Multiplicity Theorie

Abstract

In this chapter we collect the necessary facts centered around the concept of multiplicity which is connected with the simple process of “doubling” A × A ∈ ℒ(ℋ × ℋ) of an operator A ∈ ℒ(ℋ). We are mainly interested in selfadjoint operators but it is quite natural to deal with v. Neumann algebras at first. Multiplicity theory is one of the decisive foundations for an understanding of the unitary invariants of selfadjoint operators. The other foundation is the spectral theory to be presented in the next chapter. Multiplicity theory is decivise for the transition from the spectral theorem (see Chapter 3) to the spectral representation of a selfadjoint operator, that is, to a representation where the unitary invariants of the operator become evident. In principle multiplicity theory is independent of spectral theory (but usually technical tools of spectral theory appear in certain proofs of multiplicity theory).