ESSENTIAL NORMS OF COMPOSITION OPERATORS AND ALEKSANDROV MEASURES. Joseph A. Cima and Alec L. Matheson

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1 pacific journal of mathematics Vol. 179, No. 1, 1997 ESSENIAL NORMS OF COMPOSIION OPERAORS AND ALEKSANDROV MEASURES Joseph A. Cima and Alec L. Matheson he essential norm of a composition operator on H is calculated in terms of the Aleksandrov measures of the inducing holomorphic map. he argument provides a purely functiontheoretic proof of the equivalence of Sarason s compactness condition for composition operators on L 1 and Shapiro s compactness condition for composition operators on Hardy spaces. An application is given relating the essential norm to angular derivatives. 1. If φ is a holomorphic map of the unit disk D into itself, it is a consequence of Littlewood s subordination principle [5] that composition with φ induces a bounded operator C φ on each Hardy space H p. A recurring theme in the study of composition operators has been the search for function theoretic conditions on φ which guarantee the compactness of C φ on H p. It was shown by Shapiro and aylor [11] that if C φ is compact on H p for some 0 <p<, then C φ is compact on H p for all 0 <p<, and so it is enough to study compactness on H. In this context Shapiro [9] gave an expression for the essential norm of C φ on H in terms of the Nevanlinna counting function of φ, thus providing a complete function theoretic characterization of compact composition operators on H. In a different direction Sarason [7] showed how to define the composition operator C φ on the space M of complex Borel measures on the unit circle. Indeed, if u is the Poisson integral of a complex Borel measure, it is not difficult to see that u φ is also, and then that the action of C φ is bounded on M. He also showed that C φ acts boundedly on L 1, and that compactness on M is equivalent to compactness on L 1. In the process he gave a function theoretic condition on φ equivalent to compactness on L 1. Since H 1 L 1, it is evident from the above discussion that Sarason s condition implies Shapiro s. he reverse implication was established by Shapiro and Sundberg [10], and subsequently another more direct proof was found by Sundberg. However, Sarason [8] states that a direct function theoretic proof is still lacking. It is the main purpose of this note to provide such a proof. 59

2 60 JOSEPH A. CIMA & ALEC L. MAHESON Shapiro s expression for the essential norm of C φ on H is C φ e = lim sup N φ(a), log 1 a where N φ (z) is Nevanlinna s counting function for φ, given by N φ (z) = φ(ζ)=z log 1 ζ. In particular C φ is compact on H N if and only if lim sup φ (a) log 1 a the course of proving this Shapiro established the inequality C φ e lim sup C φ f a lim sup 1 a N φ (a), log 1 a =0. In where f a (z) = is the normalized kernel function for a D. his, 1 az together with the rest of his proof, shows that C φ e = lim sup C φ f a. It is important to note that although Shapiro s proof of this equation is not purely function theoretic, his methods can be used to provide such a proof. In the next section Sarason s condition will be derived from the alternate condition on the kernel functions. In the process a third expression for the essential norm of C φ will be derived in terms of the singular parts of the Aleksandrov measures of φ. An application related to angular derivatives will be given in Section 3.. Sarason s compactness condition can be given in two equivalent formulations. In the first instance if f L 1 has harmonic extension u to the unit disk, then C φ f is the boundary function of the harmonic function u φ. he Poisson formula gives u(φ(z)) = 1 φ(z) ζ φ(z) f(ζ)dm(ζ), z D, where m denotes the normalized Lebesgue measure on the unit circle. Sarason proceeds by analyzing the kernel 1 φ(ξ) ζ φ(ξ), ζ,ξ.

3 ESSENIAL NORMS OF COMPOSIION OPERAORS 61 He shows that C φ is compact on L 1 if and only if (.1) 1 φ(ξ) dm(ξ) =1 ζ φ(ξ) for all ζ, at least when φ(0) = 0. he main ingredient in his proof is a theorem of Dunford and Pettis which asserts that a sequence of functions (f n )inl 1 converges in norm to f L 1 if f n f and f n 1 f 1. It is not difficult to show that in general C φ is compact on L 1 if and only if (.) 1 φ(ξ) ( ) ζ+φ(0) ζ φ(ξ) dm(ξ) =R ζ φ(0) for all ζ. he other formulation, which is easily seen to be equivalent, results from consideration of measures studied ( by ) Aleksandrov []. For each α, since φ 1, u α (z) =R α+φ(z) is a positive harmonic function, and α φ(z) so, by Herglotz s theorem, is the Poisson integral ( of ) a positive measure τ α. his measure has total variation τ α = R α+φ(0) = u α φ(0) α (0) and Lebesgue decomposition dτ α = h α dm + dσ α, where h α L 1 and σ α m. Since h α (ξ) = lim r 1 u α (rξ) = 1 φ(ξ) α φ(ξ), for almost every ξ, it follows from (.) that C φ is compact on L 1 if and only if σ α = 0 for all α, or, what is the same thing, if and only if the Aleksandrov measures τ α are all absolutely continuous. Sarason calls this the absolute continuity condition [8]. It follows from the Lebesgue decomposition of τ α that σ α = τ α h α (ξ)dm(ξ) On the other hand C φ f rα = = (.3) = R = R ( ) α + φ(0) 1 φ(ξ) α φ(0) α φ(ξ) dm(ξ). 1 r α rφ(ξ) dm(ξ) 1 r φ(ξ) dm(ξ) r α rφ(ξ) ) r ( α + rφ(0) α rφ(0) 1 φ(ξ) α rφ(ξ) dm(ξ) 1 φ(ξ) α rφ(ξ) dm(ξ).

