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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 ...
Let X be a Banach space and let : be a closed linear operator on X with dense domain ().There are several definitions of the essential spectrum, which are not equivalent. [1]The essential spectrum , is the set of all λ such that is not semi-Fredholm (an operator is semi-Fredholm if its range is closed and its kernel or its cokernel is finite-dimensional).
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.
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