8.1 Operators on Hilbert Spaces

In the previous chapter, we studied operators on Banach spaces and their spectra and we were particularly successful in describing the spectrum of a compact operator. However, in order to say more, one has to impose further restrictions on the operator and the underlying Banach space.

Given a Banach space E and a bounded operator T ∊ B(E), we have defined a transpose operator T' ∊ B(E*). If E is a Hilbert space, then there is a conjugate-linear isomorphism Δ : E → E*. Therefore the transpose of the operator may be thought of as an operator on E itself. This operator is called the adjoint of T. This adjoint operation introduces a further structure on B(E) and is the focus of our attention in this section. An abstraction of this structure leads us to the notion of a C*-algebra, which we will study through the remainder of the chapter.

Throughout this section, H and K will denote complex Hilbert spaces and we will use 〈·, ·〉 to denote the inner product in either space. As before, B(H,K) will denote the collection of bounded operators from H to K and B(H) will denote B(H,H).