Spectra of subdivision operators

Let a := {a(k)}_k∈ℤ be a sequence of complex numbers and a(k) = 0 except for finitely many k. The subdivision operator 50 associated with a is the bi-infinite matrix S_a := (a(j - 2k))_j,k∈ℤ. This operator plays an important role in wavelet analysis and subdivision algorithms. As the adjoint it is closely related to the well-known transfer operators (also called Ruelle operator). In this paper we show that for any 1 ≤ p ≤ ∞, the spectrum of S_a in ℓ_p(ℤ) is always a closed disc centered at the origin. Moreover, except for finitely many points, all the points in the open disc of the spectrum lie in the residual spectrum. © 2000 American Mathematical Society.