In mathematics, an analytic semigroup is particular kind of

strongly continuous semigroup In mathematics, a ''C''0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary diffe ...

. Analytic semigroups are used in the solution of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

s; compared to strongly continuous semigroups, analytic semigroups provide better regularity of solutions to

initial value problem In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or o ...

s, better results concerning perturbations of the infinitesimal generator, and a relationship between the type of the semigroup and the

spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...

of the infinitesimal generator.

Definition

Let Γ(''t'') = exp(''At'') be a strongly continuous one-parameter semigroup on a

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...

(''X'', , , ·, , ) with infinitesimal generator ''A''. Γ is said to be an analytic semigroup if * for some 0 < ''θ'' < π/ 2, the

continuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded line ...

exp(''At'') : ''X'' → ''X'' can be extended to ''t'' ∈ Δ_''θ'' , ::

\Delta_ = \ \cup \,

: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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

in ''t''; * and, for all ''t'' ∈ Δ_''θ'' \ , exp(''At'') is analytic in ''t'' in the sense of the

uniform operator topology In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space . Introduction Let (T_n)_ be a sequence of linear operators on the Banach spac ...

Characterization

The infinitesimal generators of analytic semigroups have the following characterization: A closed,

densely defined In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". ...

linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...

''A'' on a Banach space ''X'' is the generator of an analytic semigroup

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

there exists an ''ω'' ∈ R such that the half-plane Re(''λ'') > ''ω'' is contained in the

resolvent set In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism. Definitions L ...

of ''A'' and, moreover, there is a constant ''C'' such that :

\,  R_ (A) \,  \leq \frac

for Re(''λ'') > ''ω'' and where

R_\lambda(A)

is the resolvent of the operator ''A''. Such operators are called '' sectorial''. If this is the case, then the resolvent set actually contains a sector of the form :

\left\

for some ''δ'' > 0, and an analogous resolvent estimate holds in this sector. Moreover, the semigroup is represented by :

\exp (At) = \frac1 \int_ e^ ( \lambda \mathrm - A )^ \, \mathrm \lambda,

where ''γ'' is any curve from ''e''^−''iθ''∞ to ''e''^+''iθ''∞ such that ''γ'' lies entirely in the sector :

\big\,

with π/ 2 < ''θ'' < π/ 2 + ''δ''.

References

* {{cite book , last = Renardy , first = Michael , author2=Rogers, Robert C. , title = An introduction to partial differential equations , series = Texts in Applied Mathematics 13 , edition = Second , publisher = Springer-Verlag , location = New York , year = 2004 , pages = xiv+434 , isbn = 0-387-00444-0 , mr = 2028503 Functional analysis Partial differential equations Semigroup theory