In mathematical analysis, a C<sub>0</sub>-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. 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 <math display="inline">(\mathbb R_{\ge 0}, +)</math> on some Banach space <math display="inline">X</math> that is continuous in the strong operator topology.
Formal definition
A strongly continuous semigroup on a Banach space <math>X</math> is a map
<math> T : \mathbb{R}_{\ge 0} \to L(X) </math> (where <math>L(X)</math> is the space of bounded operators on <math>X</math>)
such that
- <math> T(0) = I </math>, (the identity operator on <math>X</math>)
- <math>\forall t,s \ge 0 : \ T(t + s) = T(t) T(s)</math>
- <math>\forall x_0 \in X: \ \|T(t) x_0 - x_0\| \to 0</math>, as <math>t\downarrow 0</math>.
The first two axioms are algebraic, and state that <math>T</math> is a representation of the semigroup <math>{(\mathbb{R}_{\ge 0},+)}</math>; the last is topological, and states that the map <math>T</math> is continuous in the strong operator topology.
Infinitesimal generator
The infinitesimal generator <math display="inline">A</math> of a strongly continuous semigroup <math display="inline">T</math> is defined by
: <math> A\,x = \lim_{t\downarrow0} \frac{(T(t)- I)\,x}{t} </math>
whenever the limit exists. The domain of <math display="inline">A</math>, <math display="inline">D(A)</math>, is the set of <math display="inline">x \in X</math> for which this limit does exist; <math display="inline">D(A)</math> is a linear subspace and <math display="inline">A</math> is linear on this domain. The operator <math display="inline">A</math> is closed, although not necessarily bounded, and the domain is dense in <math display="inline">X</math>.
The strongly continuous semigroup <math display="inline">T</math> with generator <math display="inline">A</math> is often denoted by the symbol <math>e^{At}</math> (or, equivalently, <math>\exp(At)</math>). 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).
Uniformly continuous semigroup
A uniformly continuous semigroup is a strongly continuous semigroup <math display="inline">T</math> such that
:<math> \lim_{t \to 0^+} \| T(t) - I \| = 0 </math>
holds. In this case, the infinitesimal generator <math display="inline">A</math> of <math display="inline">T</math> is bounded and we have
:<math> \mathcal{D}(A)=X </math>
and
:<math> T(t) = e^{At}:=\sum_{k=0}^\infty\frac{A^k}{k!}t^k. </math>
Conversely, any bounded operator
:<math>A \colon X \to X</math>
is the infinitesimal generator of a uniformly continuous semigroup given by
:<math> T(t) := e^{At}</math>.
Thus, a linear operator <math display="inline">A</math> is the infinitesimal generator of a uniformly continuous semigroup if and only if <math display="inline">A</math> is a bounded linear operator. If <math display="inline">X</math> is a finite-dimensional Banach space, then any strongly continuous semigroup is a uniformly continuous semigroup. For a strongly continuous semigroup which is not a uniformly continuous semigroup the infinitesimal generator <math display="inline">A</math> is not bounded. In this case, <math>e^{At}</math> does not need to converge.
Examples
Multiplication semigroup
Consider the Banach space <math>C_0(\mathbb{R}):=\{f:\mathbb{R}\rightarrow \mathbb{C} \text{ continuous}:
\forall \epsilon >0 ~\exists c>0 \text{ such that } \vert f(x) \vert \leq \epsilon ~
\forall x\in \mathbb{R} \setminus [-c,c] \}</math> endowed with the sup norm <math>\Vert f\Vert := \text{sup}_{x\in \mathbb {R\vert f(x) \vert</math>. Let <math>q: \mathbb{R} \rightarrow \mathbb{C}</math> be a continuous function with <math>\text{sup}_{s\in \mathbb{R\text{Re}(q(s))<\infin</math>. The operator <math>M_qf:=q\cdot f</math> with domain <math>D(M_q):=\{f\in C_0(\mathbb{R}): q\cdot f \in C_0(\mathbb{R}) \}</math> is a closed densely defined operator and generates the multiplication semigroup <math>(T_q(t))_{t\geq 0}</math> where <math>T_q(t)f:= \mathrm{e}^{qt}f.</math> Multiplication operators can be viewed as the infinite dimensional generalisation of diagonal matrices and a lot of the properties of <math>M_q</math> can be derived by properties of <math>q</math>. For example <math>M_q</math> is bounded on <math>C_0(\mathbb{R)}</math> if and only if <math>q</math> is bounded.
