ABSTRACT
Spectral functions and spectral measures on H? are also related in a one-to-one manner by Theorem. However, spectral functions on an infinite-dimensional Hilbert space may not be piecewise-constant. Generally the spectrum of A can be divided into two parts: the discrete part which consists of all points of discontinuous growth and the continuous part which consists of the rest. This chapter considers operators with a purely discrete spectrum. This is followed by a section on operators with a purely continuous spectrum. For operators with a continuous spectrum, e.g., the position and momentum operators, the situation is different. The integral expression does not reduce to a sum. The discussions on commuting selfadjoint operators need to be generalised in order to apply to infinite-dimensional Hilbert spaces due to the unbounded nature of some operators. It shows there are complications arising from the operators involved having different domains.
