Discrete spectrum (mathematics)

In mathematics, specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank of the corresponding Riesz projector is finite.of a closed linear operatorin the Banach spaceis said to belong to discrete spectrumσif the following two conditions are satisfied:[1] Hereis the identity operator in the Banach spaceis a smooth simple closed counterclockwise-oriented curve bounding an open regionλis the only point of the spectrum of= { λ } .The discrete spectrumcoincides with the set of normal eigenvalues of: In general, the rank of the Riesz projector can be larger than the dimension of the root linealof the corresponding eigenvalue, and in particular it is possible to haveSo, there is the following inclusion: In particular, for a quasinilpotent operator one hasThe discrete spectrumis not to be confused with the point spectrum, which is defined as the set of eigenvalues ofWhile each point of the discrete spectrum belongs to the point spectrum, the converse is not necessarily true: the point spectrum does not necessarily consist of isolated points of the spectrum, as one can see from the example of the left shift operator,For this operator, the point spectrum is the unit disc of the complex plane, the spectrum is the closure of the unit disc, while the discrete spectrum is empty:
spectral theoryisolated pointsRiesz projectorspectrumBanach spaceidentity operatornormal eigenvaluesquasinilpotent operatorpoint spectrumeigenvaluesSpectrum (functional analysis)Decomposition of spectrum (functional analysis)Normal eigenvalueEssential spectrumSpectrum of an operatorResolvent formalismFredholm operatorOperator theoryFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraC*-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