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. 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 that is continuous in the strong operator topology.

Formal definition

A strongly continuous semigroup on a Banach space is the space of bounded operators) such that

1. , (the identity operator)

2. .

The first two axioms are algebraic; the last is topological is continuous in the strong operator topology.

Infinitesimal generator

The infinitesimal generator is defined by

whenever the limit exists. The domain is a linear subspace is linear on this domain. is closed, although not necessarily bounded, and the domain is dense.

). 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

such that

holds. In this case, the infinitesimal is bounded and we have

and

Conversely, any bounded operator

is the infinitesimal generator of a uniformly continuous semigroup given by

1. .

Thus, a linear operator is a bounded linear operator. is a finite-dimensional does not need to converge.

Examples

Multiplication semigroup

endowed with the sup norm be a continuous function is a closed densely defined operator Multiplication operators can be viewed as the infinite dimensional generalisation of diagonal matrices is bounded.

Translation semigroup

be the space of bounded, uniformly continuous.

.

Abstract Cauchy problems

Consider the abstract Cauchy problem:

is a closed operator on a Banach space. There are two concepts of solution of this problem:

• a continuously differentiable is called a classical solution and it satisfies the initial value problem,

• 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.

The following theorem connects abstract Cauchy problems and strongly continuous semigroups.

Theorem:. The following assertions are equivalent:

1. there exists a unique mild solution of the abstract Cauchy problem,

2. generates a strongly continuous semigroup,

3. the resolvent set is nonempty there exists a unique classical solution of the Cauchy problem.

.

Generation theorems

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 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

is called uniformly continuous.

The generator of a uniformly continuous semigroup is a bounded operator.

Analytic semigroups

Contraction semigroups

A C0 is called a quasicontraction semigroup. It is called a contraction semigroup.

Differentiable semigroups

is called eventually differentiable is immediately differentiable.

Every analytic semigroup is immediately differentiable.

can be chosen to be zero.

Compact semigroups

is called eventually compact is a compact operator (equivalently) . The semigroup is called immediately compact.

Norm continuous semigroups

A strongly continuous semigroup is called eventually norm continuous. The semigroup is called immediately norm continuous can be chosen to be zero.

(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 is the constant

It is so called as this number is also the infimum of all real numbers with

.

The following are equivalent:

1. ,

2. The semigroup converges to zero in the uniform operator topology,

3. ,

4. such that the spectral radius is strictly smaller than 1,

5. ,

A semigroup that satisfies these equivalent conditions is called exponentially stable or uniformly stable conditions are equivalent to exponential stability is called the Datko-Pazy theorem.

is a Hilbert space there is another condition that is equivalent to exponential stability in terms of the resolvent operator of the generator: with positive real part belongs to the Hardy space. This is called the Gearhart-Pruss theorem.

The spectral bound of an operator A is the constant

1. ,

if the spectrum of A is empty.

The growth bound of a semigroup and the spectral bound of its generator are related by. There are examples is said to satisfy the spectral determined growth condition. Eventually norm-continuous semigroups satisfy the spectral determined growth condition. 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.

Strong stability

is called strongly stable or asymptotically stable.

finite-dimensional).

The following sufficient condition for strong stability is called the Arendt–Batty–Lyubich–Phong theorem: Assume that

1. ,

2. has not point spectrum on the imaginary axis, and

3. located on the imaginary axis is countable.

is strongly stable.

is reflexive has no eigenvalues is strongly stable.

See also

• Hille–Yosida theorem

• Lumer–Phillips theorem

• Trotter–Kato theorem

• Analytic semigroup

• Contraction (operator theory)

• Matrix exponential

• Strongly continuous family of operators

• Abstract differential equation

Notes

References