Norms of Composition Operators with Rational Symbol

Abstract

We compute the norms of composition operators with rational symbols that satisfy certain properties, extending Christopher Hammond’s methods on operators with linear fractional symbols. This leads to a host of new examples of composition operators whose norms are calculable. 1. Introduction Let D be the open unit disk in the complex plane. The Hardy space H is the familiar Hilbert space of analytic functions on D with square-summable Taylor coe¢ cients. For ' an analytic self-map of D, C' denotes the composition operator de…ned by C'f = f '. Littlewood’s Subordination Principle, which can be found in [7], guarantees that C' is a bounded operator on H. We are interested in calculating the norm of C'. This is a di¢ cult problem in general, so we restrict our attention to the case when ' is rational. We now introduce several concepts that we will use frequently in this paper. De…nition 1.1. For z 2 D, let Kz : D! C be given by Kz( ) = 1 1 z : It is easy to check that Kz 2 H and that this function has the property that for any f 2 H, hf;Kzi = f(z). For this reason Kz is called the reproducing kernel at z. Also useful in the study of analytic functions on the disk is the following: De…nition 1.2. An analytic ' : D ! D is called inner if j'(e )j = 1 for almost every 2 [0; 2 ]. We now de…ne a simple and fundamental class of inner functions. De…nition 1.3. For z 2 D, the function z : D! D is de…ned as z( ) = z 1 z : Note that z is an automorphism of the disk that vanishes at z. Date : December 14, 2005. 1991 Mathematics Subject Classi…cation. 47B33. Key words and phrases. Composition Operator, Hardy Space, Norm. This material is based upon work partially supported by the National Science Foundation under Grant No. DMS-0353622 as part of the REU program at Cal Poly, San Luis Obispo. 1 2 SEAN EFFINGER-DEAN, ALAN JOHNSON, JOSEPH REED, AND JONATHAN SHAPIRO De…nition 1.4. An isometry is an operator A on a Hilbert space H with the property that for all f; g 2 H, hAf;Agi = hf; gi. If C' is an isometry, we say that ' is an isometry-inducing function. Our goal is to calculate the exact norm of composition operators whose symbols are in a certain special class of rational functions. At present, there is a very limited collection of self-maps ' for which kC'k is known exactly. These include inner functions, for which kC'k = q 1+j'(0)j 1 j'(0)j , constant maps ' a; for which kC'k = q 1 1 jaj , and even all linear maps '(z) = sz + t with jtj < 1 and jsj+ jtj 1: In this case (see [2] or [3, p. 324]), kC'k = s 2 1 + jsj2 jtj2 + p (1 jsj2 + jtj2)2 4jtj2 : C. Hammond, in [4] and [5], and, with P. Bourdon, E. Fry, and C. Spo¤ord in [1], developed techniques to compute the norm of a composition operator, in many cases, with linear fractional symbol. In this paper we extend the methods of these earlier papers to allow us to compute composition operator norms when the symbol is in a special class of (higher order) rational functions. If ' = are all analytic self-maps of the disk, then C' = C C . If is an isometry-inducing function then it is clear that kC'k = kC k. The set of isometryinducing functions is precisely the set of inner functions which …x the origin, see [6] or [3, pp. 123-124]. This allows us to extend our collection of composition operators with calculable norms in a somewhat trivial way, for example: Let '(z) = z +1 2 . We can write ' = for (z) = z+1 2 and (z) = z , an isometry-inducing function. We then compute kC'k = kC k = p 2 by the formula above. When we …nd new examples of ' with calculable norm, we will prove that there do not exist simpler and isometry-inducing with ' = . For notational convenience, we introduce the following function: De…nition 1.5. : C ! C (where C denotes the extended complex plane) is de…ned by (z) = 1=z. Note that 1 = and for z 2 @D, (z) = z. 2. Rational Functions with Calculable Composition Operator Norms The main reason we restrict ourselves to rational ' is that C ' can then be written in terms of an integral of a meromorphic function. This allows us to investigate the behavior of C 'C' more closely and, in some cases, to compute its eigenvalues. As long as C' is norm-attaining, kC 'C'k = kC'k is an eigenvalue of C 'C'. We will require the following lemmas before we prove the main result. These lemmas and the ensuing proofs appear in Hammond’s papers, [4] and [5], but we would like to include them here for completeness. Lemma 2.1. Let T be a self-adjoint operator on a Hilbert space H with a closed subspace W that is invariant under T . Then for any eigenvalue of T , there exists a corresponding eigenfunction in W or in W?. Proof. Let be an eigenvalue of T with corresponding eigenfunction g. Then there is a unique decomposition g = g1 + g2, with g1 2W and g2 2W?. Then Tg = g = g1 + g2: NORMS OF COMPOSITION OPERATORS 3 Also, Tg = Tg1 + Tg2. The subspace W? is also invariant under T because the operator is self-adjoint. Hence Tg1 2 W and Tg2 2 W?. Since the decomposition of Tg is unique, we have Tg1 = g1 and Tg2 = g2. Because either g1 or g2 is non-zero, at least one represents an eigenfunction of T with eigenvalue . Lemma 2.2. Let T be a bounded operator on a Hilbert space H. Let g be a maximizing vector for T T , i.e., a function with the property that kT Tgk = kT Tk kgk. Then g is a maximizing vector for T . Proof. We have the well-known identities kT k = kTk and kT Tk = kTk. Therefore we have kTkkgk = kT Tgk kT k kTgk = kTk kTgk: Hence kTgk kTk kgk. Clearly, kTgk kTk kgk, so kTgk = kTk kgk. Therefore, g is a maximizing vector for T . Lemma 2.3. Let ' : D ! D be a non-inner analytic function and let g be a maximizing vector for C': Then g is non-vanishing on D. Proof. Suppose that g vanishes at the point z0 2 D. Then let h = g=Bz0 , where Bz0 is the Blaschke factor which vanishes at z0. Then h is analytic, and for z 2 @D, jh(z)j = jg(z)j, so khk = kgk. Also, since we may assume that g is not identically zero, we have jh(z)j > jg(z)j almost everywhere in D. Because ' is noninner, jh('(z))j > jg('(z))j on a subset of @D which has positive measure. Hence kC'hk > kC'gk, contradicting the assumption that g is norm-attaining. Theorem 2.4. Suppose ' : D ! D extends to a non-inner rational function on C and assume that C' is norm-attaining. Let A = f kgk=1 D denote the set of roots of the function h( ) = 1 '(0) (' )( ) . Suppose that each of these roots has multiplicity 1 and that '(A) f0; '(0)g. Now let

Cite this paper

@inproceedings{EFFINGERDEAN2006NormsOC, title={Norms of Composition Operators with Rational Symbol}, author={SEAN EFFINGER-DEAN and Alan Johnson and Joseph A Reed and Christopher Hammonds}, year={2006} }