ON THE NORM OF A COMPOSITION OPERATOR WITH LINEAR FRACTIONAL SYMBOL

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1 ON THE NORM OF A COMPOSITION OPERATOR WITH LINEAR FRACTIONAL SYMBOL CHRISTOPHER HAMMOND Abstract. For any analytic map ϕ : D D, the composition operator C ϕ is bounded on the Hardy space H 2, but there is no known procedure for precisely computing its norm. This paper considers the situation where ϕ is a linear fractional map. We determine the conditions under which C ϕ is given by the action of either C ϕ or Cϕ on the normalized reproducing kernel functions of H 2. We also introduce a new set of conditions on ϕ under which we can calculate C ϕ ; moreover, we identify the elements of H 2 on which such an operator C ϕ attains its norm. Several specific examples are provided.. Introduction For p <, the Hardy space H p is the collection of all analytic functions f on D {z C : z < } with f p p sup 0<r< 2π 0 f re iθ p dθ 2π <. Under this norm, H p is a Banach space for all such p and a Hilbert space for p 2. For any analytic map ϕ : D D, the composition operator C ϕ on H p is defined by the rule C ϕ f f ϕ. Every composition operator is bounded, with /p. ϕ0 2 C ϕ : H p H p /p + ϕ0. ϕ0 These inequalities are sharp; for any value of ϕ0, there are particular examples of ϕ for which C ϕ equals the upper bound and examples for which C ϕ equals the lower bound. In general, though, there is no known procedure for precisely 2000 Mathematics Subject Classification. 47B33. This paper is part of the author s doctoral thesis, written at the University of Virginia under the direction of Barbara D. MacCluer.

2 2 C. HAMMOND computing the norm of C ϕ. We see from expression. that C ϕ whenever ϕ0 0. exactly; for example There are only a few other cases where we can determine the norm ϕ is inner; that is, lim r ϕ re iθ for almost all θ in [0, 2π, 2 ϕz az + b where a + b, 3 ϕz r+sz+ s r sz++sr where 0 < s < and 0 r. These results appear in [2, [6, and [7 respectively. Cowen and MacCluer [8 provide a comprehensive treatment of this material, as well as a thorough overview of results relating to composition operators. A straightforward argument involving Blaschke products shows that the following norm relationship holds for all p : C ϕ : H p H p p Cϕ : H 2 H 2 2. Therefore it suffices to focus our attention on the Hilbert space H 2. When studying this space, it is often helpful to consider the reproducing kernel functions {K w } w D, defined by the property that f, K w fw for all f in H 2. These functions have the form K w z wz ; hence K w 2 K w, K w K w w w 2. Throughout this paper, we write k w to denote the normalized kernel function k w z K wz w 2 K w 2 wz. For a subset W of D, let K W denote the closed linear span of the kernel functions {K w } w W. Observe that the orthogonal complement K W functions in H 2 that vanish on W. is precisely the set of all The kernel functions provide a valuable tool for the study of composition operators, in part because of the property that C ϕk w K ϕw for any adjoint C ϕ. Several authors have explored the connection between the kernel functions and the norm of C ϕ. In the cases where we know C ϕ, the norm is given by the action of the operator on the set of normalized kernel functions. This situation, however, is not true in general, a fact first proved by Appel, Bourdon, and Thrall [.

3 ON THE NORM OF A COMPOSITION OPERATOR 3 The main results of this paper pertain to the situation where ϕ : D D is a linear fractional map. In this case, we determine the conditions under which C ϕ is given by the action of either C ϕ or Cϕ on the normalized reproducing kernel functions Theorem 4.4. We also introduce a new set of conditions on ϕ under which, at least in principle, we can calculate C ϕ Theorem 5.5. For such ϕ, we identify the elements of H 2 on which C ϕ attains its norm, each of which is a finite linear combination of kernel functions. 2. Preliminaries Let T be a bounded operator on a Hilbert space H. One reasonable strategy for determining T is to investigate the spectrum of the operator T T. Since T T is self-adjoint, its spectral radius equals T T T 2. The following observation underscores the connection between the spectrum of T T and the norm of T. Proposition 2.. Let h be an element of H; then T h T h if and only if T T h T 2 h. This proposition can be proved with a straightforward Hilbert space argument, or can be deduced from other well-known results e.g. [0, p. 92. Whenever T h T h for h 0, we say that the operator T attains its norm on the element h. Let e, r, and r e denote respectively the essential norm, the spectral radius, and the essential spectral radius of an operator. Here the adjective essential signifies that a particular quantity is taken with respect to the Calkin algebra. In light of Proposition 2., our next observation follows easily. Proposition 2.2. If T e < T, then T attains its norm on an element of H. Proof. Consider the positive operator T T ; observe that r e T T T T e T 2 e < T 2 T T rt T. Therefore the largest element of the spectrum of T T does not belong to the essential spectrum, meaning that it is an eigenvalue of finite multiplicity. Consequently T T has an eigenvector corresponding to T 2, on which the operator T attains its norm.

