SPECTRA OF COMPOSITION OPERATORS ON ALGEBRAS OF ANALYTIC FUNCTIONS ON BANACH SPACES P. GALINDO 1, T.W. GAMELIN, AND M. LINDSTROMÄ 2 Abstract. Let E be a Banach space, with unit ball BE . We study the spectrum and the essential spectrum of a composition operator on 1 H (BE ) determined by an analytic symbol with a ¯xed point in BE . We relate the spectrum of the composition operator to that of the derivative of the symbol at the ¯xed point. We extend a theorem of Zheng to the context of analytic symbols on the open unit ball of a Hilbert space. 1. Introduction Let E denote a complex Banach space with open unit ball BE and let ' : BE ! BE be an analytic map. In this paper, we consider composition operators C' de¯ned by C'(f) = f ± ', acting on the uniform algebra 1 H (BE) of bounded analytic functions on BE. Evidently C' is bounded, and jjC'jj = 1 = C'(1). We are interested in the spectrum and the essential spectrum of C'. We focus on the case in which ' has a ¯xed point z0 2 BE. Our goal is twofold. The ¯rst is to relate the spectrum of C' to that of 0 ' (z0). The second is to establish an analog for higher dimensions of Zheng's theorem on the spectrum of composition operators on H1(D). This paper is a continuation of [1], [8], and [9]. It is shown in [9] that the essential spectral radius of a composition operator on a uniform algebra is strictly less than 1 if and only if the iterates of its symbol converge in the norm of the dual to a ¯nite number of attracting cycles. In the case at hand, the attracting cycles reduce to a single ¯xed point z0 2 BE. Among other things, we show under this condition that the essential spectral radius of C' 0 coincides with that of ' (z0). 1 In [20], Zheng studies composition operators C' on H (D), where D is the open unit disk in the complex plane. Under the assumption that ' has an attracting ¯xed point in D, she proves that either C' is power compact, in which case the essential spectral radius of C' is 0, or the spectrum of C' 2000 Mathematics Subject Classi¯cation. Primary 46J10. Secondary 47B38. Key words and phrases. composition operator, spectrum, essential spectrum, interpolation. 1 Supported by Project BFM-FEDER 2003-07540 (DGI. Spain). 2 Supported partially by the Academy of Finland Project 205644 and Project BFM- FEDER 2003-07540 (DGI. Spain). 1 2 GALINDO, GAMELIN, AND LINDSTROMÄ coincides with the closed unit disk. Our generalization of Zheng's theorem applies to the unit ball of Cn and also, subject to a compactness condition, to the unit ball of Hilbert space. Before outlining the contents of the paper, we establish some notation. We denote by 'n the nfold iterate of ', so that 'n = ' ± ' ± ¢ ¢ ¢ ± ' (n times). The spectrum and the spectral radius of an operator T are denoted respectively by σ(T ) and r(T ). The essential spectrum of T is denoted by σe(T ). It is de¯ned to consist of all complex numbers ¸ such that ¸I ¡ T is not a Fredholm operator. The essential spectral radius of T is denoted by re(T ). If re(T ) = 0, then T is said to be a Riesz operator. For background information on the essential spectrum and on Fredholm operators, see [17]. For background information on analytic functions on Banach spaces and the associated tensor product spaces, see [6], [10], or [16]. References on composition operators on uniform algebras are [14], [15], [11], and [12]. The paper is organized as follows. After some preliminary lemmas on lower triangular matrices in Section 2, we treat in Section 3 the lower tri- angular representation of C' corresponding to the Taylor series expansion 1 of functions in H (BE) at the ¯xed point z0. The results are applied in Section 4 to the case where re(C') < 1. There we connect the spectrum and 0 essential spectrum of C' to that of ' (z0). In particular, we show that if 0 C' is a Riesz operator, then ' (z0) is a Riesz operator, and we describe the 0 spectrum of C' in terms of that of ' (z0). This generalizes the corresponding result obtained in [1] for compact composition operators. Sections 5 and 6 contain preparatory material for the generalization of Zheng's theorem. In Section 5 we observe that the interpolation operators constructed by B. Berndtsson [2] for the unit ball of Cn yield an interpolation theorem for sequences in the unit ball BH of Hilbert space H that tend ex- ponentially to the boundary. In Section 6 we establish a Julia-type estimate for analytic self-maps of BH as the variable tends to the boundary through an approach region that clusters on a compact subset of the unit sphere. These results are combined in Section 7 to establish the generalization of Zheng's theorem. 