LumerPhillipsTheorem

Lumer-Phillips Theorem

In functional analysis, the Lumer–Phillips theorem, named after Günter Lumer and Ralph Phillips, provides a necessary and sufficient condition for a linear operator in a Banach space to generate a contraction semigroup.

Statement of the theorem

Let $A$ be a linear operator defined on a linear subspace $D(A)$ of the Banach space $X$. Then $A$ generates a contraction semigroup if and only if:

1. $D(A)$ is dense in $X$,
2. $A$ is dissipative, and
3. $A - \lambda_0 I$ is surjective for some $\lambda_0 > 0$, where $I$ denotes the identity operator.

An operator satisfying the last two conditions is referred to as maximally dissipative.

Variants of the theorem

Reflexive spaces

In reflexive Banach spaces, the conditions that $D(A)$ is dense and that $A$ is closed are dropped, as they can be derived from the remaining two conditions.

Dissipativity of the adjoint

For a dense subspace $D(A)$ of the reflexive Banach space $X$, a linear operator $A$ generates a contraction semigroup if it is closed and both $A$ and its adjoint $A^*$ are dissipative.

Quasicontraction semigroups

An operator $A$ generates a quasi contraction semigroup if:

1. $D(A)$ is dense in $X$,
2. $A$ is closed,
3. $A$ is quasidissipative, i.e. there exists a $\omega \ge 0$ such that $A - \omega I$ is dissipative, and
4. $A - \lambda_0 I$ is surjective for some $\lambda_0 > \omega$, where $I$ denotes the identity operator.

Examples

Consider $H = L^2([0, 1]; R)$ with its usual inner product, and let $Au = u'$ with domain $D(A)$ equal to those functions $u$ in the Sobolev space $H^1([0, 1]; R)$ with $u(1) = 0$. $D(A)$ is dense. Moreover, for every $u$ in $D(A)$,

$$\langle u, A u \rangle = \int_0^1 u(x) u'(x) , dx = - \frac{1}{2} u(0)^2 \leq 0,$$

so that $A$ is dissipative. The ordinary differential equation $u' - \lambda u = f$, $u(1) = 0$ has a unique solution $u$ in $H^1([0, 1]; R)$ for any $f$ in $L^2([0, 1]; R)$, namely

$$u(x)={\rm e}^{\lambda x}\int_1^x {\rm e}^{-\lambda t}f(t),dt $$

so that the surjectivity condition is satisfied. Hence, by the reflexive version of the Lumer–Phillips theorem, $A$ generates a contraction semigroup.