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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.
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The essential spectrum () is a subset of the spectrum () and its complement is called the discrete spectrum, so = ().If is self-adjoint, then, by definition, a number is in the discrete spectrum of if it is an isolated eigenvalue of finite multiplicity, meaning that the dimension of the space
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.
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].
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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A spectrum may be constructed out of a space. The suspension spectrum of a space , denoted is a spectrum = (the structure maps are the identity.) For example, the suspension spectrum of the 0-sphere is the sphere spectrum discussed above.