The essential norm of a composition operator on Bloch spaces

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.