Spectra of certain operators and iterative functional equations

Abstract

We discuss spectral properties of the operator $F : D -> F|D|$, defined by \[F|ϕ|f(x) := \sum_{k=0}^∞ \frac{1}{2^k}ϕ(2^kx)\] $D$ is the vector space of real functions $ϕ$ such that the sum above converges for all $x ∈ R$. The point spectrum and the eigenspaces of $F$ and of its restriction to the vector space $U$ of ultimately bounded functions are given. Moreover we compute the point spectrum and eigenspaces, the continuous spectrum and the residual spectrum of $F$ , restricted to the Banach spaces $B$ of bounded functions and $C$ of bounded and continuous functions.