Spectra of subdivision operators

Abstract

Let a := {a(k)}kEz be a sequence of complex numbers and a(k) = 0 except for finitely many k. The subdivision operator Sa associated with a is the bi-infinite matrix Sa := (a(j 2k))j,kE. 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 < oo, the spectrum of Sa in fp(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.