Further Results
If T is a compact operator, then it can be shown that any nonzero λ in the spectrum is an eigenvalue. In other words, the spectrum of such an operator, which was defined as a generalization of the concept of eigenvalues, consists in this case only of the usual eigenvalues, and possibly 0.
If X is a Hilbert space and T is a normal operator, then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example).
Read more about this topic: Spectrum (functional Analysis)
Famous quotes containing the word results:
“... dependence upon material possessions inevitably results in the destruction of human character.”
—Agnes E. Meyer (1887–1970)
“Life and language are alike sacred. Homicide and verbicide—that is, violent treatment of a word with fatal results to its legitimate meaning, which is its life—are alike forbidden.”
—Oliver Wendell Holmes, Sr. (1809–1894)