**The Gleason-Kahane-Żelazko theorem in function spaces**

Let T: H^{p} → H^{p} be a linear mapping (no continuity assumption). What can we say about T if we assume that “it preserves outer functions”? Similarly, under what conditions does it preserve the inner functions? Another related question is to consider linear functionals T: H^{p} → C (again, no continuity assumption) and ask about those functionals whose kernels do not include any outer function. We study such questions via an abstract result which can be interpreted as the generalized Gleason–Kahane-Żelazko theorem for modules. In particular, we see that continuity of endomorphisms and functionals is a part of the conclusion. We discuss similar questions in other function spaces, e.g., Bergman, Dirichlet, Besov, the little Bloch, and VMOA.

This is a joint work with T. Ransford.