Spectral theory

Spectrum (functional analysis)

In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number is said to be in the spectrum of a bounded linear operator if is not invertible, where is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the right shift operator R on the Hilbert space ℓ2, This has no eigenvalues, since if Rx=λx then by expanding this expression we see that x1=0, x2=0, etc. On the other hand, 0 is in the spectrum because the operator R − 0 (i.e. R itself) is not invertible: it is not surjective since any vector with non-zero first component is not in its range. In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum. The notion of spectrum extends to unbounded (i.e. not necessarily bounded) operators. A complex number λ is said to be in the spectrum of an unbounded operator defined on domain if there is no bounded inverse defined on the whole of If T is closed (which includes the case when T is bounded), boundedness of follows automatically from its existence. The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim. (Wikipedia).

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Hahn–Banach theorem | Inverse function | Riesz projector | Accumulation point | Unital algebra | Functional analysis | Vector space | Pseudospectrum | Self-adjoint operator | Subsequence | Spectral theorem | Bounded inverse theorem | Fredholm operator | Liouville's theorem (complex analysis) | Bounded operator | Banach space | Normal eigenvalue | Hydrogen atom | Decomposition of spectrum (functional analysis) | Ionization | Sequence | Banach algebra | Empty set | Complexification | Multiplication operator | Complex plane | Dimension (vector space) | Bounded set | Mathematics | Normal operator | Unbounded operator | Unit (ring theory) | Isometry | Discrete spectrum (mathematics) | Neumann series | Holomorphic function | Strictly singular operator | Resolvent set | Spectral radius | Compact operator | Eigenfunction | Hilbert space | Complex number | Lp space | Closed graph theorem | Hermitian adjoint | Matrix (mathematics) | Essential spectrum | Resolvent formalism | Closed set | Spectral theory