Spectral Theory

In this chapter we will use the theory developed so far to introduce the concept of eigenvalues and eigenvectors for linear operators on Hilbert spaces. From linear algebra we know how a self-adjoint matrix can be diagonalized; this means that there is an orthonormal basis {e_j } for the (finite dimensional) vector space such that the linear mapping f : C^k → C^k corresponding to the matrix can be expressed as f ( x )   =   ∑ j = 1 k λ j ( x ,   e j ) e j . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203755426/d9706a82-faf3-4225-933c-afa32800295e/content/eq432.tif"/>