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...

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

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...

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...

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...

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

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

follows. A bounded operator T : X → Y between Banach spaces X and Y is Fredholm if and only if it is invertible modulo compact operators, i.e., if there...

(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...

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...

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

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;...

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

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...

||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...

First notice that K is in L2(X, m), therefore T is compact. By the spectral properties of compact operators, any nonzero λ in σ(T) is an eigenvalue. But it...

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...

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...

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

not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the...

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) ...

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...

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

The compact disc (CD) is a digital optical disc data storage format that was co-developed by Philips and Sony to store and play digital audio recordings...

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

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

of an arbitrary square matrix does generalize to compact operators. Every compact operator on a complex Banach space has a nest of closed invariant subspaces...

compact operator acting on a Banach space of functions. Depending on the situation, the kernel is then variously referred to as the Fredholm operator...