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:
“There is ... in every child a painstaking teacher, so skilful that he obtains identical results in all children in all parts of the world. The only language men ever speak perfectly is the one they learn in babyhood, when no one can teach them anything!”
—Maria Montessori (1870–1952)
“If family communication is good, parents can pick up the signs of stress in children and talk about it before it results in some crisis. If family communication is bad, not only will parents be insensitive to potential crises, but the poor communication will contribute to problems in the family.”
—Donald C. Medeiros (20th century)