2. Spectra of Lower Triangular Operators We begin with the following elementary fact. Lemma 2.1. A lower triangular square matrix with entries in a unital ring is invertible and has a lower triangular inverse if and only if the diagonal entries of the matrix are invertible. Proof. Backsolve. ¤ We will be interested in operators on a direct sum X = X1 © ¢ ¢ ¢ © Xn of Banach spaces. Such an operator S leaves invariant each direct subsum SPECTRA OF COMPOSITION OPERATORS 3 Xk © ¢ ¢ ¢ © Xn, 2 · k · n, if and only if S has a lower triangular matrix representation 0 1 S11 0 0 ::: 0 B C BS21 S22 0 ::: 0 C B . C (2.1) S = B . C ; B . C @ :::::::::Sn¡1;n¡1 0 A Sn1 Sn2 :::Sn;n¡1 Snn where Sjk : Xj ! Xk. From the preceding lemma, applied to the lower triangular matrix operator ¸I ¡S, we see that if each of the diagonal entries ¸I ¡ Sjj is an invertible operator on its space Xj, then ¸I ¡ S is invertible. Similarly, if we apply the lemma to the quotient ring of operators modulo compact operators, we see that if each of the diagonal entries ¸I ¡ Sjj is a Fredholm operator on its space Xj, then ¸I ¡ S is a Fredholm operator on X. This yields the following lemma. Lemma 2.2. Let X = X1 © ¢ ¢ ¢ © Xn be a direct sum of Banach spaces, and let S be an operator on X with lower triangular matrix representation (2.1). Then (2.2) σ(S) ⊆ σ(S11) [ ¢ ¢ ¢ [ σ(Snn); and (2.3) σe(S) ⊆ σe(S11) [ ¢ ¢ ¢ [ σe(Snn); where σ(Sjj) and σe(Sjj) are respectively the spectrum and essential spec- trum of Sjj operating on Xj. It can occur that the inclusions in (2.2) and (2.3) are strict. Nevertheless we have the following. Lemma 2.3. Let X = X1 © ¢ ¢ ¢ © Xn be a direct sum of Banach spaces, and let S be an operator on X with lower triangular matrix representation (2.1). Let ­ be the unbounded component of the complement of σe(S11) [ ¢ ¢ ¢ [ σe(Sn¡1;n¡1) in the complex plane C. Then (2.4) σ(S) \ ­ = (σ(S11) [ ¢ ¢ ¢ [ σ(Snn)) \ ­; (2.5) σe(S) \ ­ = σe(Snn) \ ­: Further, @­ ⊆ σe(S). Proof. Let ¸0 2 ­nσ(S). To establish (2.4), we must show that ¸0 2= σ(S11)[¢ ¢ ¢[σ(Snn), that is, we must show that each ¸0I ¡Sjj is invertible. We break the argument into two cases. Suppose ¯rst that ¸0 2= σ(S11) [ ¢ ¢ ¢ [ σ(Sn¡1;n¡1). We must show that ¸0 2= σ(Snn). For this, let U be the lower triangular (n ¡ 1) £ (n ¡ 1) matrix obtained by striking out the last column and the bottom row of S, so that µ ¶ U 0 S = : VSnn 4 GALINDO, GAMELIN, AND LINDSTROMÄ By Lemma 2.1, ¸0I ¡ U is invertible and its inverse is lower triangular. The inverse of ¸0I ¡ S then has the form µ ¶ (¸ I ¡ U)¡1 R (¸ I ¡ S)¡1 = 0 : 0 TW Multiplying by ¸0I ¡S on the left, we ¯nd that the column vector R satis¯es (¸0I ¡ U)R = 0. Since ¸0I ¡ U is invertible, R = 0. Consequently ¸0I ¡ S ¡1 has a lower triangular inverse, and W = (¸0I ¡ Snn) . In particular, ¸0 2= σ(Snn), as required. For the remaining case, suppose that ¸0 2 σ(S11) [ ¢ ¢ ¢ [ σ(Sn¡1;n¡1). We will show that this leads to a contradiction. For this, we apply the ¯rst part of the proof to ¸'s in a punctured neighborhood of ¸0. Since σ(S11) [ ¢ ¢ ¢ [ σ(Sn¡1;n¡1) meets ­ in a discrete subset, by what we have shown there is a punctured neighborhood of ¸0 on which ¸I ¡ S has a lower triangular inverse. Letting ¸ ! ¸0, we see that the inverse of ¸0I ¡ S is also lower triangular. Hence ¸0 2= σ(Sjj) for 1 · j · n. This is a contradiction, and we conclude that (2.4) holds. The same argument as in the ¯rst case above, applied to the quotient algebra of operators modulo compact operators, shows that if ¸ 2 ­ and ¸2 = σe(S), then ¸2 = σe(Snn). Hence (2.5) holds, and further the inverse of ¸I ¡ S modulo the compacts has a lower triangular matrix representation when ¸ 2 ­nσe(S). Now let ¸0 2 @­, and suppose that ¸0 2= σe(S). By the preceding re- mark, there are ¸ 2 ­ near ¸0 for which the inverse of ¸I ¡ S modulo the compacts has a lower triangular matrix representation. Letting ¸ ! ¸0, we see that the inverse of ¸0I ¡ S modulo the compacts has a lower triangular matrix representation.