In functional analysis, a branch of mathematics, a compact operator is a linear operator T : X → Y {\displaystyle T:X\to Y} , where X , Y {\displaystyle...

compact operator 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, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert...

space. This ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space. It was Maurice Fréchet who, in 1906...

Hilbert–Schmidt operator T : H → H is a compact operator. A bounded linear operator T : H → H is Hilbert–Schmidt if and only if the same is true of the operator | T...

(A)} such that R ( z ; A ) {\displaystyle R(z;A)} is a compact operator, we say that A has compact resolvent. The spectrum σ ( A ) {\displaystyle \sigma...

holds for compact operators on a Banach space. One restricts to compact operators because every point x in the spectrum of a compact operator T is an eigenvalue;...

Fredholm if and only if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator S : Y → X {\displaystyle S:Y\to X} such that...

of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators. In quantum mechanics...

{\displaystyle \mathbf {M} .} ⁠ Compact operators on a Hilbert space are the closure of finite-rank operators in the uniform operator topology. The above series...

||x||Y ≤ C||x||X for all x in X; and The embedding of X into Y is a compact operator: any bounded set in X is totally bounded in Y, i.e. every sequence...

functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues...

a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue. If V...

Lebesgue measure on [0, ∞). Compact operator on Hilbert space Unbounded operator Hermitian adjoint Normal operator Positive operator Helffer–Sjöstrand formula...

Hilbert–Schmidt operator, hence in particular is compact. V has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum...

mathematics, a symmetrizable compact operator is a compact operator on a Hilbert space that can be composed with a positive operator with trivial kernel to...

bounded. This operator is in fact a compact operator. The compact operators form an important class of bounded operators. The Laplace operator Δ : H 2 ( R...

contain them Compact operator, a linear operator that takes bounded subsets to relatively compact subsets, in functional analysis Compact space, a topological...

mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may...

T {\displaystyle T} is then a compact operator, and one has the canonical form for compact operators. Compact operators are trace class only if the series...

Compactness can refer to: Compact space, in topology Compact operator, in functional analysis Compactness theorem, in first-order logic Compactness measure...

reference to operators on a Hilbert space. C*-algebras are now an important tool in the theory of unitary representations of locally compact groups, and...

integral operator defines a compact operator (convolution operators on non-compact groups are non-compact, since, in general, the spectrum of the operator of...

singular operators can be viewed as a generalization of compact operators, as every compact operator is strictly singular. These two classes share some important...

{\displaystyle |T|:={\sqrt {(T^{*}T)}}} , using the operator square root. If T {\displaystyle T} is compact and H 1 , H 2 {\displaystyle H_{1},\,H_{2}} are...

operator on a Banach space has an invariant subspace. However, the upper-triangularization of an arbitrary square matrix does generalize to compact operators...

compact operators. The map K ↦ TK is injective. TK is a non-negative symmetric compact operator on L2[a,b]; moreover K(x, x) ≥ 0. To show compactness...

that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of...

called eventually compact if there exists a t0 > 0 such that T(t0) is a compact operator (equivalently if T(t ) is a compact operator for all t ≥ t0) ...

class of polynomially compact operators (operators T {\displaystyle T} such that p ( T ) {\displaystyle p(T)} is a compact operator for a suitably chosen...