4 4 C. HAMMOND It is helpful to remember Proposition 2.2 when studying composition operators, especially since we have a formula due to Joel Shapiro [5 for the essential norm of C ϕ on H 2. As it happens, in cases and 3 where we know C ϕ, the operators have the property that C ϕ e C ϕ, a condition sometimes called extremal noncompactness. The remaining results in this section are specific to composition operators on the Hardy space H 2. Proposition 2.3. Suppose that the operator C ϕ : H 2 H 2 attains its norm on an element g of H 2. If ϕ is not an inner function, then g cannot vanish at any point of D. Proof. Suppose that gw 0 for some w in D. Then the function gz gz b w z gz w z wz belongs to H 2, with g 2 g 2. Since ϕ is not an inner function, neither is the composition b w ϕ. Therefore lim r g ϕ re iθ b w ϕ re iθ > lim g ϕ re iθ r for θ in a set of positive measure. Hence C ϕ g 2 > C ϕ g 2, contradicting our choice of g. Corollary 2.4. Suppose that ϕ is not inner; if g and g 2 are functions on which C ϕ attains its norm, then one is a scalar multiple of the other. Proof. Both g and g 2 are eigenfunctions for CϕC ϕ : H 2 H 2 corresponding to the eigenvalue C ϕ 2 ; moreover, g 0 and g 2 0 are both nonzero. If g g 0/g 2 0 g 2 were not identically 0, then it would be an eigenfunction corresponding to C ϕ 2, in other words a function on which C ϕ attains its norm, that vanishes at 0. Therefore g g 0/g 2 0 g 2, as we had hoped to show. We end this section with a straightforward, but remarkably useful observation. Let λ be an eigenvalue for C ϕc ϕ with a corresponding eigenfunction g; since C ϕ

5 ON THE NORM OF A COMPOSITION OPERATOR 5 fixes the constant function K 0 z, we see that 2. gϕ0 C ϕ g, K 0 C ϕ g, C ϕ K 0 C ϕc ϕ g, K0 λ g, K0 λg0. In particular, this result holds for λ C ϕ 2 if C ϕ attains its norm on g. Let 3. The operator C ϕc ϕ ϕz az + b cz + d be a nonconstant linear fractional self-map of D. Cowen [6 proved that the adjoint operator C ϕ may be written T γ C σ T η, with 3. σz az c bz + d, γz bz + d, ηz cz + d, where T γ and T η denote the corresponding Toeplitz operators. Hence C ϕc ϕ f Tγ C σ T η C ϕ f for any f in H 2. Recalling that T z is the backward shift on H 2, we see that 3.2 C ϕ C ϕ f z γz c fϕσz fϕ0 σz + dfϕσz c d [fϕσz fϕ0 + az c bz + d fϕσz for all z in D not equal to σ 0 c a. We rewrite this expression simply as 3.3 C ϕ C ϕ f z ψzfτz + χzfϕ0, where τ denotes the composition ϕ σ and ad bc z ψz az c c and χz bz + d az + c. Equation 3.3 holds for all points except z σ 0, which only belongs to D if c < a. investigate its spectrum. Having such a concrete representation for C ϕc ϕ makes it easier to

