| In this paper we describe the spectral properties of composition operators of the form Cϕ(f) = f ∘ ϕ, f ∈ H infinity(U). Here, U is a certain type of infinitely connected domain called an L-domain, and Hinfinity(U) denotes the algebra of bounded analytic functions on U. These domains are interesting in part because M. Behrens [2] used them to provide the first known examples of infinitely connected domains for which a corona theorem holds.; To obtain our results we use the Behrens isomorphism theorem which relates Hinfinity(U) to H infinity( DxN ), the algebra of bounded analytic functions on a sequence of disks. In certain cases we obtain a correspondence between a composition operator Cϕ on Hinfinity( U) and a "shift-composition" operator on H infinity( DxN ) of the form Cphi(F) = F ∘ phi where phi(z, n) = (ϕn(z), n + 1).; In the case where phi(z, n) has the relatively simple form phi(gammaz, n + 1) for 0 < gamma < 1, we can completely describe the spectrum and essential spectrum of Cphi on Hinfinity ( DxN ).; Theorem 0.1. Let phi(z, n) = (gammaz, n + 1), with 0 < gamma < 1, and let Cphi be the corresponding composition operator on H0infinity( DxN ). Then sCF =l≤g . If gammam+1 < |lambda| < gammam then lambda is an eigenvalue of multiplicity m. Furthermore, seCF =0∪⋃ m=1infinity l=gm , and for gammam+1 < |lambda| < gammam, the Fredholm index of lambdaI - Cphi on H 0infinity( DxN ) is m.; We describe an L-domain U such that ϕ( z) = alphaz is a self-map of U into U. Let U=D\0 \ ⋃n=1infinity Dn, where, Dn = {lcub}|z - r n| ≤ rn{rcub}, cn = alphan-1c 1 and rn = betan. In this case we get the following result concerning the essential spectrum of Cϕ on Hinfinity (U).; Theorem 0.2. Let U be an L-domain of the type described above, and let ϕ(z) = alpha z. Let Cϕ be the corresponding composition operator on Hinfinity(U). If (beta/alpha)m+1 < |&lgr;| < (beta/alpha)m, then &lgr; I - Cϕ is a Fredholm operator, and the dimension of the null space of &lgr;I - C ϕ is m. Furthermore seC4 =0∪⋃ n=0infinity l=b an . |
