Spectrum (functional analysis)

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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. Specifically, a complex number λ {\displaystyle \lambda } is said to be in the spectrum of a bounded linear operator T {\displaystyle T} if T − λ I {\displaystyle T-\lambda I} either has no set-theoretic inverse; or the set-theoretic inverse is either unbounded or defined on a non-dense subset. Here, I {\displaystyle I} is the identity operator. By the closed graph theorem, λ {\displaystyle \lambda } is in the spectrum if and only if the bounded operator T − λ I : V →...

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