6 6 C. HAMMOND 4. The quantities S ϕ and S ϕ Let ϕ be an analytic self-map of D. Bourdon and Retsek [4 defined the quantities { } Cϕ K w S ϕ sup 2 { } sup Cϕ k w w D K w 2 2 w D and S ϕ sup w D { C ϕk w 2 K w 2 } { sup Cϕk w } 2. w D Among other results, they proved that S ϕ S ϕ for all ϕ and that S ϕ S ϕ C ϕ whenever ϕ0 0 or ϕ has the form ϕz az + b; moreover, when ϕ0 0 and ϕz az + b, they showed that S ϕ cannot equal C ϕ unless C ϕ e C ϕ. The quantities S ϕ and S ϕ were also studied, with different notation, by Avramidou and Jafari [2. In this section, we determine the conditions under which either S ϕ C ϕ or S ϕ C ϕ when ϕ : D D is a linear fractional map. We begin with a few observations which hold for any analytic ϕ : D D. If {w j } is a sequence of points converging to w in D, then the normalized kernel functions { kwj } converge to kw in the norm of H 2. Therefore, since C ϕ is a bounded operator, either S ϕ C ϕ k w 2 for a particular w in D or S ϕ lim sup w C ϕ k w 2. The analogous result holds for S ϕ. Cima and Matheson [5 observed that C ϕ e lim sup C ϕ k w 2, w a fact which follows from the proof of Shapiro s essential norm formula [5. In the case where ϕ is univalent, Shapiro s formula may be expressed w 2 C ϕ e lim sup 2 lim sup C ϕk w w ϕw 2. w Therefore S ϕ C ϕ e for any ϕ and S ϕ C ϕ e whenever ϕ is univalent. Before proving our results for linear fractional ϕ, we need the following pair of general lemmas. The first, which pertains to the lower bound in expression., appears with a different proof in a current paper of David Pakorny and Jonathan Shapiro [3. Lemma 4.. If ϕ : D D is a nonconstant analytic map with ϕ0 0, then C ϕ > ϕ0 2.

7 ON THE NORM OF A COMPOSITION OPERATOR 7 Proof. The Hardy space H 2 has the property that f0 < f 2 for any nonconstant element f. Observe that the function k ϕ0 ϕ is nonconstant; therefore C ϕ C ϕ kϕ0 2 > k ϕ0 ϕ 0 ϕ0 2, as we had hoped to show. Lemma 4.2. Suppose that ϕ : D D is a nonconstant analytic map with ϕ0 0. If the operator C ϕ attains its norm on a normalized kernel function k w, then w > ϕ0. Proof. Suppose that C ϕ attains its norm on k w ; then K w is an eigenfunction for C ϕc ϕ corresponding to C ϕ 2. Appealing to equation 2., we see that It follows from Lemma 4. that meaning that w > ϕ0. wϕ0 K wϕ0 C ϕ 2 K w 0 C ϕ 2. wϕ0 > ϕ0 2, Now we turn our attention to the situation where ϕ is a linear fractional map. Proposition 4.3. Let ϕ : D D be a linear fractional map with ϕ0 0 and which does not have the form ϕz az + b. For any point w in D, C ϕ > C ϕ k w 2. Proof. Suppose, to the contrary, that C ϕ attains its norm on some normalized kernel function k w ; then K w is an eigenfunction for C ϕc ϕ. Hence the subspace K {w} {αk w : α C} is invariant under C ϕc ϕ. Since C ϕc ϕ is self-adjoint, the orthogonal complement K {w} {f H2 : fw 0} is also invariant under the operator; this observation will give rise to a contradiction. Lemma 4.2 tells us that w cannot equal 0 or ϕ0. Suppose then that w is the problematic point σ 0 c a. Applying L Hôpital s rule to expression 3.2, we obtain C ϕ C ϕ f σ 0 c a f ϕ0τ σ 0 + ad ad bc fϕ0, which must equal 0 for any f in K {w}. Consider the function f z z ϕ0z w in K {w}. The assumption that ϕz az + b guarantees that c 0; since

