Spectra of subdivision operators

Let $a:=\{ a(k)\}_{k\in \mathbb{Z}}$ be a sequence of complex numbers and $a(k)=0$ except for finitely many $k$. The subdivision operator $S_{a}$ associated with $a$ is the bi-infinite matrix $S_{a}:= \left ( a(j-2k)\right )_{j, k\in \mathbb{Z}}$. 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 $1\le p\le \infty$, the spectrum of $S_{a}$ in $\ell _{p}(\mathbb{Z})$ 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.