In the field of functional analysis, Spectral Theory studies the spectrum of bounded linear operators, which is a generalisation of the set of eigenvalues of a matrix.

The spectrum of an operator on a finite-dimensional space , is just the set of eigenvalues, that is, a complex number  is in the spectrum of the operator , if  for some non-zero element  in , in other word, the operator  is not one-to-one. Eigenvalues and their corresponding eigenvectors are important for understanding linear transformations, and what it does to elements in the space. For example, if the matrix  represents a rotation in our three-dimensional space, then the eigenvectors are the vectors in the space that do not change direction under this transformation. Further, in applications, one is often required to solve system of linear equations , for some  in , and the eigenvalues of  provides criteria for when the equation is uniquely solvable.

However, an operator on infinite-dimensional space may have no eigenvalues. For example, consider the infinite-dimensional sequence space , and an operator  that acts on it by shifting every coordinate to the right:

This operator  has no eigenvalues, because if , for some complex number  then the only solution is for .

Hence, we are motivated to define the spectrum to generalised the notion of the eigenvalues. More precisely, the spectrum of the operator is defined as the set of complex numbers , such that  is either not one-to-one, or not onto.

Historically, the study of spectral theory originated with David Hilbert when he studied integral equations in the twentieth century, and he introduced the name “spectrum” without any connection to physics in mind. He was surprised to find later, application of this theory in quantum mechanics to explain features of the actual atomic spectra.

Enxi Lin
The University of New South Wales