Fredholm composition operators on Riemann surfaces (Q2581217)

The author considers a class of maps between function spaces defined on (not necessarily compact) Riemann surfaces. Let \(M,N\) be the Riemann surfaces in question and let \(\rho:M \rightarrow N\) be an analytic map. Let \(u\) be a bounded measurable function on \(M\). Let \(\Lambda_2^{1}(M)\) be the space of measurable 1-forms on \(M\) with finite square norm. Then the author defines two maps, \(C_\rho:\Lambda_2^{1}(N)\rightarrow \Lambda_2^{1}(M)\) and \(M_u:\Lambda_2^{1}(M)\rightarrow \Lambda_2^{1}(M)\). The first is the natural induced map, the second multiplication. The author shows that the following three statements are equivalent: 1. \(M_uC_\rho\) is Fredholm. 2. \(M_uC_\rho\) is invertible. 3. \(M_u\) and \(C_\rho\) are both invertible. This is proved by a closer analysis of the two maps \(M_u\) and \(C_\rho\) separately.