are normed vector spaces, with the property that
are Banach, but the definition can be extended to more general spaces.
that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting.
is a Hilbert space, it is true that any compact operator is a limit of finite-rank operators,[1] so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the norm topology.
Whether this was true in general for Banach spaces (the approximation property) was an unsolved question for many years; in 1973 Per Enflo gave a counter-example, building on work by Alexander Grothendieck and Stefan Banach.
[2] The origin of the theory of compact operators is in the theory of integral equations, where integral operators supply concrete examples of such operators.
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.
between two topological vector spaces is said to be compact if there exists a neighborhood
Then the following statements are equivalent, and some of them are used as the principal definition by different authors[4] If in addition
is Banach, these statements are also equivalent to: If a linear operator is compact, then it is continuous.
denotes the space of compact operators
is the adjoint or transpose of T. A crucial property of compact operators is the Fredholm alternative, which asserts that the existence of solution of linear equations of the form
(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 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).
An important example of a compact operator is compact embedding of Sobolev spaces, 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.
[8] 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 in the 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 algebra, known as the Calkin algebra, is simple.
For Hilbert spaces, another equivalent definition of compact operators is given as follows.
are orthonormal sets (not necessarily complete), and
is a sequence of positive numbers with limit zero, called the singular values of the operator, and the series on the right hand side converges in the operator norm.
A bounded linear operator T : X → Y is called completely continuous if, for every weakly convergent sequence
Compact operators on a Banach space are always completely continuous.
If X is a reflexive Banach space, then every completely continuous operator T : X → Y is compact.
Somewhat confusingly, compact operators are sometimes referred to as "completely continuous" in older literature, even though they are not necessarily completely continuous by the definition of that phrase in modern terminology.