Hilbert–Schmidt operator

In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operatorthat acts on a Hilbert spaceand has finite Hilbert–Schmidt normdef{\displaystyle \|A\|_{\operatorname {HS} }^{2}\ {\stackrel {\text{def}}{=}}\ \sum _{i\in I}\|Ae_{i}\|_{H}^{2},}is an orthonormal basis.[1][2] The index setHowever, the sum on the right must contain at most countably many non-zero terms, to have meaning.[3] This definition is independent of the choice of the orthonormal basis.In finite-dimensional Euclidean space, the Hilbert–Schmidt normis identical to the Frobenius norm.The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis.As for any bounded operator,in the first formula, obtainAn important class of examples is provided by Hilbert–Schmidt integral operators.Every bounded operator with a finite-dimensional range (these are called operators of finite rank) is a Hilbert–Schmidt operator.The identity operator on a Hilbert space is a Hilbert–Schmidt operator if and only if the Hilbert space is finite-dimensional., which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator; moreover, for any bounded linear operatorTr ⁡{\displaystyle \operatorname {Tr} \left(A\left(x\otimes y\right)\right)=\left\langle Ax,y\right\rangle }is a bounded compact operator with eigenvalues, where each eigenvalue is repeated as often as its multiplicity, then, in which case the Hilbert–Schmidt norm ofis a measure space, then the integral operator[5] The product of two Hilbert–Schmidt operators has finite trace-class norm; therefore, if A and B are two Hilbert–Schmidt operators, the Hilbert–Schmidt inner product can be defined as{\displaystyle \langle A,B\rangle _{\text{HS}}=\operatorname {Tr} (A^{*}B)=\sum _{i}\langle Ae_{i},Be_{i}\rangle .}The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on H. They also form a Hilbert space, denoted by BHS(H) or B2(H), which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaceswhere H∗ is the dual space of H. The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space).[4] The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with the Hilbert–Schmidt norm).[4] The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, H is finite-dimensional.
mathematicsDavid HilbertErhard Schmidtbounded operatorHilbert spaceorthonormal basisEuclidean spaceHilbert–Schmidt integral operatorsidentity operatortrace-class norm*-idealBanach algebranaturallytensor product of Hilbert spacesdual spacenorm topologycompact operatornuclear operatorscompact operatorsnuclear operatortrace class operatorSchatten normFrobenius inner productSazonov's theoremTrace classEncyclopedia of MathematicsEMS PressConway, John B.Schaefer, Helmut H.Hilbert spacesAdjointInner productL-semi-inner productPrehilbert spaceOrthogonal complementBessel's inequalityCauchy–Schwarz inequalityRiesz representationHilbert projection theoremParseval's identityPolarization identityCompact operator on Hilbert spaceDensely definedNormalSelf-adjointSesquilinear formUnitaryCn(K) with K compact & n<∞Segal–Bargmann FTopological tensor productsnuclear spacesAuxiliary normed spacesNuclear spaceTensor productTopological tensor productof Hilbert spacesTopologiesInductive tensor productInjective tensor productProjective tensor productFredholm determinantFredholm kernelHypocontinuityIntegralNuclearbetween Banach spacesGrothendieck trace theoremSchwartz kernel theoremFunctional analysistopicsglossaryBanachFréchetHilbertHölderOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluBoundedCompactTransposeUnboundedC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theory