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| In [[mathematics]], an '''analytic semigroup''' is particular kind of [[C0 semigroup|strongly continuous semigroup]]. Analytic semigroups are used in the solution of [[partial differential equations]]; compared to strongly continuous semigroups, analytic semigroups provide better [[smooth function|regularity]] of solutions to initial value problems, better results concerning perturbations of the [[C0 semigroup#Infinitesimal generator|infinitesimal generator]], and a relationship between the type of the semigroup and the [[spectrum (functional analysis)|spectrum]] of the infinitesimal generator.
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| ==Definition==
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| 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
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| * for some 0 < ''θ'' < ''π'' ⁄ 2, the [[continuous linear operator]] exp(''At'') : ''X'' → ''X'' can be extended to ''t'' ∈ Δ<sub>''θ''</sub>,
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| ::<math>\Delta_{\theta} = \{ 0 \} \cup \{ t \in \mathbb{C} : | \mathrm{arg}(t) | < \theta \},</math>
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| :and the usual semigroup conditions hold for ''s'', ''t'' ∈ Δ<sub>''θ''</sub>: exp(''A''0) = id, exp(''A''(''t'' + ''s'')) = exp(''At'')exp(''As''), and, for each ''x'' ∈ ''X'', exp(''At'')''x'' is [[continuous function|continuous]] in ''t'';
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| * and, for all ''t'' ∈ Δ<sub>''θ''</sub> \ {0}, exp(''At'') is [[analytic function|analytic]] in ''t'' in the sense of the [[uniform topology|uniform operator topology]].
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| ==Characterization== | |
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| The infinitesimal generators of analytic semigroups have the following characterization:
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| A [[closed operator|closed]], [[dense set|densely-defined]] [[linear operator]] ''A'' on a Banach space ''X'' is the generator of an analytic semigroup [[if and only if]] there exists an ''ω'' ∈ '''R''' such that the [[half-plane]] Re(''λ'') > ''ω'' is contained in the [[resolvent formalism|resolvent set]] of ''A'' and, moreover, there is a constant ''C'' such that
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| :<math>\| R_{\lambda} (A) \| \leq \frac{C}{| \lambda - \omega |}</math> | |
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| for Re(''λ'') > ''ω'' and where <math>R_\lambda(A)</math> is the [[Resolvent_formalism|resolvent]] of the operator ''A''. If this is the case, then the resolvent set actually contains a sector of the form
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| :<math>\left\{ \lambda \in \mathbf{C} : | \mathrm{arg} (\lambda - \omega) | < \frac{\pi}{2} + \delta \right\}</math>
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| for some ''δ'' > 0, and an analogous resolvent estimate holds in this sector. Moreover, the semigroup is represented by
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| :<math>\exp (At) = \frac1{2 \pi i} \int_{\gamma} e^{\lambda t} ( \lambda \mathrm{id} - A )^{-1} \, \mathrm{d} \lambda,</math> | |
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| where ''γ'' is any curve from ''e''<sup>−''iθ''</sup>∞ to ''e''<sup>+''iθ''</sup>∞ such that ''γ'' lies entirely in the sector
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| :<math>\big\{ \lambda \in \mathbf{C} : | \mathrm{arg} (\lambda - \omega) | \leq \theta \big\},</math>
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| with ''π'' ⁄ 2 < ''θ'' < ''π'' ⁄ 2 + ''δ''.
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| ==References== | |
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| * {{cite book
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| | last = Renardy
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| | first = Michael
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| | coauthors = Rogers, Robert C.
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| | title = An introduction to partial differential equations
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| | series = Texts in Applied Mathematics 13
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| | edition = Second edition
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| | publisher = Springer-Verlag
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| | location = New York
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| | year = 2004
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| | pages = xiv+434
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| | isbn = 0-387-00444-0
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| | mr = 2028503
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| }}
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| [[Category:Functional analysis]]
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| [[Category:Partial differential equations]]
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| [[Category:Semigroup theory]]
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