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In mathematics, an analytic semigroup is particular kind of strongly continuous semigroup. Analytic semigroups are used in the solution of partial differential equations; compared to strongly continuous semigroups, analytic semigroups provide better regularity of solutions to initial value problems, better results concerning perturbations of the infinitesimal generator, and a relationship between the type of the semigroup and the spectrum of the infinitesimal generator. ==Definition== Let Γ(''t'') = exp(''At'') be a strongly continuous one-parameter semigroup on a Banach space (''X'', ||·||) with infinitesimal generator ''A''. Γ is said to be an analytic semigroup if * for some 0 < ''θ'' < ''π'' ⁄ 2, the continuous linear operator exp(''At'') : ''X'' → ''X'' can be extended to ''t'' ∈ Δ''θ'', :: :and the usual semigroup conditions hold for ''s'', ''t'' ∈ Δ''θ'': exp(''A''0) = id, exp(''A''(''t'' + ''s'')) = exp(''At'')exp(''As''), and, for each ''x'' ∈ ''X'', exp(''At'')''x'' is continuous in ''t''; * and, for all ''t'' ∈ Δ''θ'' \ , exp(''At'') is analytic in ''t'' in the sense of the uniform operator topology. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Analytic semigroup」の詳細全文を読む スポンサード リンク
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