On continuous and residual spectra of operators connected with iterative functional equations

Abstract

The sum type operator $F$, given by
\[
F \lbrack \varphi \rbrack (x) : = \displaystyle\sum^{\infty}_{\nu = 0} 2^{-\nu} \varphi (2^{\nu} x),
\]
will be considered on the space $D$ of bounded real functions, equipped with the supremum norm and on its three proper closed subspaces. All the according restrictions are Banach space automorphisms. In their spectral theory some iterative functional equations arise in a natural way. We determine in all four cases the resolvent set, the point spectrum, the continuous spectrum and the residual spectrum.