In mathematical analysis, a C0-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 differential equations, strongly continuous semigroups provide solutions of linear constant coefficient ordinary differential equations in Banach spaces.
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.
The infinitesimal generator A of a strongly continuous semigroup T is defined by whenever the limit exists.
[1] The operator A is closed, although not necessarily bounded, and the domain is dense in X.
[2] The strongly continuous semigroup T with generator A is often denoted by the symbol
This notation is compatible with the notation for matrix exponentials, and for functions of an operator defined via functional calculus (for example, via the spectral theorem).
In this case, the infinitesimal generator A of T is bounded and we have and Conversely, any bounded operator is the infinitesimal generator of a uniformly continuous semigroup given by Thus, a linear operator A is the infinitesimal generator of a uniformly continuous semigroup if and only if A is a bounded linear operator.
is a closed densely defined operator and generates the multiplication semigroup
Multiplication operators can be viewed as the infinite dimensional generalisation of diagonal matrices and a lot of the properties of
be the space of bounded, uniformly continuous functions on
[5] Consider the abstract Cauchy problem: where A is a closed operator on a Banach space X and x∈X.
[6] The following theorem connects abstract Cauchy problems and strongly continuous semigroups.
Theorem:[7] Let A be a closed operator on a Banach space X.
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.
A complete characterization of operators that generate exponentially bounded strongly continuous semigroups is given by the Hille–Yosida theorem.
Of more practical importance are however the much easier to verify conditions given by the Lumer–Phillips theorem.
The generator of a uniformly continuous semigroup is a bounded operator.
A C0-semigroup Γ(t), t ≥ 0, is called a quasicontraction semigroup if there is a constant ω such that ||Γ(t)|| ≤ exp(ωt) for all t ≥ 0.
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 t1 ≥ 0 such that for all x ∈ X the solution u of the abstract Cauchy problem is differentiable on (t1, ∞).
The semigroup is called immediately norm continuous if t0 can be chosen to be zero.
[10] 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:[11] 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 Lp conditions are equivalent to exponential stability is called the Datko-Pazy 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:[12] 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
The spectral bound of an operator A is the constant with the convention that s(A) = −∞ if the spectrum of A is empty.
If s(A) = ω0(T ), then T is said to satisfy the spectral determined growth condition.
Eventually norm-continuous semigroups satisfy the spectral determined growth condition.
[15] This gives another equivalent characterization of exponential stability for these semigroups: Note that eventually compact, eventually differentiable, analytic and uniformly continuous semigroups are eventually norm-continuous so that the spectral determined growth condition holds in particular for those semigroups.
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.