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

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

$\backslash frac\{\backslash mathrm\; dx\}\{\backslash mathrm\; dt\}\; =\; f(x,t)\; ;~\; x(0)\; =\; x\_0\; ~,\backslash qquad\backslash rm(CP)$

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

$\backslash Gamma(t)\backslash ,x\_0\; =\; x(t)$ , 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

$\backslash lim\_\{t\backslash to0\}\; \backslash |\backslash Gamma(t)\backslash ,x\_0\; -\; x\_0\; \backslash |\; =\; 0$

Formal definition

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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

$A\backslash ,x\; =\; \backslash lim\_\{t\backslash to0\}\; \backslash frac1t\backslash ,(\backslash Gamma(t)-\; I)\backslash ,x$

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

$\backslash omega\; =\; \backslash lim\_\{t\backslash to0\}\; \backslash frac1t\; \backslash log\; \backslash |\; \backslash Gamma(t)\; \backslash |$ .

The semigroup is exponentially stable, i.e.

$\backslash exists\; K,a\; >\; 0,~\; \backslash forall\; t\backslash ge0:\; \backslash |\; \backslash Gamma(t)\; \backslash |\; \backslash le\; K\backslash ,e^\{-\; a\backslash ,t\}$

if and only if its growth bound is negative.

One has the following:

Theorem: A semigroup is exponentially stable if and only if for every $x\; \backslash in\; H$ there is $C\; >\; 0$ such that

.

See also

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• Schrödinger semigroups

References

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• 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.

Functional analysis