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:
“I have no doubt that it was a principle they fought for, as much as our ancestors, and not to avoid a three-penny tax on their tea; and the results of this battle will be as important and memorable to those whom it concerns as those of the battle of Bunker Hill, at least.”
—Henry David Thoreau (1817–1862)
“Pain itself can be pleasurable accidentally in so far as it is accompanied by wonder, as in stage-plays; or in so far as it recalls a beloved object to one’s memory, and makes one feel one’s love for the thing, whose absence gives us pain. Consequently, since love is pleasant, both pain and whatever else results from love, in so far as they remind us of our love, are pleasant.”
—Thomas Aquinas (c. 1225–1274)