)!
8
=(n
_2)
,c,•
{[•'(0)]'•-3•'"(0)
(0)]
n--4
['•Dt'
(0)]
2}'
(n-2)!c•-2 (n-3)!3!+ (n-4)! 8
This reduces
to •b"'(0)= C•--c,%b"'(0)
(usingthe factthat •b"(0)= Cr•c•
2
-- 1
for all n).
Proceedingthrough all the coefficients
of zm in Formula (3) yields
•b(•)(0)=
c,•p(•)(0) forall k,n withk < n.
Cr•--k + l
Case 1A. a- b- O, •'(0) - •b'(O)• O, both •
•b"(0)• 0 and %b"(0)
• 0.
and •p are compact,
Onecantakeeiø- •"(o)
•"(o)_ c•_•
c• foranyn. Thenforallk,
k-1
On-k-l-1
UNITARILY
EQUIVALENT
COMPACT
COMPOSITION
OPERATORS
195
sothatd(•)(0): e-(•-•)iø!b(•)(0)
andd(z): eiø!b(e-•øz
) byLemma
A.
Case lB. a = b = 0, 9•(0) = !b•(0)• 0, bothC• andC½are compact,
= ½"(0)=0.
If 4"(0) = 0 thenthis gap(andany subsequent
ones)cancausedif-
cultyinproving
that•(k)(0)= e-(k-•)•øtb(k)
(0) (sothatonecanuseLemma
A). Tothat end,consider
œ= {j- 1' •(•)(0) • 0 andj- 1 isnota multiple
of somesmallerelementof œ}.œis a finitesetwhichgenerates
(by taking
integerlinear combinations
of elementsof œ) the samealgebraicideal 27
that is generated
by {j - 1 ß9½•)(0)
• 0}. Let G bethegreatest
common
factor
ofœ(also
theGCFof27).Forj - 1inœ' •pO')
•(j)(ø)
_ c•_j+•
c• foranyn.
(0)
Since G is some integer linear combinationof elementsof œ it followsthat
c, is constantfor any n. Thus Cr•--G
c"can be expressed
as ½Giofor some
Cr•--G
•. Thenforanyk,•(•)(0) • 0 implies
that
Cn-k+l
= ½(k-1)iO•(k)(O),
sothat•(k)(0)= e-(k-•)iø!b(k)(0).
ByLemma
A, •(z)= eiø!b(e-iøz
).
Note that in Case 1A, 0 is uniquely determined,while in Case lB
several(G to be precise)valuesof 0 sufficeto make½GiO
constantand
•(z) = e•ø!b(e-•øz).
If oneconstructed
the setœin Case1A, it wouldbe
œ= {1}, whichgenerates
the ideal27= g.
Case 2. a = b • 0, 9•(a): !b•(a)• 0, bothC• andC½arecompact.
Letf• = b•+ •=o z•,•b•denote
a generator
oftheone-dimensional
eigenspace
of C• corresponding
to ,X= 9•(a)'• [•, page[}3]andlet #,• =
b•+ Y•=o!/•,•b•denote
a generator
ofthe,X= 9•(a)• eigenspace
ofC}.
One
can
use
Formula
(1)forb•interms
ofK?) and
Formula
(2)for
to find that fo = go = boand f• = #• = b• + abo.It is not necessary
to
completelycalculatethe higherordereigenvectors
in orderto completethis
proof. Someof thesecalculationswill be broughtup as needed.
C4 unitarilyequivalent
to C½implies
that G : U*C•U forsome
unitary operator U and Uf,, = c,,#,, with c• constant. Sincef0 = go =
bo, Icol= 1. In fact,onecanassume
withoutlossof generality
that co= 1
(otherwisedivideU by co).
196
RANDALL
K. CAMPBELL-WRIGHT
One can now prove by induction that c• = 1 and the' derivatives
•b(•)(a)= •p(•)(a)forall n, sothatU = I and•b= •p.Thiswillcomplete
the proof.It, is alreadyknownthat •b(ø)(a)
= a = •b(ø)(a),
andco= 1
by assumption.For n = 1, it hasbeenobserved
that •b•(a)= •p•(a).Also,
a = (b• +abo,bo) = (f•,bo) = (Ufo, Ubo)= (c•b• + c•abo,bo)= cxa.Thus,
cl=l.
