The essential norm of a composition operator on Bloch spaces

Pacific Journal of Mathematics THE ESSENTIAL NORM OF A COMPOSITION OPERATOR ON BLOCH SPACES Alfonso Montes-Rodr ´ ıguez Volume 188 No. 2 April 1999 PACIFIC JOURNAL OF MATHEMATICS Vol. 188, No. 2, 1999 THE ESSENTIAL NORM OF A COMPOSITION OPERATOR ON BLOCH SPACES Alfonso Montes-Rodr ´ ıguez We express the essential norm of a composition operator on the Bloch space and the little Bloch space as the asymptotic upper bound of a quantity involving the inducing map and the Pick-Schwarz Lemma. As a consequence, we obtain a new proof of a recently obtained characterization of the compact composition operators on Bloch spaces. 1. Introduction. Let Ddenote the unit disk in the complex plane. A function fanalytic on the unit disk is said to belong to the Bloch space Bif sup D (1 −|z|2)|f0(z)|<∞ and to the little Bloch space B0if lim |z|→1− (1 −|z|2)|f0(z)|= 0. It is well known and easy to prove that Bis a Banach space under the norm kfk=|f(0)|+ sup D (1 −|z|2)|f0(z)| and that B0is a closed subspace of B. Good sources for results and references about Bloch functions are the papers of Anderson-Clunie-Pommerenke [ACP], Fern´andez [Fe], Pommerenke [Po], and the book of Zhu [Zh, Chapter 5]. If ϕis an analytic function on Dwith ϕ(D)⊂D, then the equation Cϕf =f◦ϕdefines a composition operator Cϕon the space of all holomorphic functions on D. The Pick-Schwarz Lemma (see [CM, p. 47], for instance) asserts that (1) 1−|z|2 1−|ϕ(z)|2|ϕ0(z)| ≤ 1. 339 340 ALFONSO MONTES-RODR´ IGUEZ As noticed in [MM] this and the chain rule give an easy proof of the fact that Cϕacts boundedly on the Bloch space. In fact we have (1 −|z|2)|(f◦ϕ)0(z)|= (1 −|z|2)|f0(ϕ(z))||ϕ0(z)| =1−|z|2 1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)|f0(ϕ(z))| ≤sup D (1 −|ϕ(z)|2)|f0(ϕ(z))| = sup ϕ(D) (1 −|w|2)|f0(w)| ≤sup D (1 −|z|2)|f0(z)|. In addition, if Cϕacts boundedly on B0then ϕmust belong to B0. This follows from the fact that Cϕz=ϕ. Conversely, if ϕ∈ B0, then from the estimates above it is easy to show that ϕinduces a continuous operator on B0(see [MM]). The main goal of this paper is to compute the essential norm of Cϕin terms of an asymptotic bound involving the quantity 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|. We recall that the essential norm of a continuous linear operator Tis the distance from Tto the compact operators, that is, kTke= inf{kT−Kk:Kis compact}. Notice that kTke= 0 if and only if Tis compact, so that estimates on kTke lead to conditions for Tto be compact. Thus we will obtain a different proof of a recent result of Madigan and Matheson [MM] in which they characterize those ϕwhich induces compact composition operators on B and B0. The fundamental ideas of the proof are those used by J.H. Shapiro [Sh1] to obtain the essential norm of a composition operator on Hilbert spaces of analytic functions (Hardy and weighted Bergman spaces) in terms of natural counting functions associated with ϕ. However, since neither B nor B0are Hilbert spaces our method differs in some interesting details from those of Shapiro. Before going further, we want to say a word about the well-known heuristic principle which states that if a “big-oh” condition describes a class of bounded operators, then the corresponding “little-oh” condition picks out the subclass of compact operators. An excellent example of this principle in action can be seen in the paper of J.H. Shapiro [Sh1] mentioned above. The “big-oh” condition on Bloch spaces is given by (1). Madigan and Matheson were able to prove the “little-oh” condition, that is, that a composition COMPOSITION OPERATORS 341 operator Cϕon B0is compact if and only if lim |z|→1− 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|= 0. They also obtained (with a different proof) that Cϕis compact on Bif and only if for every ε > 0 there exists r, 0 < r < 1, such that sup |ϕ(z)|>r 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|< ε. As we will see later the conditions of compactness on Band B0are actually the same. In fact, the essential norm of a composition operator is independent of the underlying space Bor B0. This should not cause any surprise. The fact that Bis isometrically isomorphic to the second dual of B0and the inclusion B0⊂ B corresponds to the canonical imbedding of B0into B?? 0 (see [ACP]) does not affect the computation of the essential norm. This is exactly what happens if we consider a bounded diagonal operator defined by a bounded sequence {an}on the sequence spaces l∞and c0, respectively. Then its essential norm equals lim sup anand this quantity is independent of the underlying space. In fact the proof of the main result in the following section is done simultaneously for both Band B0. Before proceeding further, the author would like to thank Nigel J. Kalton who indicated the proof of Proposition 2.3. The author would also like to thank Joel H. Shapiro for providing the proof of Theorem 2.5, some references and helpful comments. 2. Main result. Main Theorem 2.1. Suppose that Cϕdefines a continuous operator on B (or on B0).Then (1) kCϕke= lim s→1− sup |ϕ(z)|>s 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|. In particular, Cϕis compact on B(or B0)if and only if lim s→1− sup |ϕ(z)|>s 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|= 0. It is understood that if {z:|ϕ(z)|> s}is the empty set for some 0 < s < 1 the supremum equals zero. This happens when ϕ(D) is a relatively compact subset of Dand in this case it is easy to show that Cϕis a compact operator. If ϕhas an angular derivative at a point ξ∈∂D, then we can apply the Julia Carath´eodory Theorem (see [Sh2, p. 57]) and the Pick-Schwarz 342 ALFONSO MONTES-RODR´ IGUEZ Lemma to obtain 1 = lim inf 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 immediate consequence of Theorem 2.1 we have kCϕke= 1 whenever ϕhas a finite angular derivative. Before proving Theorem 2.1 let us show that for the little Bloch space B0 there is an equivalent formula in terms of another quantity. This a simple consequence of the following proposition: Proposition 2.2. Suppose that Cϕdefines a continuous operator 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)|. Proof. As remarked in the introduction the fact that Cϕacts boundedly on B0implies that ϕ∈ B0. If ϕ(D) is a relatively compact subset of D, then both limits in (2) are zero and coincide. So we may suppose that ϕ(D) is not a relatively compact subset of D. Let 0 < sn<1 be any increasing sequence tending to 1. We set tn= inf{t:|ϕ(z)|> snfor some zwith |z|> t}. By continuity {tn}also tends to 1. Since {z:|z|> tn}={z:|ϕ(z)|> snand |z|> tn}∪{z:|ϕ(z)| ≤ snand |z|> tn}we find that the left hand side of (2) is less than or equal to the right hand side of (2). On the other hand, we can always find a sequence {zn}for 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 either there is a subsequence {znk}such that {|ϕ(znk)|} → 1 as k→ ∞, or for every positive integer nwe have |ϕ(zn)| ≤ s0for some 0 < s0<1. Clearly, in the former case both limits in (2) coincide. For the latter case we find that the limit in (3) is zero because ϕ∈ B0. Since this limit is greater than or equal to the limit on the left hand side of (2), we find that they are the same again. The proof is now finished.  