8 8 C. HAMMOND f ϕ0 0 and τ ϕ σ is univalent, the term f ϕ0 must equal 0, which is not the case. Therefore w cannot equal σ 0. Hence equation 3.3 is valid at w, meaning that 0 C ϕc ϕ f w ψwfτw + χwfϕ0 for all f in K {w}. Again consider the function f. Observe that f ϕ0 0; since w 0, the term ψw is nonzero. Hence f τw 0, meaning that τw equals either w or ϕ0. If τw ϕ0, then w σ 0, which is not the case; therefore τw w. Now take f 2 z z w in K {w}. Since f 2τw f 2 w 0 and χw c c aw 0, we see that f 2ϕ0 0. Therefore ϕ0 must equal w, which is a contradiction. We now state main result of this section: Theorem 4.4. Let ϕ : D D be a linear fractional map with ϕ0 0 and which does not have the form ϕz az + b. Then S ϕ C ϕ if and only if C ϕ e C ϕ ; likewise S ϕ C ϕ if and only if C ϕ e C ϕ. Proof. Recall that C ϕ e S ϕ S ϕ C ϕ for any univalent ϕ; if C ϕ e C ϕ, then all of these quantities are equal. On the other hand, suppose that C ϕ e < C ϕ. Since C ϕ k w 2 < C ϕ for all w in D, it follows from our characterization of S ϕ that S ϕ < C ϕ. Since S ϕ S ϕ, our result follows. As a consequence of this theorem, we see that S ϕ C ϕ if and only if S ϕ C ϕ. We should mention, though, that there are linear fractional ϕ such that S ϕ S ϕ < C ϕ ; for example, Retsek [4 showed that the map ϕz 4 5 z has this property. Theorem 4.4 no longer holds if we eliminate the hypothesis that ϕ be linear fractional. In light of the aforementioned results of Bourdon and Retsek, we see that our assertion for S ϕ holds whenever ϕ is univalent an observation also made by Retsek [4. On the other hand, Bourdon and Retsek [4 proved that S ϕ < C ϕ e C ϕ whenever ϕ is a non-univalent inner function with ϕ0 0. Extremal noncompactness implies that S ϕ C ϕ for any ϕ. It is not difficult, however, to find further examples of analytic ϕ with ϕ0 0 and C ϕ e < S ϕ C ϕ. To that end, let ν be an inner function that fixes the origin; then as shown by Nordgren [2 the

9 ON THE NORM OF A COMPOSITION OPERATOR 9 composition operator C ν is an isometry of H 2. Hence, for any analytic ϕ : D D, the operator C ϕ ν C ν C ϕ has the same norm as C ϕ ; moreover, S ϕ ν C ϕ ν if and only if S ϕ C ϕ. Consider the map ϕz az + b, where both a and b are nonzero and a + b <. We know that S ϕ C ϕ, and that both of the operators C ϕ and C ϕ ν are compact. Hence C ϕ ν e 0 < S ϕ ν C ϕ ν ; in particular, this result holds if we take νz z m for some integer m, so that ϕ νz az m + b. 5. The spectrum of C ϕc ϕ Let ϕ : D D be a nonconstant linear fractional map, as discussed in Section 3; let σ be defined as in 3.. Our goal now is to find a set of conditions under which we can determine C ϕ. For a non-negative integer j, let τ j denote the jth iterate of τ ϕ σ; that is, τ 0 is the identity map on D and τ j+ τ τ j. Throughout the next two sections, we make the following assumption: There is some integer n 0 such that τ n ϕ0 0. In effect, this condition is a generalization of the case where ϕ0 0. To avoid a triviality, we also assume that ϕ does not have the form ϕz az. These assumptions guarantee that τ j ϕ0 never equals σ 0 and that τ j ϕ0 0 for j n, as we can see from arguments involving fixed points. Furthermore, these conditions exclude all disk automorphisms and all maps of the form ϕz az + b. Let W denote the set of points {τ j ϕ0} n j0 ; recall that K W H 2 consisting of all functions that vanish on W. We claim that K W is the subspace of is invariant under the operator CϕC ϕ. Suppose that f belongs to KW ; it follows from equation 3.3 that C ϕ C ϕ f τ j ϕ0 ψτ j ϕ0fτ j+ ϕ0 + χτ j ϕ0fϕ0 ψτ j ϕ0fτ j+ ϕ0. For 0 j n, the term fτ j+ ϕ0 equals 0; for j n, the term ψτ j ϕ0 ψ0 vanishes. Therefore C ϕc ϕ f also belongs to K W. Since C ϕc ϕ : K W K W looks like a weighted composition operator, we can deduce a good deal of information about its spectrum. For example, if C ϕ : H 2 H 2 is compact, then the spectrum of C ϕc ϕ : K W K W is precisely

