Category

Operator theory

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inner product space that is metrically complete; a Banach space whose norm induces an inner product (follows the parallelogram identity)

Cauchy–Schwarz inequality

a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics

linear subspace

subset of a vector space that forms a vector space itself

differential operator

typically linear operator defined in terms of differentiation of functions

Cholesky decomposition

matrix decomposition

operator in physics

function acting on the space of physical states in physics

Hermitian adjoint

continuous dual of a Hermitian operator

unitary operator

surjective bounded operator on a Hilbert space preserving the inner product

compact operator

type of continuous linear operator

map that assigns a length or size to any operator in a function space

operator theory

mathematical study of linear operators on function spaces, such as differential operators and integral operators

von Neumann algebra

unital *-algebra of bounded operators on a Hilbert space closed in the weak operator topology

bounded operator

linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X

polar decomposition

representation of invertible matrices as unitaries multiplying a Hermitian operator

self-adjoint operator

densely defined operator on a Hilbert space whose domain coincides with that of its adjoint and which equals its adjoint; symmetric operator whose adjoint's domain equals its own domain

normal operator

(on a complex Hilbert space) continuous linear operator

Sturm–Liouville theory

theory of 2nd‐order linear ODEs that are eigenvalue equations of the operator 𝑤(𝑥)⁻¹((d∕d𝑥)𝑝(𝑥)d∕d𝑥+𝑞(𝑥))

space of holomorphic functions on the unit disk or upper half plane

Krylov subspace

operator algebra

branch of functional analysis

trace class operator

compact operator for which a finite trace can be defined

invariant subspace

in a vector space

in mathematics, the square root of an eigenvalue of a nonnegative self-adjoint operator

Hilbert–Schmidt operator

nuclear operator of order 2; a bounded operator A on a Hilbert space H such that tr(A*A) is finite

continuous linear operator

Function between topological vector spaces

Gelfand–Naimark theorem

mathematics theorem in functional analysis

positive element

self-adjoint (or Hermitian) element A of a C*-algebra A is called positive if its spectrum σ (A) consists of non-negative real numbers

Hausdorff locally convex space such that tensor product with any other Hausdorff locally convex space is uniquely defined (i.e. the projective tensor product is canonically isomorphic to the injective tensor product)

Finite-rank operator

linear operator in functional analysis

discrete Laplace operator

analog of the continuous Laplace operator

Partial isometry

linear map between Hilbert spaces

inequality for real numbers

unbounded operator

linear operator defined on a dense linear subspace

Positive-definite kernel

generalization of a positive-definite matrix

Von Neumann's inequality

Banach–Stone theorem

transfer operator

pushforward on the space of measurable functions

Min-max theorem

variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces

mathematical norm

mutually unbiased bases

a set of orthonormal bases, where each vector of a given basis has the same overlap with all the vectors of other bases

nuclear operator

operators on Banach spaces with properties similar to finite-dimensional operators

Toeplitz operator

compression of a multiplication operator on the circle to the Hardy space

Riesz–Thorin theorem

tensor product of Hilbert spaces

tensor product space endowed with a special inner product