Paci ic
Jou nal o
Ma hema ics
THE ESSENTIAL NORM OF A COMPOSITION
OPERATOR ON BLOCH SPACES
Al onso Mon es-Rod
´
ıguez
Volume 188 No. 2 Ap il 1999
PACIFIC JOURNAL OF MATHEMATICS
Vol. 188, No. 2, 1999
THE ESSENTIAL NORM OF A COMPOSITION
OPERATOR ON BLOCH SPACES
Al onso Mon es-Rod
´
ıguez
We exp ess he essen ial no m o a composi ion ope a o on
he Bloch space and he li le Bloch space as he asymp o ic
uppe bound o a quan i y in ol ing he inducing map and
he Pick-Schwa z Lemma. As a consequence, we ob ain a new
p oo o a ecen ly ob ained cha ac e iza ion o he compac
composi ion ope a o s on Bloch spaces.
1. In oduc ion.
Le Ddeno e he uni disk in he complex plane. A unc ion analy ic on
he uni disk is said o belong o he Bloch space Bi
sup
D
(1 −|z|2)| 0(z)|<∞
and o he li le Bloch space B0i
lim
|z|→1−
(1 −|z|2)| 0(z)|= 0.
I is well known and easy o p o e ha Bis a Banach space unde he
no m
k k=| (0)|+ sup
D
(1 −|z|2)| 0(z)|
and ha B0is a closed subspace o B. Good sou ces o esul s and e e -
ences abou Bloch unc ions a e he pape s o Ande son-Clunie-Pomme enke
[ACP], Fe n´andez [Fe], Pomme enke [Po], and he book o Zhu [Zh, Chap-
e 5].
I ϕis an analy ic unc ion on Dwi h ϕ(D)⊂D, hen he equa ion Cϕ
= ◦ϕde ines a composi ion ope a o Cϕon he space o all holomo phic
unc ions on D. The Pick-Schwa z Lemma (see [CM, p. 47], o ins ance)
asse s ha
(1) 1−|z|2
1−|ϕ(z)|2|ϕ0(z)| ≤ 1.
339
340 ALFONSO MONTES-RODR´
IGUEZ
As no iced in [MM] his and he chain ule gi e an easy p oo o he ac
ha Cϕac s boundedly on he Bloch space. In ac we ha e
(1 −|z|2)|( ◦ϕ)0(z)|= (1 −|z|2)| 0(ϕ(z))||ϕ0(z)|
=1−|z|2
1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)| 0(ϕ(z))|
≤sup
D
(1 −|ϕ(z)|2)| 0(ϕ(z))|
= sup
ϕ(D)
(1 −|w|2)| 0(w)|
≤sup
D
(1 −|z|2)| 0(z)|.
In addi ion, i Cϕac s boundedly on B0 hen ϕmus belong o B0. This
ollows om he ac ha Cϕz=ϕ. Con e sely, i ϕ∈ B0, hen om he
es ima es abo e i is easy o show ha ϕinduces a con inuous ope a o on
B0(see [MM]). The main goal o his pape is o compu e he essen ial
no m o Cϕin e ms o an asymp o ic bound in ol ing he quan i y
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|.
We ecall ha he essen ial no m o a con inuous linea ope a o Tis he
dis ance om T o he compac ope a o s, ha is,
kTke= in {kT−Kk:Kis compac }.
No ice ha kTke= 0 i and only i Tis compac , so ha es ima es on kTke
lead o condi ions o T o be compac . Thus we will ob ain a di e en
p oo o a ecen esul o Madigan and Ma heson [MM] in which hey
cha ac e ize hose ϕwhich induces compac composi ion ope a o s on B
and B0. The undamen al ideas o he p oo a e hose used by J.H. Shapi o
[Sh1] o ob ain he essen ial no m o a composi ion ope a o on Hilbe
spaces o analy ic unc ions (Ha dy and weigh ed Be gman spaces) in e ms
o na u al coun ing unc ions associa ed wi h ϕ. Howe e , since nei he B
no B0a e Hilbe spaces ou me hod di e s in some in e es ing de ails om
hose o Shapi o.
Be o e going u he , we wan o say a wo d abou he well-known heu is-
ic p inciple which s a es ha i a “big-oh” condi ion desc ibes a class o
bounded ope a o s, hen he co esponding “li le-oh” condi ion picks ou
he subclass o compac ope a o s. An excellen example o his p inciple in
ac ion can be seen in he pape o J.H. Shapi o [Sh1] men ioned abo e. The
“big-oh” condi ion on Bloch spaces is gi en by (1). Madigan and Ma he-
son we e able o p o e he “li le-oh” condi ion, ha is, ha a composi ion
COMPOSITION OPERATORS 341
ope a o Cϕon B0is compac i and only i
lim
|z|→1−
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|= 0.
