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...

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...

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional)...

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

especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : H → H that commutes with its Hermitian...

relatively compact in C([a,b]) with the uniform norm and a fortiori in L2[a,b]. Now apply the spectral theorem for compact operators on Hilbert spaces to TK...

countable with the origin as limit point, and hence a compact operator (see compact operator on Hilbert space). The case p = 1 is often referred to as the nuclear...

operator Essential spectrum Spectral density Topologies on the set of operators on a Hilbert space norm topology ultrastrong topology strong operator...

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

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...

mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain Ω in n-dimensional Euclidean space Rn, then the...

algebras of operators on a separable Hilbert space, endowed with the operator norm topology. In the case of operators on a Hilbert space, the Hermitian...

of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces). Note that...

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

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...

*-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type...

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...

-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)}...

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...

"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...

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...

transpose). The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces H {\displaystyle H} . The definition has...

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...

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

context. In the above we can interpret A {\displaystyle A} as a compact operator on Hilbert spaces, and x {\displaystyle x} and b {\displaystyle b} as elements...

and therefore is a Banach space. If p = 2 {\displaystyle p=2} then ℓ 2 {\displaystyle \ell ^{2}} is also a Hilbert space when endowed with its canonical...

\alpha \geq 0.} Exactly the same argument shows that an operator T {\displaystyle T} on a Hilbert space H {\displaystyle H} is of rank 1 {\displaystyle 1}...

the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is...

that if the operator T {\displaystyle T} on a Hilbert space is polynomially compact (in other words p ( T ) {\displaystyle p(T)} is compact for some non-zero...