10 0 C. HAMMOND 0 { ψw 0 τ w 0 j} j0, where w 0 denotes the Denjoy-Wolff point of τ. This fact, however, does not help us to determine C ϕ and is not proved here; details appear in the author s thesis [9. Now consider K W, the span of the kernel functions { n K τj ϕ0} ; observe that j0 it has dimension n +. The subspace K W is also invariant under the self-adjoint operator C ϕc ϕ : H 2 H 2. Our strategy for finding C ϕ centers around determining the spectrum, namely the eigenvalues, of the operator C ϕc ϕ : K W K W. The next several results pertain to the eigenvalues and eigenfunctions of C ϕc ϕ. The following proposition serves as a generalization of equation 2.. Proposition 5.. Let λ be an eigenvalue of C ϕc ϕ : H 2 H 2 with a corresponding eigenfunction g. For every integer j 0, the following relationship holds: [ j [ j k λ j+ g0 ψτ m ϕ0 gτ j ϕ0+ χτ k ϕ0 ψτ m ϕ0 λ j k g0, m0 k0 where we take m0 to equal and k0 to equal 0. Proof by induction. Since λg0 gϕ0, the claim holds for j 0. For any j 0, equation 3.3 dictates that λgτ j ϕ0 C ϕc ϕ g τj ϕ0 m0 5. ψτ j ϕ0gτ j+ ϕ0 + χτ j ϕ0λg0. Now assume that our claim holds for the index j. Multiplying the consequent equation by λ and substituting expression 5. for λgτ j ϕ0, we obtain λ j+2 g0 [ j m0 j + ψτ m ϕ0 χτ k ϕ0 k0 [ψτ j ϕ0gτ j+ ϕ0 + χτ j ϕ0λg0 [ k m0 ψτ m ϕ0 [ j ψτ m ϕ0 gτ j+ ϕ0 m0 k0 m0 λ j+ k g0 [ j k + χτ k ϕ0 ψτ m ϕ0 λ j+ k g0. Hence our claim also holds for the index j +.

11 ON THE NORM OF A COMPOSITION OPERATOR Since both K W and K W are invariant under C ϕc ϕ : H 2 H 2, each eigenvalue λ of C ϕc ϕ has an eigenfunction belonging to one of the two subspaces. The next proposition provides a distinguishing characteristic for eigenfunctions in K W. Proposition 5.2. Let g be an eigenfunction for C ϕc ϕ : H 2 H 2 ; then g belongs to K W if and only if g0 0. Proof. If g belongs to K W, then by definition g0 gτ nϕ0 equals 0. Conversely, suppose that g is an eigenfunction for C ϕc ϕ with g0 0. In this case, Proposition 5. dictates that 0 λ j+ g0 [ j m0 ψτ m ϕ0 gτ j ϕ0 for all j 0. Since ψτ m ϕ0 is nonzero for 0 m n, the function g must vanish on the entire set {τ j ϕ0} n j0. In other words, g belongs to the subspace K W. Corollary 5.3. Suppose that g and g 2 are eigenfunctions for C ϕc ϕ which belong to K W ; if they correspond to the same eigenvalue, then one is a scalar multiple of the other. Proof. We appeal to the proof of Corollary 2.4, bearing in mind that no eigenfunction in K W can vanish at 0. Consequently every eigenspace of C ϕc ϕ : K W K W has dimension. Since C ϕc ϕ : K W K W is a self-adjoint operator on a finite dimensional space, we know that K W is spanned by eigenfunctions of C ϕc ϕ. Since K W has dimension n +, the operator C ϕc ϕ : K W K W must have n + distinct eigenvalues. We return to the result of Proposition 5.. Taking j n and observing that χτ n ϕ0 χ0, we obtain the expression [ n k λ n+ g0 χτ k ϕ0 ψτ m ϕ0 λ n k g0. k0 m0 Suppose that the eigenfunction g belongs to K W ; then g0 0 and [ n k λ n+ χτ k ϕ0 ψτ m ϕ0 λ n k. k0 m0

