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.