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:
“In the works of man, everything is as poor as its author; vision is confined, means are limited, scope is restricted, movements are labored, and results are humdrum.”
—Joseph De Maistre (1753–1821)
“Being a parent is unlike any previous job—the results of any one action are not clearly visible for a long time, if at all.”
—Anonymous Mother. As quoted in Between Generations by Ellen Galinsky, ch. 2 (1981)