*C*_{0}-semigroup

It has been suggested that Quasicontraction semigroup be merged into this article. (Discuss) Proposed since November 2023. |

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 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.

## Formal definition[edit]

A **strongly continuous semigroup** on a Banach space is a map
such that

- , (the identity operator on )
- , as .

The first two axioms are algebraic, and state that is a representation of the semigroup ; the last is topological, and states that the map is continuous in the strong operator topology.

## Infinitesimal generator[edit]

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; *D*(*A*) is a linear subspace and *A* is linear on this domain.^{[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 (or, equivalently, ). 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[edit]

A uniformly continuous semigroup is a strongly continuous semigroup *T* such that

holds. 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.^{[3]} If *X* 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 *A* is not bounded. In this case, does not need to converge.

## Examples[edit]

### Multiplication semigroup[edit]

Consider the Banach space endowed with the sup norm . Let be a continuous function with . The operator with domain is a closed densely defined operator and generates the multiplication semigroup where Multiplication operators can be viewed as the infinite dimensional generalisation of diagonal matrices and a lot of the properties of can be derived by properties of . For example is bounded on if and only if is bounded.^{[4]}

### Translation semigroup[edit]

Let be the space of bounded, uniformly continuous functions on endowed with the sup norm. The (left) translation semigroup is given by .

Its generator is the derivative with domain .^{[5]}

## Abstract Cauchy problems[edit]

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:

- a continuously differentiable function
*u*: [0, ∞) →*X*is called a**classical solution**of the Cauchy problem if*u*(*t*) ∈*D*(*A*) for all*t*> 0 and it satisfies the initial value problem, - a continuous function
*u*: [0, ∞) →*X*is called a**mild solution**of the Cauchy problem if

Any classical solution is a mild solution. A mild solution is a classical solution if and only if it is continuously differentiable.^{[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*. The following assertions are equivalent:

- for all
*x*∈*X*there exists a unique mild solution of the abstract Cauchy problem, - the operator
*A*generates a strongly continuous semigroup, - the resolvent set of
*A*is nonempty and for all*x*∈*D*(*A*) there exists a unique classical solution of the Cauchy problem.

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*.

## Generation theorems[edit]

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 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.

## Special classes of semigroups[edit]

### Uniformly continuous semigroups[edit]

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.

### Analytic semigroups[edit]

### Contraction semigroups[edit]

### Differentiable semigroups[edit]

A strongly continuous semigroup *T* is called **eventually differentiable** if 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.

### Compact semigroups[edit]

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^{[8]} 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.

### Norm continuous semigroups[edit]

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.^{[9]}

## Stability[edit]

### Exponential stability[edit]

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:^{[10]}

- There exist
*M*,*ω*>0 such that for all*t*≥ 0: - The growth bound is negative:
*ω*_{0}< 0, - The semigroup converges to zero in the uniform operator topology: ,
- There exists a
*t*_{0}> 0 such that , - There exists a
*t*_{1}> 0 such that the spectral radius of*T*(*t*_{1}) is strictly smaller than 1, - There exists a
*p*∈ [1, ∞) such that for all*x*∈*X*: , - For all
*p*∈ [1, ∞) and all*x*∈*X*:

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

**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:^{[11]} 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 **Gearhart-Pruss 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^{[12]} *s*(*A*) ≤ *ω*_{0}(*T* ). There are examples^{[13]} where *s*(*A*) < *ω*_{0}(*T* ). If *s*(*A*) = *ω*_{0}(*T* ), then *T* is said to satisfy the **spectral determined growth condition**. Eventually norm-continuous semigroups satisfy the spectral determined growth condition.^{[14]} This gives another equivalent characterization of exponential stability for these semigroups:

- An eventually norm-continuous semigroup is exponentially stable if and only if
*s*(*A*) < 0.

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.

### Strong stability[edit]

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 infinite-dimensional (it is true for *X* finite-dimensional).

The following sufficient condition for strong stability is called the **Arendt–Batty–Lyubich–Phong theorem**:^{[15]}^{[16]} Assume that

*T*is bounded: there exists a*M*≥ 1 such that ,*A*has not residual spectrum on the imaginary axis, and- The spectrum of
*A*located on the imaginary axis is countable.

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.

## See also[edit]

## Notes[edit]

**^**Partington (2004) page 23**^**Partington (2004) page 24**^**Pazy, A. (1983),*Semigroups of Linear Operators and Applications to Partial Differential Equations*, New York: Springer-Verlag, p. 2, ISBN 0-387-90845-5**^**Klaus-Jochen Engel (2006),*A short course on operator semigroups*(in German), New York, N.Y.: Springer, pp. 20ff, ISBN 0-387-36619-9**^**Klaus-Jochen Engel (2006),*A short course on operator semigroups*(in German), New York, N.Y.: Springer, p. 51, ISBN 0-387-36619-9**^**Arendt et al. Proposition 3.1.2**^**Arendt et al. Theorem 3.1.12**^**Engel and Nagel Lemma II.4.22**^**Engel and Nagel (diagram II.4.26)**^**Engel and Nagel Section V.1.b**^**Engel and Nagel Theorem V.1.11**^**Engel and Nagel Proposition IV2.2**^**Engel and Nagel Section IV.2.7, Luo et al. Example 3.6**^**Engel and Nagel Corollary 4.3.11**^**Arendt, Wolfgang; Batty, Charles (1988), "Tauberian theorems and stability of one-parameter semigroups",*Transactions of the American Mathematical Society*,**306**(2): 837–852, doi:10.1090/S0002-9947-1988-0933321-3**^**Lyubich, Yu; Phong, Vu Quoc (1988), "Asymptotic stability of linear differential equations in Banach spaces",*Studia Mathematica*,**88**(1): 37–42, doi:10.4064/sm-88-1-37-42

## References[edit]

- E Hille, R S Phillips:
*Functional Analysis and Semi-Groups*. American Mathematical Society, 1975. - R F Curtain, H J Zwart:
*An introduction to infinite dimensional linear systems theory*. Springer Verlag, 1995. - E.B. Davies:
*One-parameter semigroups*(L.M.S. monographs), Academic Press, 1980, ISBN 0-12-206280-9. - Engel, Klaus-Jochen; Nagel, Rainer (2000),
*One-parameter semigroups for linear evolution equations*, Springer - Arendt, Wolfgang; Batty, Charles; Hieber, Matthias; Neubrander, Frank (2001),
*Vector-valued Laplace Transforms and Cauchy Problems*, Birkhauser - Staffans, Olof (2005),
*Well-posed linear systems*, Cambridge University Press - Luo, Zheng-Hua; Guo, Bao-Zhu; Morgul, Omer (1999),
*Stability and Stabilization of Infinite Dimensional Systems with Applications*, Springer - Partington, Jonathan R. (2004),
*Linear operators and linear systems*, London Mathematical Society Student Texts, Cambridge University Press, ISBN 0-521-54619-2