C0-semigroup

DISPLAYTITLE:C0-semigroup In mathematics, 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. Thus, strictly speaking, a strongly continuous semigroup is not a semigroup, but rather a continuous representation of a very particular semigroup. 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. 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 (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). 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.