Translation semigroup
Let <math>C_{ub}(\mathbb{R})</math> be the space of bounded, uniformly continuous functions on <math>\mathbb{R}</math> endowed with the sup norm. The (left) translation semigroup <math>(T_l(t))_{t\geq 0}</math> is given by <math>T_l(t)f(s):=f(s+t), \quad s,t\in \mathbb{R}</math>.
Its generator is the derivative <math>Af:=f'</math> with domain <math>D(A):=\{f\in C_{ub}(\mathbb{R}): f \text{ differentiable with }f'\in C_{ub}(\mathbb{R})\}</math>.
Abstract Cauchy problems
Consider the abstract Cauchy problem:
:<math>u'(t)=Au(t),\quad u(0)=x,</math>
where <math display="inline">A</math> is a closed operator on a Banach space <math display="inline">X</math> and <math display="inline">x \in X</math>. There are two concepts of solution of this problem:
- a continuously differentiable function <math display="inline">u : [0, \infty) \to X</math> is called a classical solution of the Cauchy problem if <math display="inline">u(t) \in D(A)</math> for all <math display="inline">t > 0</math> and it satisfies the initial value problem,
- a continuous function <math display="inline">u : [0, \infty) \to X</math> is called a mild solution of the Cauchy problem if
::<math>\int_0^t u(s)\,ds\in D(A)\text{ and }A \int_0^t u(s)\,ds=u(t)-x.</math>
Any classical solution is a mild solution. A mild solution is a classical solution if and only if it is continuously differentiable.
The following theorem connects abstract Cauchy problems and strongly continuous semigroups.
Theorem: Let <math display="inline">A</math> be a closed operator on a Banach space <math display="inline">X</math>. The following assertions are equivalent:
- for all <math display="inline">x \in X</math> there exists a unique mild solution of the abstract Cauchy problem,
- the operator <math display="inline">A</math> generates a strongly continuous semigroup,
- the resolvent set of <math display="inline">A</math> is nonempty and for all <math display="inline">x \in D(A)</math> there exists a unique classical solution of the Cauchy problem.
When these assertions hold, the solution of the Cauchy problem is given by <math display="inline">u(t) \in T(t)x</math> with <math display="inline">T</math> the strongly continuous semigroup generated by <math display="inline">A</math>.
Generation theorems
In connection with Cauchy problems, usually a linear operator <math display="inline">A</math> 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 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.
For families of unitary operators on a Hilbert space that are not just semigroups but groups, i.e. are defined for negative values of <math display="inline">t</math> as well, Stone's theorem on one-parameter unitary groups describes their generators as precisely the self-adjoint operators. This has applications in the time evolution of a quantum mechanical system.
Special classes of semigroups
Uniformly continuous semigroups
The strongly continuous semigroup <math display="inline">T</math> is called uniformly continuous if the map <math display="inline">t \to T(t)</math> is continuous from <math display="inline">[0,\infty)</math> to <math display="inline">L(X)</math>.
The generator of a uniformly continuous semigroup is a bounded operator.
Analytic semigroups
Contraction semigroups
A C<sub>0</sub>-semigroup <math display="inline">(T(t))_{t \ge 0}</math> is called a quasicontraction semigroup if there is a constant <math display="inline">\omega</math> such that <math display="inline">\|T(t)\| \le \exp(\omega t)</math> for all <math display="inline">t \ge 0</math>. It is called a contraction semigroup if <math display="inline">\|T(t)\| \le 1</math> for all <math display="inline">t \ge 0</math>.
Differentiable semigroups
A strongly continuous semigroup <math display="inline">T</math> is called eventually differentiable if there exists a <math display="inline">t_0 > 0</math> such that <math display="inline">T(t_0)X \subset D(A)</math> (equivalently: <math display="inline">T(t)X \subset D(A)</math> for all <math display="inline">t \ge t_0</math>) and <math display="inline">T</math> is immediately differentiable if <math display="inline">T(t)X \subset D(A)</math> for all <math display="inline">t > 0</math>.