They also ob ained (wi h a di e en p oo ) ha Cϕis compac on Bi and
only i o e e y ε > 0 he e exis s , 0 < < 1, such ha
sup
|ϕ(z)|>
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|< ε.
As we will see la e he condi ions o compac ness on Band B0a e ac ually
he same. In ac , he essen ial no m o a composi ion ope a o is indepen-
den o he unde lying space Bo B0. This should no cause any su p ise.
The ac ha Bis isome ically isomo phic o he second dual o B0and
he inclusion B0⊂ B co esponds o he canonical imbedding o B0in o B??
0
(see [ACP]) does no a ec he compu a ion o he essen ial no m. This is
exac ly wha happens i we conside a bounded diagonal ope a o de ined
by a bounded sequence {an}on he sequence spaces l∞and c0, espec i ely.
Then i s essen ial no m equals lim sup anand his quan i y is independen
o he unde lying space. In ac he p oo o he main esul in he ollowing
sec ion is done simul aneously o bo h Band B0.
Be o e p oceeding u he , he au ho would like o hank Nigel J. Kal on
who indica ed he p oo o P oposi ion 2.3. The au ho would also like
o hank Joel H. Shapi o o p o iding he p oo o Theo em 2.5, some
e e ences and help ul commen s.
2. Main esul .
Main Theo em 2.1. Suppose ha Cϕde ines a con inuous ope a o on B
(o on B0).Then
(1) kCϕke= lim
s→1−
sup
|ϕ(z)|>s
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|.
In pa icula , Cϕis compac on B(o B0)i and only i
lim
s→1−
sup
|ϕ(z)|>s
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|= 0.
I is unde s ood ha i {z:|ϕ(z)|> s}is he emp y se o some 0 < s < 1
he sup emum equals ze o. This happens when ϕ(D) is a ela i ely compac
subse o Dand in his case i is easy o show ha Cϕis a compac ope a o .
I ϕhas an angula de i a i e a a poin ξ∈∂D, hen we can apply
he Julia Ca a h´eodo y Theo em (see [Sh2, p. 57]) and he Pick-Schwa z
342 ALFONSO MONTES-RODR´
IGUEZ
Lemma o ob ain
1 = lim in
z→ξ
1−|z|2
1−|ϕ(z)|2|ϕ0(z)| ≤ lim
s→1−
sup
|ϕ(z)|>s
1−|z|2
1−|ϕ(z)|2|ϕ0(z)| ≤ 1.
Thus, as an immedia e consequence o Theo em 2.1 we ha e kCϕke= 1
whene e ϕhas a ini e angula de i a i e.
Be o e p o ing Theo em 2.1 le us show ha o he li le Bloch space B0
he e is an equi alen o mula in e ms o ano he quan i y. This a simple
consequence o he ollowing p oposi ion:
P oposi ion 2.2. Suppose ha Cϕde ines a con inuous ope a o on B0.
Then
(2) lim
s→1−
sup
|ϕ(z)|>s
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|= lim sup
|z|→1−
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|.
P oo . As ema ked in he in oduc ion he ac ha Cϕac s boundedly on
B0implies ha ϕ∈ B0. I ϕ(D) is a ela i ely compac subse o D, hen
bo h limi s in (2) a e ze o and coincide. So we may suppose ha ϕ(D) is no
a ela i ely compac subse o D. Le 0 < sn<1 be any inc easing sequence
ending o 1. We se n= in { :|ϕ(z)|> sn o some zwi h |z|> }.
By con inui y { n}also ends o 1. Since {z:|z|> n}={z:|ϕ(z)|>
snand |z|> n}∪{z:|ϕ(z)| ≤ snand |z|> n}we ind ha he le hand
side o (2) is less han o equal o he igh hand side o (2). On he o he
hand, we can always ind a sequence {zn} o which
lim
n→∞
1−|zn|2
1−|ϕ(zn)|2|ϕ0(zn)|= lim
s→1−
sup
|z|>s
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|
= lim sup
|z|→1−
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|.(3)
Then ei he he e is a subsequence {znk}such ha {|ϕ(znk)|} → 1 as k→ ∞,
o o e e y posi i e in ege nwe ha e |ϕ(zn)| ≤ s0 o some 0 < s0<1.
Clea ly, in he o me case bo h limi s in (2) coincide. Fo he la e case we
ind ha he limi in (3) is ze o because ϕ∈ B0. Since his limi is g ea e
han o equal o he limi on he le hand side o (2), we ind ha hey a e
he same again. The p oo is now inished. 
Now we u n o he p oo o ou main esul .
The lowe es ima e. Fi s we show ha :
(4) kCϕke≥lim
s→1−
sup
|ϕ(z)|≥s
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|.
