Editing Compact operator (section)

==Origins in integral equation theory==

A crucial property of compact operators is the [[Fredholm alternative]], which asserts that the existence of solution of linear equations of the form

<math>(\lambda K + I)u = f </math>

(where ''K'' is a compact operator, ''f'' is a given function, and ''u'' is the unknown function to be solved for) behaves much like as in finite dimensions. The [[spectral theory of compact operators]] then follows, and it is due to [[Frigyes Riesz]] (1918). It shows that a compact operator ''K'' on an infinite-dimensional Banach space has spectrum that is either a finite subset of '''C''' which includes 0, or the spectrum is a [[Countable set|countably infinite]] subset of '''C''' which has 0 as its only [[limit point]]. Moreover, in either case the non-zero elements of the spectrum are [[eigenvalue]]s of ''K'' with finite multiplicities (so that ''K'' − λ''I'' has a finite-dimensional [[kernel (algebra)#Linear operators|kernel]] for all complex λ ≠ 0).

An important example of a compact operator is [[compact embedding]] of [[Sobolev space]]s, which, along with the [[Gårding inequality]] and the [[Lax–Milgram theorem]], can be used to convert an [[elliptic boundary value problem]] into a Fredholm integral equation.<ref name="mclean">William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000</ref> Existence of the solution and spectral properties then follow from the theory of compact operators; in particular, an elliptic boundary value problem on a bounded domain has infinitely many isolated eigenvalues. One consequence is that a solid body can vibrate only at isolated frequencies, given by the eigenvalues, and arbitrarily high vibration frequencies always exist.

The compact operators from a Banach space to itself form a two-sided [[ideal (ring theory)|ideal]] in the [[algebra over a field|algebra]] of all bounded operators on the space. Indeed, the compact operators on an infinite-dimensional separable Hilbert space form a maximal ideal, so the [[quotient associative algebra|quotient algebra]], known as the [[Calkin algebra]], is [[simple algebra|simple]].  More generally, the compact operators form an [[operator ideal]].