4 6 JOSEPH A. CIMA & ALEC L. MAHESON ( ) ( ) Clearly, since φ(0) < 1, lim r 1 R α+rφ(0) = R α+φ(0) uniformly in α. α rφ(0) α φ(0) Nowif0 <r<s 1 and w 1, it is geometrically obvious that 1 s w < 1 r w, and so (.4) It follows that r 1 φ(ξ) α rφ(ξ) every ξ, and so (.5) Hence lim r 1 r r 1 rw < s 1 sw. increases monotonically to 1 φ(ξ) α φ(ξ) 1 φ(ξ) α rφ(ξ) dm(ξ) = 1 φ(ξ) α φ(ξ) dm(ξ). for almost (.6) In particular lim C φf rα r 1 = σ α. (.7) σ α lim sup C φ f a = C φ e for all α, and so Shapiro s compactness condition implies Sarason s. In order to prove the reverse inequality set A = lim sup C φ f a and fix ɛ>0. For each r, 0<r<1, let { ( ) α+φ(0) E r = α R r α φ(0) 1 φ(ξ) } α rφ(ξ) dm(ξ) A ɛ. By continuity each E r is a closed set. Since r 1 φ(ξ) dm(ξ) isanincreasing function of r for each α, it follows that E r E s whenever α rφ(ξ) r<s<1. Choose r 0 so that ( ) α + φ(0) R R α φ(0) ( ) α+rφ(0) <ɛ α rφ(0) for all α if r 0 r<1. Now if r 0 r<1, there exists r 1, r r 1 < 1, and α such that C φ f r1α >A ɛ, and so α E r1 E r. In particular each E r is nonempty. By compactness there exists α 0 0<r<1 E r. Hence, passing to the limit, σ α0 A ɛ. Combining this with (.7) yields (.8) C φ e = sup α σ α.

5 ESSENIAL NORMS OF COMPOSIION OPERAORS Equation (.8) leads quickly to a lower bound for C φ e in terms of angular derivatives. he angular derivative φ (ζ) for ζ with φ(ζ) = 1 is the limit lim z ζ φ(ζ) φ(z) ζ z, provided the limit exists nontangentially. Let S α = { ζ φ(ζ) =α}for each α. he Julia-Carathéodory theorem asserts that for each ζ S α, φ (ζ) exists or is infinite. In any case the proof of the Julia-Carathéodory theorem on p. 11 of [1] shows that τ α ({ζ}) = 1 φ (ζ) for each ζ S α (with the usual convention that 1 φ (ζ) =0ifφ (ζ)= ). In particular the quantity δ(α) = ζ S α 1 φ (ζ) is the variation of the purely atomic part of τ α and hence is finite. Now (.8) yields (3.1) C φ e sup δ(α), α an estimate first obtained by Cowen [3, 4], who also provided the upper bound sup α δ(α) ifφ is continuous on D. Actually, if τ α has continuous singular part for no α, then in fact (3.) C φ e = sup δ(α). α Since σ α is supported on S α, this happens in particular whenever S α is a finite set for each α. A theorem of Novinger and Oberlin [6] shows that this is the case if φ satisfies a Lipschitz condition of order 1 (see also [13]). Hence in this case (3.) holds, improving Cowen s upper bound. It should be remarked that Shapiro has used his calculation of C φ e and the Julia- Carathéodory theorem to give a proof of the result of Novinger and Oberlin. Finally it would be of interest to see a direct proof of (.7). Since it is relatively easy to prove that C φ e σ α for each α, this is a question of providing a proof of the inequality sup α σ α C φ e which does not use Nevanlinna s counting function. References [1] L.V. Ahlfors, Conformal invariants, McGraw-Hill, New York, [] A.B. Aleksandrov, Multiplicity of boundary values of inner functions, Izv. Akad. Nauk Armyan. SSR, Ser. Mat., (1987), (Russian). [3] Carl C. Cowen, Composition operators on H, J. Operator heory, 9 (1983), [4] Carl C. Cowen and Ch. Pommerenke, Inequalities for the angular derivative of an analytic function in the unit disc, J. London Math. Soc., 6 (1978), [5] J.E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc., 3 (195),

6 64 JOSEPH A. CIMA & ALEC L. MAHESON [6] W.P. Novinger and D.M. Oberlin, Peak sets for Lipschitz functions, Proc. Amer. Math. Soc., 68 (1978), [7] D. Sarason, Composition operators as integral operators, in Analysis and Partial Differential Equations, Marcel Dekker, New York, [8], Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas lecture notes in the mathematical sciences, Vol. 10, John Wiley & Sons, Inc., New York, [9] J.H. Shapiro, he essential norm of a composition operator, Annals of Math., 15 (1987), [10] J.H. Shapiro and C. Sundberg, Compact composition operators on L 1, Proc. Amer. Math. Soc., 108 (1990), [11] J.H. Shapiro and P.D. aylor, Compact, nuclear, and Hilbert-Schmidt composition operators on H, Indiana Univ. Math. J., 15 (1973), [1] Charles S. Stanton, Counting functions and majorizaton for Jensen measures, Pacific J. Math., 15 (1986), [13] B.A. aylor and D.L. Williams, he peak sets of A m, Proc. Amer. Math. Soc., 4 (1970), Received September 6, 1995 and revised March 9, he second author was partially supported by National Science Foundation grant DMS University of North Carolina Chapel Hill, NC address: cima@math.unc.edu and Lamar University Beaumont, X address: matheson@math.lamar.edu