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...

finite-rank operators in an infinite-dimensional setting. When Y {\displaystyle Y} is a Hilbert space, it is true that any compact operator is a limit...

In mathematics, a Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the...

Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A : H → H {\displaystyle A\colon H\to H} that acts on a...

are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian operator H ^ {\displaystyle {\hat {H}}} defined...

functional analysis, a normal operator on a complex Hilbert space H {\displaystyle H} is a continuous linear operator N : H → H {\displaystyle N\colon...

linear operators on a Banach space X. Let ( T n ) n ∈ N {\displaystyle (T_{n})_{n\in \mathbb {N} }} be a sequence of linear operators on the Banach space X...

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

properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite-dimensional Euclidean spaces. They were introduced...

"Banach space" and Banach in turn then coined the term "Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet...

be dubbed 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...

countable with the origin as limit point, and hence a compact operator (see compact operator on Hilbert space). Matrix norms Fan, Ky. (1951). "Maximum properties...

k\Vert _{L^{2}}.} Hilbert–Schmidt integral operators are both continuous and compact. The concept of a Hilbert–Schmidt integral operator may be extended...

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

linear operators on a complex Hilbert space with two additional properties: A is a topologically closed set in the norm topology of operators. A is closed...

general weakly compact in Hilbert spaces (consider the set consisting of an orthonormal basis in an infinite-dimensional Hilbert space which is closed...

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

analytic form of the Hilbert curve, however, is more complicated than Peano's. Let C {\displaystyle {\mathcal {C}}} denote the Cantor space 2 N {\displaystyle...

Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The spectral...

kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H {\displaystyle...

of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces). Let H {\displaystyle...

Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible bounded...

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...

Theorem. Let G be a locally compact group. If U is a strongly continuous unitary representation of G on a Hilbert space H, then π U ( f ) = ∫ G f ( g...

the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert...

paracompact. The group of unitary operators U ( H ) {\displaystyle \mathbb {U} ({\mathcal {H}})} on a separable Hilbert space H {\displaystyle {\mathcal {H}}}...

-1}(B-A)(B-zI)^{-1}\,.} When studying a closed unbounded operator A: H → H on a Hilbert space H, if there exists z ∈ ρ ( A ) {\displaystyle z\in \rho (A)}...

generalized to the trace class of compact operators on Hilbert spaces, and the analog of the Frobenius norm is called the Hilbert–Schmidt norm. If K {\displaystyle...

certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used...

arisen to cover this case, since the space is a Hilbert space: H k = W k , 2 . {\displaystyle H^{k}=W^{k,2}.} The space H k {\displaystyle H^{k}} can be defined...