Hille–Yosida theorem

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In functional analysis, the Hille–Yosida theorem characterizes one-parameter semigroups of linear operators on Banach spaces satisfying certain continuity restrictions. The theorem is useful for solving certain differential equations such as the heat equation. The theorem is named after the mathematicians Einar Hille and Kosaku Yosida who independently discovered the result around 1948.

A special case of the Hille-Yosida theorem is Stone's theorem characterizing strongly continuous one-parameter groups of operators on a Hilbert space.

1 Formal definitions
2 Statement of the theorem
3 Connection to the Laplace transform
4 References

[edit] Formal definitions

If E is a Banach space, a one-parameter semigroup of operators on E is a family of operators indexed on the non-negative real numbers {T_t} _{t ∈ [0,∞)} such that

$T_0= I \quad$
$T_{s+t}= T_s \circ T_t, \quad \forall t,s \geq 0$

The semigroup is said to be strongly continuous, also called a (C₀) semigroup, if and only if the mapping

$t \mapsto T_t \varphi$

is continuous for all φ ∈ E, where [0, ∞) has the usual topology and E has the norm topology.

Example. Consider the Banach space BUC[0, ∞) of bounded uniformly continuous complex-valued functions of the interval [0, ∞). Let

$(1) \quad [T_t \varphi](x) = \varphi(x+t), \quad x,t \in [0, \infty).$

Then {T_t} is a strongly continuous one parameter semigroup. In this case the operators T_t have norm at most 1. The strong continuity property follows from the fact that the functions in the space BUC[0, ∞) are uniformly continuous. In fact, the family of translation operators defined by (1) on the larger space BC[0, ∞) of bounded continuous complex-valued functions on [0, ∞) is a one-parameter semigroup but fails to be strongly continuous.

The infinitesimal generator of a one-parameter semigroup {T_t} _{t ∈ [0,∞)} is an operator A defined on a possibly proper subspace of E as follows:

The domain of A is the set of ψ ∈ E such that

$h^{-1}\bigg(T_h \psi - \psi\bigg)$

approaches a limit as h approaches 0.

The value of A ψ is the value of the above limit. In other words A ψ is the derivative at 0 of the function

$t \mapsto T_t \psi.$

Theorem. The infinitesimal generator of a strongly continuous one-parameter semigroup is a closed linear operator defined on a dense subspace of E.

The Hille-Yosida theorem provides a necessary and sufficient condition for a closed linear operator A on a Banach space to be the infinitesimal generator of a strongly continuous one-parameter semigroup.

[edit] Statement of the theorem

We first state the theorem for the special case of the contraction semigroup {T_t} _{t ∈ [0,∞)}, that is, the semigroup in which all operators T_t have norm at most 1.

Theorem. Let A be a (partially defined) operator on the Banach space E. A necessary and sufficient condition for A to be the infinitesimal generator of a strongly continuous contraction semigroup on E is that

A is densely defined;
For all λ > 0, the operator λ I − A has an everywhere defined inverse R(λ, A) such that

$\| \lambda \operatorname{R}(\lambda, A) \| \leq 1.$

To say λ I − A has an everywhere defined inverse means that the operator λ I − A is injective on the domain of A and that its range is all of E.

If A is the generator of a strongly continuous contraction semigroup, it is the generator of only one such semigroup. That unique semigroup is called the semigroup generated by A.

The operator-valued function R(λ, A) is called the resolvent of A. The resolvent is related to the semigroup {T_t} _{t ∈ [0,∞)} generated by A as follows:

Theorem. Suppose A generates {T_t} _{t ∈ [0,∞)}. Then

$\operatorname{R}(\lambda, A) \varphi = \int_0^\infty e^{-\lambda t} T_t \varphi \ dt, \quad \forall \varphi \in E.$

The above integral is defined as a Bochner integral.

[edit] Connection to the Laplace transform

By definitions of the infinitesimal generator and the semigroup we have that

$\frac{d}{dt} T_t\varphi=\lim_{h\to 0}\frac{T_{t+h}\varphi-T_t\varphi}{h}=AT_t \varphi$

If we formally do the Laplace transform of the semigroup

$\mathcal{L} T_t\varphi=\int_0^\infty e^{-\lambda t} T_t \varphi dt$

we get by integration by parts

$\mathcal{L} \frac{d}{dt} T_t \varphi = \lambda \mathcal{L} T_t\varphi- T_0\varphi$

When applied to the differential equation above we get

$\lambda \mathcal{L} T_t\varphi-\varphi=A\mathcal{L} T_t\varphi$

$\mathcal{L} T_t\varphi=(\lambda I-A)^{-1} \varphi=\operatorname{R}(\lambda,A)\varphi$

If A and $\operatorname{R}(\lambda,A)$ satisfy the conditions of Hille-Yosida theorem, then we can invert the Laplace transform and A generates the semigroup