Now we turn to the proof of our main result. The lower estimate. First we show that: (4) kCϕke≥lim s→1− sup |ϕ(z)|≥s 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|. Instead of the reproducing kernels used by Shapiro for the Hardy and Bergman spaces we will use the sequence {zn}n≥2. This sequence converges COMPOSITION OPERATORS 343 uniformly on compact subsets of the unit disk. An elementary computation shows that kznk= max D(1 −|z|2)|nzn−1|=2n n+ 1 n−1 n+ 1(n−1)/2 . Observe that for each n≥2 the above maximum is attained at any point on the circle centered at the origin and of radius rn=n−1 n+1 1/2. These maxima form a decreasing sequence which tends to 2/e. Therefore, the sequence {zn}n≥2is bounded away from zero. Now we consider the normalized sequence {fn=zn kznk}which also tends to zero uniformly on compact subsets of the unit disk. For each n≥2 we define the closed annulus An={z∈D:rn≤ |z| ≤ rn+1}and compute min An (1 −|z|2)|f0 n(z)|= (1 −r2 n+1)|f0 n(rn+1)| =n+ 1 n+ 2 n2+n n2+n−2(n−1)/2 .(5) Observe that these minima tend to 1 as n→ ∞ and for each n≥2 the minimum above is attained at any point of the circle centered at the origin and of radius rn+1. For the moment fix any compact operator Kon B0or B. The uniform convergence on compact subsets of the sequence {fn}to zero and the compactness of Kimply that kKfnk → 0. It is easy to show that if a bounded sequence that is contained in B0converges uniformly on compact subsets of the unit disk, then it also converges weakly to zero in B0as well as in B. Thus kCϕ−Kk ≥ lim sup nk(Cϕ−K)fnk ≥lim sup n(kCϕfnk−kKfnk) = lim sup nkCϕfnk. Upon taking the infimum of both sides of this inequality over all compact operators K, we obtain the lower estimate: kCϕke≥lim sup nkCϕfnk = lim sup nsup D (1 −|z|2)|f0 n(ϕ(z))||ϕ0(z)| = lim sup nsup D 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)|f0 n(ϕ(z))|.(6) Now (6) is greater than or equal to (7) lim sup nsup ϕ(z)∈An 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)|f0 n(ϕ(z))| 344 ALFONSO MONTES-RODR´ IGUEZ and (7) is greater than or equal to (8) lim sup nsup ϕ(z)∈An 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|min ϕ(z)∈An (1 −|ϕ(z)|2)|f0 n(ϕ(z))|. If ϕ(D) is a relatively compact subset of Dboth sides of (4) are zero and there is nothing to prove. Otherwise we find that minϕ(z)∈An(1−|ϕ(z)|2)|f0 n(ϕ(z))| = minAn(1−|z|2)|f0 n(z)|because the minimum in (5) is attained at any point on the circle centered at the origin and of radius rn+1. Since these minima tend to 1 as n→ ∞, it follows that (8) is equal to (9) lim sup nsup ϕ(z)∈An 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|. Finally, an easy exercise shows that (9) coincides with the right hand side of (4). To obtain the upper estimate in the case of the Hardy and Bergman spaces, Shapiro [Sh1] used the operators Pnwhich take fto the nth partial sum of its Taylor series. On the Hardy space these operators satisfy: i) Each Pnis compact, ii) (I−Pn)ftends to zero uniformly on compact subsets for any fin the Hardy space, and iii) for each nthe norm in the Hardy space of I−Pnequals 1. Although each Pnis also compact in the Bloch space, and (I−Pn)ftends to zero uniformly on compact subsets for each function f∈ B, this sequence does not satisfy anything analogous to iii) above. In fact, kPnk ≥ Clog nwhere Cis a universal constant (see [ACP, p. 18-19]). Therefore, by the reverse triangle inequality kI−Pnk ≥ Clog n−1. One of the issues here is