Extended Spectra for Some Composition Operators on Weighted Hardy Spaces

Let $\alpha$ be a complex scalar, and let $A$ be a bounded linear operator on a Hilbert space $H$. We say that $\alpha$ is an extended eigenvalue of $A$ if there exists a nonzero bounded linear operator $X$ such that $AX=\alpha XA$. In weighted Hardy spaces invariant under automorphisms, we completely compute the extended eigenvalues of composition operators induced by linear fractional self-mappings of the unit disk $\mathbb{D}$ with one fixed point in $\mathbb{D}$ and one outside $\overline{\mathbb{D}}$. Such classes of transformations include elliptic and loxodromic mappings as well as a hyperbolic nonautomorphic mapping.