Spectrum of a composition operator

Abstract:

A composition operator is a linear operator induced on a subspace of ${K^X}$ by a point transformation $\phi$ on a set X (where K denotes the scalar field) by the formula $Tf(x) = f \circ \phi (x)$. Familiar examples include translation operators on the real line and on topological groups, analytic functions which preserve the class of harmonic functions (and Green’s functions), ergodic transformations which induce unitary operators on ${L^2}$, shift and weighted shift operators. The spectrum, approximate point spectrum, and point spectrum of an ${L^p}$-composition operator have circular symmetry about 0, except on the unit circle, where they form unions of subgroups; certain consequences are derived from this.