that in general it is not easy to compute exactly either the norms of Bloch functions, or the norms of operators defined on Bloch spaces. To obtain the upper estimate we need the operators Kn,n≥2, which take each function f(z) to f(n−1 nz). Every operator Knis compact on B (or B0). We also have that (I−Kn)ftends to zero uniformly on compact subsets of the unit disk for every f∈ B, and (although we do not know if limn→∞ kI−Knk= 1) we have the following proposition, whose proof is delayed. Proposition 2.3. There exists a sequence of convex combinations Lnof Kn(Ln=Pm≥ncn,mKmwith cm,n >0and Pm≥ncn,m = 1) such that limn→∞ kI−Lnk= 1. The upper estimate. The goal now is to show that (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 Proposition 2.3. Since each Lnis compact so is CϕLn. Therefore kCϕke≤ kCϕ−CϕLnk=kCϕ(I−Ln)k. On the other hand, we have kCϕ(I−Ln)k = sup kfk=1 kCϕ(I−Ln)fk = sup kfk=1 sup |z|<1 (1 −|z|2)|((I−Ln)f)0(ϕ(z))||ϕ0(z)|(11) = sup kfk=1 sup |z|<1 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)|((I−Ln)f)0(ϕ(z))|. Now fix 0 < s < 1. Then the right hand side of (11) is less than or equal to sup kfk=1 sup |ϕ(z)|≤s 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)|((I−Ln)f)0(ϕ(z))|(12) + sup kfk=1 sup |ϕ(z)|>s 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|(1 −|ϕ(z)|2)|((I−Ln)f)0(ϕ(z))|. By applying the Pick-Schwarz Lemma in the first term, and the fact that for fin the unit ball sup |ϕ(z)|>s (1 −|ϕ(z)|2)|((I−Ln)f)0(ϕ(z))| ≤sup |z|<1 (1 −|z|2)|((I−Ln)f)0(z)| ≤ kI−Lnk to the second term, we find that (12) is less than or equal to sup kfk=1 sup |w|≤s (1 −|w|2)|((I−Ln)f)0(w)|(13) +kI−Lnksup |ϕ(z)|>s 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|. Let us prove that the first term in (13) tends to zero as n→ ∞. By the triangle inequality we have that the first term in (13) is less than or equal to (14) X m≥n cn,m sup kfk=1 sup |w|≤s (1 −|w|2)|((I−Km)f)0(w)|. 346 ALFONSO MONTES-RODR´ IGUEZ By the triangle inequality again we find that (1 −|w|2)|((I−Km)f)0(w)|is less than or equal to (15) sup kfk=1 sup |w|≤s (1 −|w|2)f0(w)−f01−1 mw +1 msup kfk=1 sup |w|≤s (1 −|w|2)f01−1 mw. By integrating f00 along the radial segment [(1 −1/m)w, w] it is easy to see that the first term in (15) is less than or equal to (16) 1 msup kfk=1 sup |w|≤s (1 −|w|2)|w||f00(ξ(w))|, where ξ(w) belongs to the radial segment [(1 −1/m)w, w] that is still contained in the closed disk of radius s. The Cauchy inequalities applied to a circle C(ξ(w)) centered at ξ(w) and of any fix radius 0 < R < 1−syields that (16) is less than or equal to (17) 1 mR sup kfk=1 sup |w|≤s (1 −|w|2)|w|max |z|=s+R|f0(z)|. On the other hand, on the unit ball of B(or B0) we have max|z|=s+R|f0(z)| ≤ 1 1−(s+R)2. So we find that (17) is less than or equal to 1 mR sup |w|≤s (1 −|w|2)|w|1 1−(s+R)2≤1 mR s 1−(s+R)2. Since the second term in (15) is less than 1/m we find that (15) is ≤C/m, where Conly depends on s. Therefore, we find that (14) is less than or equal to X m≥n cn,m C m≤X m≥n cn,m C n=C n which tends to zero as n→ ∞. Hence, letting n→ ∞ in (13), applying Proposition 2.3 and putting everything together, the following inequality follows kCϕke≤sup |ϕ(z)|>s 1−|z|2 1−|ϕ(z)|2|ϕ0(z)|. Since swas arbitrary inequality (10) holds. Remarks. 1. By the triangle inequality we have kI−Knk ≤ kIk+kKnk= 2. Therefore, if we use the sequence {Kn}instead of the sequence {Ln}in the proof of the upper estimate, then we obtain twice the upper estimate. However, that is enough to characterize the compact composition operators on Bloch spaces without requiring Proposition 2.3.