Spectral theory is the study of the spectrum of a linear operator and the ways in which spectral
data determine the operator. For a bounded
operator 
on a complex Banach space, its spectrum
is
 |
In finite dimensions this is the set of eigenvalues, but an operator on an infinite-dimensional space may have spectral values which are not eigenvalues.
The spectral theorem represents a normal operator on a Hilbert space by a projection-valued measure. For a compact self-adjoint operator, this reduces to an orthogonal expansion in eigenvectors, closely paralleling the diagonalization of a Hermitian matrix.
