C0-semigroup

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 (R₊, +) on some Banach space X that is continuous in the strong operator topology.

A strongly continuous semigroup on a Banach space ${\displaystyle X}$ is a map ${\displaystyle T:\mathbb {R} _{+}\to L(X)}$ (where ${\displaystyle L(X)}$ is the space of bounded operators on ${\displaystyle X}$) such that

The first two axioms are algebraic, and state that ${\displaystyle T}$ is a representation of the semigroup ${\displaystyle {(\mathbb {R} _{+},+)}}$; the last is topological, and states that the map ${\displaystyle T}$ is continuous in the strong operator topology.

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. The operator A is closed, although not necessarily bounded, and the domain is dense in X.

The strongly continuous semigroup T with generator A is often denoted by the symbol ${\displaystyle e^{At}}$ (or, equivalently, ${\displaystyle \exp(At)}$). 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).

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