In functional analysis, a branch of mathematics, a compact operator is a linear operator , where are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets of (subsets with compact closure in ). Such an operator is necessarily a bounded operator, and so continuous.[1] Some authors require that are Banach, but the definition can be extended to more general spaces.
Any bounded operator 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. When 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 Grothendieck and 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.
In functional analysis, a branch of mathematics, a compactoperator is a linear operator T : X → Y {\displaystyle T:X\to Y} , where X , Y {\displaystyle...
compactoperator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators...
In functional analysis, compactoperators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert...
contain them Compactoperator, a linear operator that takes bounded subsets to relatively compact subsets, in functional analysis Compact space, a topological...
mathematics, a symmetrizable compactoperator is a compactoperator on a Hilbert space that can be composed with a positive operator with trivial kernel to...
space. This ultimately led to the notion of a compactoperator as an offshoot of the general notion of a compact space. It was Maurice Fréchet who, in 1906...
holds for compactoperators on a Banach space. One restricts to compactoperators because every point x in the spectrum of a compactoperator T is an eigenvalue;...
follows. A bounded operator T : X → Y between Banach spaces X and Y is Fredholm if and only if it is invertible modulo compactoperators, i.e., if there...
of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compactoperators. In quantum mechanics...
||x||Y ≤ C||x||X for all x in X; and The embedding of X into Y is a compactoperator: any bounded set in X is totally bounded in Y, i.e. every sequence...
(A)} such that R ( z ; A ) {\displaystyle R(z;A)} is a compactoperator, we say that A has compact resolvent. The spectrum σ ( A ) {\displaystyle \sigma...
a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compactoperator is an eigenvalue. If V...
{\displaystyle \mathbf {M} .} Compactoperators on a Hilbert space are the closure of finite-rank operators in the uniform operator topology. The above series...
integral operator defines a compactoperator (convolution operators on non-compact groups are non-compact, since, in general, the spectrum of the operator of...
class of polynomially compactoperators (operators T {\displaystyle T} such that p ( T ) {\displaystyle p(T)} is a compactoperator for a suitably chosen...
of an arbitrary square matrix does generalize to compact operators. Every compactoperator on a complex Banach space has a nest of closed invariant subspaces...
bounded. This operator is in fact a compactoperator. The compactoperators form an important class of bounded operators. The Laplace operator Δ : H 2 ( R...
is a centred Gaussian measure, C is also a nuclear operator. In particular, it is a compactoperator of trace class, that is, it has finite trace. Even...
mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may...
compactoperator acting on a Banach space of functions. Depending on the situation, the kernel is then variously referred to as the Fredholm operator...
singular operators can be viewed as a generalization of compactoperators, as every compactoperator is strictly singular. These two classes share some important...
for compact self-adjoint operators is virtually the same as in the finite-dimensional case. Theorem — Suppose A is a compact self-adjoint operator on a...
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