Экспоненциалды функцияның жалпыламасы: C0-жартылай топтар

Generalization of the exponential function

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.

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.

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 .

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.

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.

A strongly continuous semigroup T is called eventually differentiable if there exists a t0 > 0 such that T(t0)X ⊂ D(A) (equivalently: T(t )X ⊂ D(A) for all t ≥ t0) 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 t1 ≥ 0 such that for all x ∈ X the solution u of the abstract Cauchy problem is differentiable on (t1, ∞). The semigroup is immediately differentiable if t1 can be chosen to be zero.

A strongly continuous semigroup T is called eventually compact if there exists a t0 > 0 such that T(t0) is a compact operator (equivalently if T(t ) is a compact operator for all t ≥ t0) The semigroup is called immediately compact if T(t ) is a compact operator for all t > 0.

A strongly continuous semigroup is called eventually norm continuous if there exists a t0 ≥ 0 such that the map t → T(t ) is continuous from (t0, ∞) to L(X). The semigroup is called immediately norm continuous if t0 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.