In mathematics, a C_{0}semigroup, also known as a strongly continuous oneparameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equations, strongly continuous semigroups provide solutions of linear constant coefficient ordinary differential equations in Banach spaces. Such differential equations in Banach spaces arise from e.g. delay differential equations and partial differential equations.
Formally, a strongly continuous semigroup is a representation of the semigroup (R_{+},+) on some Banach space X that is continuous in the strong operator topology. Thus, strictly speaking, a strongly continuous semigroup is not a semigroup, but rather a continuous representation of a very particular semigroup.
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A strongly continuous semigroup on a Banach space X is a map
such that
The first two axioms are algebraic, and state that T is a representation of the semigroup (R_{+},+); the last is topological, and states that the map T is continuous in the strong operator topology.
Let A be a bounded operator on the Banach space X, then
is a strongly continuous semigroup (it is even continuous in the uniform operator topology). Conversely^{[1]}, any uniformly continuous semigroup is necessarily of this form for some bounded linear operator A. In particular^{[2]}, if X is a finitedimensional Banach space, then any strongly continuous semigroup is necessarily of this form for some linear operator A.
The infinitesimal generator A of a strongly continuous semigroup T is defined by
whenever the limit exists. The domain of A, D(A), is the set of x∈X for which this limit does exist.
The strongly continuous semigroup T with generator A is often denoted by the symbol e^{At}. This notation is compatible with the notation for matrix exponentials, and for functions of an operator defined via the spectral theorem.
Consider the abstract Cauchy problem:
where A is a closed operator on a Banach space X and x∈X. There are two concepts of solution of this problem:
Any classical solution is a mild solution. A mild solution is a classical solution if and only if it is continuously differentiable^{[3]}.
The following theorem connects abstract Cauchy problems and strongly continuous semigroups.
Theorem^{[4]} Let A be a closed operator on a Banach space X. The following assertions are equivalent:
When these assertions hold, the solution of the Cauchy problem is given by u(t) = T(t)x with T the strongly continuous semigroup generated by A.
In connection with Cauchy problems, usually a linear operator A is given and the question is whether this is the generator of a strongly continuous semigroup. Theorems which answer this question are called generation theorems. A complete characterization of operators that generate strongly continuous semigroups is given by the HilleYosida theorem. Of more practical importance are however the much easier to verify conditions given by the LumerPhillips theorem.
The strongly continuous semigroup T is called uniformly continuous if the map t → T(t) is continuous from [0, ∞) to L(X).
The generator of a uniformly continuous semigroup is a bounded operator^{[5]}.
A strongly continuous semigroup T is called eventually differentiable there exists a t_{0} > 0 such that T(t_{0})X⊂D(A) (equivalently: T(t)X ⊂ D(A) for all t ≥ t_{0}) and T is immediately differentiable if T(t)X ⊂ D(A) for all t > 0.
Every analytic semigroup is immediately differentiable.
An equivalent characterization in terms of Cauchy problems is the following: The strongly continuous semigroup generated by A is eventually differentiable if and only if there exists a t_{1} ≥ 0 such that for all x ∈ X the solution u of the abstract Cauchy problem is differentiable on (t_{1}, ∞). The semigroup is immediately differentiable if t_{1} can be chosen to be zero.
A strongly continuous semigroup T is called eventually compact if there exists a t_{0} > 0 such that T(t_{0}) is a compact operator (equivalently^{[6]} if T(t) is a compact operator for all t ≥ t_{0}) . The semigroup is called immediately compact if T(t) is a compact operator for all t > 0.
A strongly continuous semigroup is called eventually norm continuous if there exists a t_{0} ≥ 0 such that the map t → T(t) is continuous from (t_{0}, ∞) to L(X). The semigroup is called immediately norm continuous if t_{0} can be chosen to be zero.
Note that for an immediately norm continuous semigroup the map t → T(t) may not be continuous in t = 0 (that would make the semigroup uniformly continuous).
Analytic semigroups, (eventually) differentiable semigroups and (eventually) compact semigroups are all eventually norm continuous^{[7]}.
The growth bound of a semigroup T is the constant
It is so called as this number is also the infimum of all real numbers ω such that there exists a constant M (≥ 1) with
for all t ≥ 0.
The following are equivalent^{[8]}:
A semigroup that satisfies these equivalent conditions is called exponentially stable or uniformly stable (either of the first three of the above statements is taken as the definition in certain parts of the literature). That the L^{p} conditions are equivalent to exponential stability is called the DatkoPazy theorem.
In case X is a Hilbert space there is another condition that is equivalent to exponential stability in terms of the resolvent operator of the generator^{[9]}: all λ with positive real part belong to the resolvent set of A and the resolvent operator is uniformly bounded on the right half plane, i.e. (λI − A)^{−1} belongs to the Hardy space . This is called the GearhartPruss theorem.
The spectral bound of an operator A is the constant
with the convention that s(A) = −∞ if the spectrum of A is empty.
The growth bound of a semigroup and the spectral bound of its generator are related by^{[10]}: s(A)≤ω_{0}(T). There are examples^{[11]} where s(A) < ω_{0}(T). If s(A) = ω_{0}(T), then T is said to satisfy the spectral determined growth condition. Eventually normcontinuous semigroups satisfy the spectral determined growth condition^{[12]}. This gives another equivalent characterization of exponential stability for these semigroups:
Note that eventually compact, eventually differentiable, analytic and uniformly continuous semigroups are eventually normcontinuous so that the spectral determined growth condition holds in particular for those semigroups.
A strongly continuous semigroup T is called strongly stable or asymptotically stable if for all x ∈ X: .
Exponential stability implies strong stability, but the converse is not generally true if X is infinitedimensional (it is true for X finitedimensional).
The following sufficient condition for strong stability is called the ArendtBattyLyubichPhong theorem^{[13]}: Assume that
Then T is strongly stable.
If X is reflexive then the conditions simplify: if T is bounded, A has no eigenvalues on the imaginary axis and the spectrum of A located on the imaginary axis is countable, then T is strongly stable.
