Hilbert space H, the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional...

limit of finite-rank operators, so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in...

Sturm–Liouville theory, Integral equations, Fredholm theory Compact operators, Isospectral operators, Completeness Spectral geometry Spectral graph theory List of functional...

spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as...

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

general operators on infinite-dimensional spaces often requires a genuinely different approach. For example, the spectral theory of compact operators on Banach...

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

operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are...

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...

this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general, operator algebras are non-commutative...

eigenfunctions which form an orthonormal basis follows from the spectral theorem for compact operators. Finally, note that ( L − z ) − 1 u = α u , L u = ( z +...

pseudodifferential operators. The spectral theory of unbounded self-adjoint operators is only marginally more difficult than for bounded operators. The spectrum of an...

greater detail the structure of the proof of Mercer's theorem, particularly how it relates to spectral theory of compact operators. The map K ↦ TK is injective...

broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space...

mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm...

such operators appear in scattering theory). The spectral theorem applies only to self-adjoint operators, and not in general to symmetric operators. Nevertheless...

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...

now known as Sturm–Liouville theory. In modern language, it is an application of the spectral theorem for compact operators due to David Hilbert. In his...

quantum information theory. Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras Conway 2000, p. 41. Hall 2013, p. 138...

functional calculus Spectral theory Compact operator Laplace transform Fredholm theory Liouville–Neumann series Decomposition of spectrum (functional...

Dmitri; Vassilevich, Dmitri (2011), Operators, Geometry and Quanta: Methods of Spectral Geometry in Quantum Field Theory, Theoretical and Mathematical Physics...

integral operators are both continuous and compact. The concept of a Hilbert–Schmidt integral operator may be extended to any locally compact Hausdorff...

summable spectral triple is a spectral triple (A, H, D) such that a.D for any a in A has a compact resolvent which belongs to the class of Lp+-operators for...

Ker(T) is contained in an eigenspace of C2, which is finite-dimensional (see spectral theory of compact operators). Therefore, Ker(T) is also finite-dimensional...

areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis, spectral analysis, and neuroscience....

pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial...

of the algebra of bounded operators on some Hilbert space. Measure algebra: A Banach algebra consisting of all Radon measures on some locally compact...

theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics. Another active area of...

Compactness may be shown by invoking equicontinuity, and more specifically the theorem of Arzelà-Ascoli. As an operator, it has a spectral theory that...

Mathematical operator-value measure of interest in quantum mechanics and functional analysis Spectral theory of compact operators Spectral theorem – Result...