In mathematics, a C₀-semigroup is a continuous morphism from (R₊,+) into a topological monoid, usually L(H), the algebra of linear continuous operators on some Hilbert space H.

Thus, strictly speaking, not the C₀-semigroup, but rather its image, is a semigroup.

Example

C₀-semigroups occur for example in the context of initial value problems,

where x and f take values in a Hilbert space H.

If the solution of (CP) is unique (depending on f) for x₀ in some given domain D ⊂ H, one has the "solution operator" defined by

, where x(t) is solution of (CP).

Thus one can view Γ as an "evolution operator", and it is clear that one should have

Γ(s+t)=Γ(s) Γ(t)

on the domain D. This is just the condition of a semigroup-morphism.

Then one can study the conditions under which Γ is continuous for the topology on L(H) induced by the norm on H, which amounts to check that

Formal definition

All that follows concerns the following definition:

A (strongly continuous) C₀-semigroup on a Hilbert space H is a map

Γ : R₊ → L(H)

such that

1. Γ(0) = I := id_H ,   (identity operator on H)
2. ∀ t,s ≥ 0 : Γ(t+s) = Γ(t) Γ(s)
3. ∀ x₀ ∈ H : || Γ(t) x₀ - x₀ || → 0 , as t → 0 .

Infinitesimal generator

The infinitesimal generator A of a C₀-semigroup Γ is defined by

whenever the limit exists. The domain of A, D(A), is the set of x ∈ H for which this limit does exist.

Stability

The growth bound of a semigroup Γ (on a Hilbert space) is the constant

.

The semigroup is exponentially stable, i.e.

iff its growth bound is negative.

One has the following

Theorem: A semigroup is exponentially stable iff for every there is C < 0 such that

.

See also

• Schrödinger semigroups

References

• E Hille, R S Phillips: Functional Analysis and Semi-Groups. American Mathematical Society, 1957.
• R F Curtain, H J Zwart: An introduction to infinite dimensional linear systems theory. Springer Verlag, 1995.