Ins ead o he ep oducing ke nels used by Shapi o o he Ha dy and Be g-
man spaces we will use he sequence {zn}n≥2. This sequence con e ges

COMPOSITION OPERATORS 343
uni o mly on compac subse s o he uni disk. An elemen a y compu a ion
shows ha
kznk= max
D(1 −|z|2)|nzn−1|=2n
n+ 1 n−1
n+ 1(n−1)/2
.
Obse e ha o each n≥2 he abo e maximum is a ained a any poin
on he ci cle cen e ed a he o igin and o adius n=n−1
n+1 1/2. These
maxima o m a dec easing sequence which ends o 2/e.
The e o e, he sequence {zn}n≥2is bounded away om ze o. Now we
conside he no malized sequence { n=zn
kznk}which also ends o ze o
uni o mly on compac subse s o he uni disk. Fo each n≥2 we de ine he
closed annulus An={z∈D: n≤ |z| ≤ n+1}and compu e
min
An
(1 −|z|2)| 0
n(z)|= (1 − 2
n+1)| 0
n( n+1)|
=n+ 1
n+ 2 n2+n
n2+n−2(n−1)/2
.(5)
Obse e ha hese minima end o 1 as n→ ∞ and o each n≥2 he
minimum abo e is a ained a any poin o he ci cle cen e ed a he o igin
and o adius n+1. Fo he momen ix any compac ope a o Kon B0o
B. The uni o m con e gence on compac subse s o he sequence { n} o
ze o and he compac ness o Kimply ha kK nk → 0. I is easy o show
ha i a bounded sequence ha is con ained in B0con e ges uni o mly on
compac subse s o he uni disk, hen i also con e ges weakly o ze o in
B0as well as in B. Thus
kCϕ−Kk ≥ lim sup
nk(Cϕ−K) nk
≥lim sup
n(kCϕ nk−kK nk)
= lim sup
nkCϕ nk.
Upon aking he in imum o bo h sides o his inequali y o e all compac
ope a o s K, we ob ain he lowe es ima e:
kCϕke≥lim sup
nkCϕ nk
= lim sup
nsup
D
(1 −|z|2)| 0
n(ϕ(z))||ϕ0(z)|
= lim sup
nsup
D
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)| 0
n(ϕ(z))|.(6)
Now (6) is g ea e han o equal o
(7) lim sup
nsup
ϕ(z)∈An
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)| 0
n(ϕ(z))|
344 ALFONSO MONTES-RODR´
IGUEZ
and (7) is g ea e han o equal o
(8) lim sup
nsup
ϕ(z)∈An
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|min
ϕ(z)∈An
(1 −|ϕ(z)|2)| 0
n(ϕ(z))|.
I ϕ(D) is a ela i ely compac subse o Dbo h sides o (4) a e ze o and he e
is no hing o p o e. O he wise we ind ha minϕ(z)∈An(1−|ϕ(z)|2)| 0
n(ϕ(z))|
= minAn(1−|z|2)| 0
n(z)|because he minimum in (5) is a ained a any poin
on he ci cle cen e ed a he o igin and o adius n+1. Since hese minima
end o 1 as n→ ∞, i ollows ha (8) is equal o
(9) lim sup
nsup
ϕ(z)∈An
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|.
Finally, an easy exe cise shows ha (9) coincides wi h he igh hand side
o (4).
To ob ain he uppe es ima e in he case o he Ha dy and Be gman
spaces, Shapi o [Sh1] used he ope a o s Pnwhich ake o he n h pa ial
sum o i s Taylo se ies. On he Ha dy space hese ope a o s sa is y: i) Each
Pnis compac , ii) (I−Pn) ends o ze o uni o mly on compac subse s o
any in he Ha dy space, and iii) o each n he no m in he Ha dy space
o I−Pnequals 1. Al hough each Pnis also compac in he Bloch space,
and (I−Pn) ends o ze o uni o mly on compac subse s o each unc ion
∈ B, his sequence does no sa is y any hing analogous o iii) abo e. In
ac , kPnk ≥ Clog nwhe e Cis a uni e sal cons an (see [ACP, p. 18-19]).
The e o e, by he e e se iangle inequali y kI−Pnk ≥ Clog n−1. One
o he issues he e is ha in gene al i is no easy o compu e exac ly ei he
he no ms o Bloch unc ions, o he no ms o ope a o s de ined on Bloch
spaces.
To ob ain he uppe es ima e we need he ope a o s Kn,n≥2, which
ake each unc ion (z) o (n−1
nz). E e y ope a o Knis compac on B
(o B0). We also ha e ha (I−Kn) ends o ze o uni o mly on compac
subse s o he uni disk o e e y ∈ B, and (al hough we do no know i
limn→∞ kI−Knk= 1) we ha e he ollowing p oposi ion, whose p oo is
delayed.
