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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.
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 ...
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)} ;
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.
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem .
A pair of graphs are said to be cospectral mates if they have the same spectrum, but are non-isomorphic. The smallest pair of cospectral mates is {K 1,4, C 4 ∪ K 1}, comprising the 5-vertex star and the graph union of the 4-vertex cycle and the single-vertex graph [1].
In mathematics, the essential spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible".
The spectrum in a rainbow. A spectrum (pl.: spectra or spectrums) [1] is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word spectrum was first used scientifically in optics to describe the rainbow of colors in visible light after passing through a prism.