In functional analysis, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator, and so continuous. Any L that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalisation of the class of finite rank operators in an infinite-dimensional setting. When X = Y and is a Hilbert space, it is true that any compact operator is a limit of finite rank operators, so that the class of compact operators can be defined alternatively as the closure in the operator norm of the finite rank operators. Whether this was true in general for Banach spaces (the approximation property) was an unsolved question for many years; in the end Enflo gave a counter-example.
The origin of the theory of compact operators is in the theory of integral equations. A typical Fredholm integral equation gives rise to a compact operator K on function spaces; the compactness property is shown by equicontinuity. The method of approximation by finite rank operators is basic in the numerical solution of such equations. The abstract idea of Fredholm operator is derived from this connection.
The spectral theory for compact operators in the abstract was worked out by Frigyes Riesz (published 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 a 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 eigenvalues of K with finite multiplicities (so that K − λI has a finite-dimensional kernel for all complex λ ≠ 0).
The compact operators form a two-sided ideal in the set of all operators between two Banach spaces. Indeed, the compact operators on a Hilbert space form a maximal ideal, so the quotient algebra, known as the Calkin algebra, is simple.
Examples of compact operators include Hilbert-Schmidt operators, or more generally, operators in the Schmidt class .
