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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. 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.

5. 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 .

6. Discrete spectrum (mathematics) - Wikipedia

   en.wikipedia.org/.../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.

7. 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).