Assume
thatcj = I and&(J)(a)= •(J)(a)forallj < n. An explicit
calculationof the coefficient
x.,o in the eigenvector
f. will helpshowthat
c• = 1 and&(•)(a)= •(•)(a). (Note:othercalculations,
likex•,•_•, lead
to the sameconclusion.)
n--1
(4)
• =• + • •,•
j=0
n--1
=(-a)•(1-i•i•)•/•K•+ • x•,j(-a)J(
1-i•i•)•/•K•
j=0
+•
(terms
involving
K?))
k=l
(by Formula(1)). UsingFormul• (2) and (4) andequatingthe coe•cients
ofK• intheequality
C•f• = •(a)•f• yields
n--1
(-a)•(1- Ia•)l/•+ •xmj(-a)J(1- al•)•/•
j=O
n--1
=•,(a)•(-a)•(1
-lal•)l/•+ &'(a)
• • /•,•(-a)Y(
1-lal•)X/•,
j=O
which yields
n-1
(-•)• + • •,•(-•/= 0,
j=0
n--1
(•)
xmo
= -[(-a) • + • xmj(-a)J]
ß
j=l
UNITARILY
EQUIVALENT
COMPACT
COMPOSITION
OPERATORS
197
One can showthat c• - 1. First observethat for j < n,
Then x•,0 = c•,y•,,oand Formula(5) gives
n--1
n--1
(-a)•+ • x•,j(-a)5-- c•(-a)•+ • x•,j(-a)5.
j=l
j=l
so that c• = 1. Thus, x•,,5 = y•,,j for j < n and f• = g•. It follows
byequating
thecoefficients
ofK• inC;K•
© = C•K•
© (see
Formula
(2)) that •b(•)(a)= •h(•)(a).By induction
onn, c• = 1 and•b(•)(a)=
•h(•)(a)forall n, so•b= •h. []
3. Concluding comments.
The similaritycalculationin [4] waswritten usingeigenfunctions
of
C•, but it wouldreadthe sameif it hadbeenwritten usingeigenfunctions
of
C; andpointevaluation
kernels.
Thustheunitaryequivalence
calculation
in Case I of the proof of Theorem D in this paper can be consideredas
a specialcaseof the similaritycalculation
in [4]. Forthe similarityC• =
S-xC•S the calculations
revealthat S is lowertriangular,but onemust
examineeachdiagonalin the limit as n goesto infinity in order to reach
equalitiesbetween•b and •h. This requiresa good deal of combinatorial
bookkeeping.By contrast,in the unitaryequivalence
C4 = U*C•U it
turns out that U is diagonalwith diagonalentriesof absolutevalueone,so
that many terms drop out of the analysis. Thus the calculation in Case 1
of this papercanbe viewedasthe coreequalitythat [4]hasto approachin
the limit.
One of the obstaclesin doing explicit calculationswith composition
operatorsis missingderivatives.This paperconsiderspowercompactcom-
positionoperators
with •b•(a)• 0. If •b•(a)= 0 thenthe spectrum
of C•
reduces
to {0, 1} andmucheigenfunction
structureis lost[5],[6, Theorem
4.1]. It is clearthat differentmethodswouldbe requiredto considerequivalences
of thesecomposition
operators.Someroughprogress
is madeusing
normestimates
in [3, Lemma2.5, Theorem2.6],but nothingapproaching
the precisionof the present paper.
198
RANDALL
K.
CAMPBELL-WRIGHT
As another example of the impact of missingderivatives, in Case 1 of
TheoremD, where•b(O)= 0 and•b•(0) • O,onecancomplete
the calculation
in onestepif •b•(0) • 0 but morework(andan algebraicideal)is required
if •b"(0)= 0.
Future investigationsinvolving closecontact with the finer structure
of compositionoperatorswill surelyface more of these "missingderivative"
issues.
My regardsto the refereeof this paper, who made severalsuggestions
to improve the presentation. And my thanks to the referee of an earlier
version of this paper, who suggestedthat I do the calculation in Case 1
rather than quotingthe (muchmoredimcult)Theorem5 of [4].
REFERENCES
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formulas, graphs, and mathematical tables,Dover, New York, 1965.
2. Campbell-Wright, R.K., On the equivalenceof composition operators, Thesis, Purdue University, 1989.
3.
4. __,
, Equivalent compositionoperators,Integ. Eq. Op. Th. 14 (1991), 775-786.
Similar compactcomposition
operators,Acta Sci.Math. (Szeged)58 (1993),
473-495.
5. Caughran, J.G. and Schwartz, H.J., Spectraof compactcompositionoperators,Proc.
Amer. Math. Soc. 51 (1975), 127-130.
6. Cowen,C.C., Composition
operators
on H 2, J. OperatorTheory9 (1983),77-106.
7. Cowen, C.C. and MacCluer, B.D., Composition operatorson spacesof analytic functions, CRC press, Boca Raton, 1995.
8. Fak di Bruno, F., Note sur une nouvelle formule de calcul diffdrentiel, Quart. J.
Pure Appl. Math. i (1857), 359-360.
9. Forelli, F., The isometriesof Z p, Canad. J. Math. 16 (1964), 721-728.
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11. Schwartz, H.J., Composition operators on H v, Thesis, University of Toledo, 1969.
12. Shapiro,J.H., The essentialnorm of a compositionoperator,Ann. of Math. (2) 125
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13. •,
Composition operators and classicalfunction theory, Springer Verlag, New
York, 1993.
Received November 10, 1993
Revised version received August 1, 1994
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TAMPA, TAMPA, FLORIDA 33606
E-mail address: rcampwri@cfrvm.bitnet