Every analytic semigroup is immediately differentiable.
An equivalent characterization in terms of Cauchy problems is the following: the strongly continuous semigroup generated by <math display="inline">A</math> is eventually differentiable if and only if there exists a <math display="inline">t_1 \ge 0</math> such that for all <math display="inline">x \in X</math> the solution <math display="inline">u</math> of the abstract Cauchy problem is differentiable on <math display="inline">(t_1, \infty)</math>. The semigroup is immediately differentiable if <math display="inline">t_1</math> can be chosen to be zero.
Compact semigroups
A strongly continuous semigroup <math display="inline">T</math> is called eventually compact if there exists a <math display="inline">t_0 > 0</math> such that <math display="inline">T(t_0)</math> is a compact operator (equivalently if <math display="inline">T(t)</math> is a compact operator for all <math display="inline">t \ge t_0</math>) . The semigroup is called immediately compact if <math display="inline">T(t)</math> is a compact operator for all <math display="inline">t > 0</math>.
Norm continuous semigroups
A strongly continuous semigroup is called eventually norm continuous if there exists a <math display="inline">t_0 \ge 0</math> such that the map <math display="inline">t \mapsto T(t)</math> is continuous from <math display="inline">(t_0, \infty)</math> to <math display="inline">L(X)</math>. The semigroup is called immediately norm continuous if <math display="inline">t_0</math> can be chosen to be zero.
Note that for an immediately norm continuous semigroup the map <math display="inline">t \mapsto T(t)</math> may not be continuous in <math display="inline">t=0</math> (that would make the semigroup uniformly continuous).
Analytic semigroups, (eventually) differentiable semigroups and (eventually) compact semigroups are all eventually norm continuous.
Stability
Exponential stability
The growth bound of a semigroup <math display="inline">T</math> is the constant
: <math>\omega_0 = \inf_{t>0} \frac{1}{t} \log \| T(t) \|. </math>
It is so called as this number is also the infimum of all real numbers <math display="inline">\omega</math> such that there exists a constant <math display="inline">M > 0</math> with
: <math>\|T(t)\| \leq Me^{\omega t}</math>
for all <math display="inline">t \ge 0</math>.
The following are equivalent:
- There exist <math display="inline">M, \omega > 0</math> such that for all <math display="inline">t \ge 0</math>: <math>\|T(t)\|\leq M{\rm e}^{-\omega t},</math>
- The growth bound is negative: <math display="inline">\omega_0 < 0</math>,
- The semigroup converges to zero in the uniform operator topology: <math>\lim_{t\to\infty}\|T(t)\|=0</math>,
- There exists a <math display="inline">t_0 > 0</math> such that <math>\|T(t_0)\|<1</math>,
- There exists a <math display="inline">t_1 > 0</math> such that the spectral radius of <math display="inline">T(t_1)</math> is strictly smaller than 1,
- There exists a <math display="inline">p \in [1, \infty)</math> such that for all <math display="inline">x \in X</math>: <math>\int_0^\infty\|T(t)x\|^p\,dt<\infty</math>,
- For all <math display="inline">p \in [1, \infty)</math> and all <math display="inline">x \in X</math>: <math>\int_0^\infty\|T(t)x\|^p\,dt<\infty.</math>
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 <math display="inline">L^p</math> conditions are equivalent to exponential stability is called the Datko-Pazy theorem.
In case <math display="inline">X</math> is a Hilbert space there is another condition that is equivalent to exponential stability in terms of the resolvent operator of the generator: all <math display="inline">\lambda</math> with positive real part belong to the resolvent set of <math display="inline">A</math> and the resolvent operator is uniformly bounded on the right half plane, i.e. <math display="inline">(\lambda I - A)^{-1}</math> belongs to the Hardy space <math>H^\infty(\mathbb{C}_+;L(X))</math>. This is called the Gearhart-Pruss theorem.
The spectral bound of an operator A is the constant
:<math>s(A) := \sup\