12 2 C. HAMMOND In other words, any eigenvalue λ of C ϕc ϕ : K W K W is a solution to this polynomial equation. Since there are n + distinct eigenvalues and the equation has no more than n + roots, we conclude that every solution is an eigenvalue. In other words, [ n k 5.2 pλ λ n+ χτ k ϕ0 ψτ m ϕ0 k0 m0 λ n k is the characteristic polynomial of the operator C ϕc ϕ : K W K W. Finally, we make an observation regarding the essential norm of C ϕ. The author is indebted to Paul Bourdon for suggesting the proof of this proposition. Proposition 5.4. Under the assumptions of this section, C ϕ e <. Proof. If ϕ <, then C ϕ is compact, so our claim holds. Suppose then that ϕ ; since ϕ is not an automorphism, there is precisely one pair of points ζ and ω on D with ϕζ ω. Bourdon, Levi, Narayan, and Shapiro [3 proved in general that σω ζ and σ ω ϕ ζ ; hence τω ω and τ ω. Since the map τ n ϕ fixes the origin and τ n ϕζ ω, it follows from Lemma 7.33 in [8, together with the Julia-Carathéodory theorem, that τ n ϕ ζ >. Therefore < τ n ϕζ ϕ ζ τ n ω ϕ ζ ϕ ζ. Since ϕ is univalent on a neighborhood of the closed unit disk, Shapiro s essential norm formula [5 yields } C ϕ 2 e { ϕ max w : w ϕw ϕ ζ <, as we had hoped to show. Since C ϕ, Proposition 2.2 dictates that C ϕ : H 2 H 2 attains its norm on an element of H 2 ; that is, C ϕ 2 is an eigenvalue of CϕC ϕ : H 2 H 2. Proposition 2.3 guarantees that any corresponding eigenfunction must belong to K W. In other words, C ϕ 2 is the largest eigenvalue of CϕC ϕ : K W K W, meaning that it is the largest zero of the polynomial p. Hence we have proved the following result: Theorem 5.5. Let ϕ : D D be a linear fractional map, with ϕz az. Suppose that τ n ϕ0 0 for some integer n 0; then C ϕ 2 is the largest zero of the

13 ON THE NORM OF A COMPOSITION OPERATOR 3 polynomial p in equation 5.2, and the elements on which C ϕ attains its norm are linear combinations of the kernel functions { K τj ϕ0} n j0. Whenever n, Theorem 4.4 dictates that S ϕ < C ϕ. Assuming that we can find examples of such ϕ, this would appear to be the first case of a composition operator whose norm we can calculate, for which the norm is not given by the action of C ϕ on the normalized reproducing kernel functions of H The eigenfunctions of C ϕc ϕ Having determined a particular eigenvalue λ of C ϕc ϕ : K W K W, it is possible to find the corresponding eigenfunctions. In particular, considering Theorem 5.5, we can identify the functions on which the operator C ϕ attains its norm. Let λ be such an eigenvalue and g be its unique eigenfunction in K W with g0 gτ n ϕ0. We write gz n i0 α i τ i ϕ0z, where we hope to determine the coefficients α i. For any index 0 j n, we may appeal to Proposition 5. to find gτ j ϕ0 explicitly in terms of λ: λ j+ j k0 χτ kϕ0 gτ j ϕ0 Therefore we obtain the matrix equation [ τ i ϕ0τ j ϕ0 0 j,i n [ k m0 ψτ mϕ0 j m0 ψτ mϕ0 λ j k [α i 0 i n [gτ j ϕ0 0 j n. The n+ n+ matrix is simply the Gram matrix of the vectors { K τi ϕ0} n i0, whose determinant is positive since the kernel functions are linearly independent see [, p Hence we can use Cramer s rule to solve explicitly for the coefficients. For example, take n. Then ϕ0 2 α 0 α gϕ0 g0 λ,.

