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  2. Spectrum (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/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.

  3. Spectral theory - Wikipedia

    en.wikipedia.org/wiki/Spectral_theory

    The spectrum of T is the set of all complex numbers ζ such that R ζ fails to exist or is unbounded. Often the spectrum of T is denoted by σ(T). The function R ζ for all ζ in ρ(T) (that is, wherever R ζ exists as a bounded operator) is called the resolvent of T. The spectrum of T is therefore the complement of the resolvent set of T in ...

  4. Spectrum of a matrix - Wikipedia

    en.wikipedia.org/wiki/Spectrum_of_a_matrix

    In mathematics, the spectrum of a matrix is the set of its eigenvalues. [ 1 ] [ 2 ] [ 3 ] More generally, if T : V → V {\displaystyle T\colon V\to V} is a linear operator on any finite-dimensional vector space , its spectrum is the set of scalars λ {\displaystyle \lambda } such that T − λ I {\displaystyle T-\lambda I} is not invertible .

  5. Decomposition of spectrum (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/Decomposition_of_spectrum...

    The spectrum of T restricted to H ac is called the absolutely continuous spectrum of T, σ ac (T). The spectrum of T restricted to H sc is called its singular spectrum, σ sc (T). The set of eigenvalues of T is called the pure point spectrum of T, σ pp (T). The closure of the eigenvalues is the spectrum of T restricted to H pp.

  6. Spectral theorem - Wikipedia

    en.wikipedia.org/wiki/Spectral_theorem

    When this happens, we say that has "simple spectrum" in the sense of spectral multiplicity theory. That is, a bounded self-adjoint operator that admits a cyclic vector should be thought of as the infinite-dimensional generalization of a self-adjoint matrix with distinct eigenvalues (i.e., each eigenvalue has multiplicity one).

  7. Discrete spectrum (mathematics) - Wikipedia

    en.wikipedia.org/.../Discrete_spectrum_(Mathematics)

    A point in the spectrum of a closed linear operator: in the Banach space with domain is said to belong to discrete spectrum of if the following two conditions are satisfied: [1] λ {\displaystyle \lambda } is an isolated point in σ ( A ) {\displaystyle \sigma (A)} ;

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  9. Resolvent set - Wikipedia

    en.wikipedia.org/wiki/Resolvent_set

    The spectrum is the complement of the resolvent set σ ( L ) = C ∖ ρ ( L ) , {\displaystyle \sigma (L)=\mathbb {C} \setminus \rho (L),} and subject to a mutually singular spectral decomposition into the point spectrum (when condition 1 fails), the continuous spectrum (when condition 2 fails) and the residual spectrum (when condition 3 fails).