P oposi ion 2.3. The e exis s a sequence o con ex combina ions Lno
Kn(Ln=Pm≥ncn,mKmwi h cm,n >0and Pm≥ncn,m = 1) such ha
limn→∞ kI−Lnk= 1.
The uppe es ima e. The goal now is o show ha
(10) kCϕke≤lim
s→1−
sup
|ϕ(z)|>s
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|.
COMPOSITION OPERATORS 345
This will be accomplished by applying P oposi ion 2.3. Since each Lnis
compac so is CϕLn. The e o e
kCϕke≤ kCϕ−CϕLnk=kCϕ(I−Ln)k.
On he o he hand, we ha e
kCϕ(I−Ln)k
= sup
k k=1 kCϕ(I−Ln) k
= sup
k k=1
sup
|z|<1
(1 −|z|2)|((I−Ln) )0(ϕ(z))||ϕ0(z)|(11)
= sup
k k=1
sup
|z|<1
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)|((I−Ln) )0(ϕ(z))|.
Now ix 0 < s < 1. Then he igh hand side o (11) is less han o equal o
sup
k k=1
sup
|ϕ(z)|≤s
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)|((I−Ln) )0(ϕ(z))|(12)
+ sup
k k=1
sup
|ϕ(z)|>s
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)|((I−Ln) )0(ϕ(z))|.
By applying he Pick-Schwa z Lemma in he i s e m, and he ac ha
o in he uni ball
sup
|ϕ(z)|>s
(1 −|ϕ(z)|2)|((I−Ln) )0(ϕ(z))|
≤sup
|z|<1
(1 −|z|2)|((I−Ln) )0(z)| ≤ kI−Lnk
o he second e m, we ind ha (12) is less han o equal o
sup
k k=1
sup
|w|≤s
(1 −|w|2)|((I−Ln) )0(w)|(13)
+kI−Lnksup
|ϕ(z)|>s
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|.
Le us p o e ha he i s e m in (13) ends o ze o as n→ ∞. By he
iangle inequali y we ha e ha he i s e m in (13) is less han o equal
o
(14) X
m≥n
cn,m sup
k k=1
sup
|w|≤s
(1 −|w|2)|((I−Km) )0(w)|.
346 ALFONSO MONTES-RODR´
IGUEZ
By he iangle inequali y again we ind ha (1 −|w|2)|((I−Km) )0(w)|is
less han o equal o
(15) sup
k k=1
sup
|w|≤s
(1 −|w|2) 0(w)− 01−1
mw
+1
msup
k k=1
sup
|w|≤s
(1 −|w|2) 01−1
mw.
By in eg a ing 00 along he adial segmen [(1 −1/m)w, w] i is easy o see
ha he i s e m in (15) is less han o equal o
(16) 1
msup
k k=1
sup
|w|≤s
(1 −|w|2)|w|| 00(ξ(w))|,
whe e ξ(w) belongs o he adial segmen [(1 −1/m)w, w] ha is s ill con-
ained in he closed disk o adius s. The Cauchy inequali ies applied o a
ci cle C(ξ(w)) cen e ed a ξ(w) and o any ix adius 0 < R < 1−syields
ha (16) is less han o equal o
(17) 1
mR sup
k k=1
sup
|w|≤s
(1 −|w|2)|w|max
|z|=s+R| 0(z)|.
On he o he hand, on he uni ball o B(o B0) we ha e max|z|=s+R| 0(z)| ≤
1
1−(s+R)2. So we ind ha (17) is less han o equal o
1
mR sup
|w|≤s
(1 −|w|2)|w|1
1−(s+R)2≤1
mR
s
1−(s+R)2.
Since he second e m in (15) is less han 1/m we ind ha (15) is ≤C/m,
whe e Conly depends on s. The e o e, we ind ha (14) is less han o
equal o X
m≥n
cn,m
C
m≤X
m≥n
cn,m
C
n=C
n
which ends o ze o as n→ ∞. Hence, le ing n→ ∞ in (13), applying
P oposi ion 2.3 and pu ing e e y hing oge he , he ollowing inequali y
ollows
kCϕke≤sup
|ϕ(z)|>s
1−|z|2
1−|ϕ(z)|2|ϕ0(z)|.
Since swas a bi a y inequali y (10) holds.
Rema ks. 1. By he iangle inequali y we ha e kI−Knk ≤ kIk+kKnk= 2.
The e o e, i we use he sequence {Kn}ins ead o he sequence {Ln}in
he p oo o he uppe es ima e, hen we ob ain wice he uppe es ima e.
Howe e , ha is enough o cha ac e ize he compac composi ion ope a o s
on Bloch spaces wi hou equi ing P oposi ion 2.3.