14 4 C. HAMMOND so α 0 λ ϕ0 2 λ ϕ0 2 and α λ ϕ0 2 ϕ0 2 λ ϕ0 2. ϕ Examples It is not difficult to find examples of linear fractional ϕ : D D with τϕ0 0. In terms of the coefficients of ϕ, this condition is equivalent to d 2 b 2 a cd ab. b In this case, considering the polynomial p in equation 5.2, we see that any eigenvalue λ of C ϕc ϕ : K W K W has the form λ χϕ0 ± χϕ ψϕ0 2 acd b ± acd 2 b 4 ad bc ad 2 d 2 b 2. In particular, C ϕ 2 is the larger of these two values. For example, take ϕz 6z + 8 9z Since ϕ <, the operator C ϕ is compact. Observe that τϕ0 0, which means that C ϕ We now turn our attention to a larger class of examples. Let n be a positive integer and r a real number greater than n. Define ϕz rz n n + z + r +. It is easy to show that ϕ is a self-map of D and that ϕd D {}. Note that C ϕ 2 e ϕ r n 2 rr + nn + r n r + n +. A straightforward induction argument shows that each iterate τ j has the form τ j z r + n j + z + j jz + r + n + j +.

15 ON THE NORM OF A COMPOSITION OPERATOR 5 Consequently τ j ϕ0 j r + n j + n r+ n r+ + j + r + n + j + from which we see that τ n ϕ0 0. Observe that ψτ j ϕ0 r j n r + j +, r r + n n + j n j n r+j+ + n + n j n r+j+ r+j+ r n j n r + j + j + r + n + r + j n + + r + and χτ j ϕ0 r j n r+j+ n + + n + n + r + j + j + r + n +. Hence the characteristic polynomial for C ϕc ϕ : K W K W may be written pλ λ n+ n k0 [ k n + r + k + r n m n r + m + λ n k, k + r + n + m + r + n + r + m n + m0 and C ϕ 2 is the largest zero of this polynomial. In particular, if n then C ϕ 2 r + r r +. r + 2 r For n 2, we solve the resulting cubic equation to obtain C ϕ 2 r + r r + r + 3 r r Re 2r + 2 r i r 2, r where we take the principal branch of the cube root function. For example, if then C ϕ Re i 7 ϕz 4z 2 3z arctan 2 7 cos

16 6 C. HAMMOND References [ M. J. Appel, P. S. Bourdon and J. J. Thrall, Norms of composition operators on the Hardy space, Experiment. Math., 5 996, -7. [2 P. Avramidou and F. Jafari, On norms of composition operators on Hardy spaces, Function Spaces Edwardsville, 998, 47-54, Contemp. Math., 232, Amer. Math. Soc., Providence, 999. [3 P. S. Bourdon, D. Levi, S. Narayan and J. H. Shapiro, Which linear-fractional composition operators are essentially normal?, preprint, [4 P. S. Bourdon and D. Q. Retsek, Reproducing kernels and norms of composition operators, Acta Sci. Math. Szeged, , [5 J. A. Cima and A. L. Matheson, Essential norms of composition operators and Aleksandrov measures, Pacific J. Math., , [6 C. C. Cowen, Linear fractional composition operators on H 2, Integral Equations Operator Theory, 988, [7 C. C. Cowen and T. L. Kriete, Subnormality and composition operators on H 2, J. Funct. Anal., 8 988, [8 C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 995. [9 C. Hammond, On the Norm of a Composition Operator, Thesis, University of Virginia, [0 A. A. Kirillov and A. D. Gvishiani, Theorems and Problems in Functional Analysis, translated by H. H. McFaden, Springer-Verlag, New York, 982. [ D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 993. [2 E. A. Nordgren, Composition operators, Canadian J. Math., , [3 D. B. Pakorny and J. E. Shapiro, Continuity of the norm of a composition operator, Integral Equations Operator Theory, to appear. [4 D. Q. Retsek, The Kernel Supremum Property and Norms of Composition Operators, Thesis, Washington University, 200. [5 J. H. Shapiro, The essential norm of a composition operator, Annals of Math., , C. Hammond, Department of Mathematics, University of Virginia, Charlottesville, VA , USA address: cnh5u@virginia.edu