Spectra of selfadjoint operators in constructive analysis

The approximate point spectrum sigma(a)(T) of a selfadjoint operator T on a nontrivial separable Hilbert space is examined constructively with the help of the functional calculus for T. In particular, it is proved that sigma(a)(T) is compact if and only if parallel to f(T)parallel to can be computed for each f in C[-b,b], where b > 0 is a bound for T. The paper culminates in a full constructive analysis of the spectrum of a compact selfadjoint operator with infinite-dimensional range. Brouwerian examples show that the results are the best possible in the constructive setting.