Classes of Compact Operators

Abstract

In traditional Hilbert space quantum mechanics, the spectral representation of a (generally unbounded) selfadjoint operator plays a key role. In this chapter the main focus is on the class of compact operators and its two subsets, the trace class and the Hilbert-Schmidt class. This theory is not just an elementary introduction to the general case but is actually central e.g. in the study of the states of a quantum mechanical system. The chapter concludes with applications to tensor products of Hilbert spaces by introducing the notions of a partial trace and the Schmidt decomposition.