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Lumer–Phillips theorem
In mathematics, the Lumer–Phillips theorem, named after Günter Lumer and Ralph Phillips, is a result in the theory of strongly continuous semigroups that gives a necessary and sufficient condition for a linear operator in a Banach space to generate a contraction semigroup.
==Statement of the theorem==
Let ''A'' be a linear operator defined on a linear subspace ''D''(''A'') of the Banach space ''X''. Then ''A'' generates a contraction semigroup if and only if〔Engel and Nagel Theorem II.3.15, Arent et al. Theorem 3.4.5, Staffans Theorem 3.4.8〕
# ''D''(''A'') is dense in ''X'',
# ''A'' is closed,
# ''A'' is dissipative, and
# ''A'' − ''λ''0''I'' is surjective for some ''λ''0> 0, where ''I'' denotes the identity operator.
An operator satisfying the last two conditions is called maximally dissipative.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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