Continuity of spectral functions and the lakes of Wada (Q2266421)

The functions \(\sigma\), mapping a Hilbert space operator T into its spectrum \(\sigma\) (T), or \(\sigma_ e\) (defined by \(\sigma_ e(T)\) \(=\) essential spectrum of T), or \(\rho^ h_{s-F}(T)\) mapping T into the set of complex numbers \(\lambda\) such that \(\lambda\)-T is semi-Fredholm of index h, etc., have a very erratic behavior. They are continuous on a dense set of operators and discontinuous on another dense set of operators. It is shown that all of them are simultaneously continuous on a certain dense subset and simultaneously discontinuous on another dense subset.