Hille–Yosida theorem

Remarks

The necessity comes from the resolvent formula $(\lambda I - A)^{-1} x = \int_0^\infty e^{-\lambda t} T(t) x \, dt$, whose norm is at most $\norm{x} / \lambda$ when $\norm{T(t)} \le 1$. For the converse, one forms the Yosida approximations $A_\lambda = \lambda A (\lambda I - A)^{-1} = \lambda^2 (\lambda I - A)^{-1} - \lambda I$. They are bounded operators, the exponentials $e^{t A_\lambda}$ are contractions, and they converge strongly as $\lambda \to \infty$ to a semigroup with generator $A$. The general form concerns semigroups with $\norm{T(t)} \le M e^{\omega t}$ for constants $M \ge 1$ and $\omega \in \R$. The operator $A$ generates such a semigroup if and only if it is closed and densely defined, $(\omega, \infty)$ lies in its resolvent set, and $\norm{(\lambda I - A)^{-n}} \le M (\lambda - \omega)^{-n}$ for all $\lambda > \omega$ and $n \ge 1$. Example: on $X = \C$, the operator $A x = a x$ with $\operatorname{Re} a \le 0$ satisfies $\abs{(\lambda - a)^{-1}} \le 1/\lambda$ for $\lambda > 0$, and it generates the contractions $T(t) = e^{a t}$. If $a > 0$ is real, then $\lambda = a$ is not in the resolvent set, and indeed $e^{a t} > 1$ for $t > 0$, so no contraction semigroup has this generator. The Lumer–Phillips theorem replaces condition 2 by dissipativity and a range condition, which are often easier to check.

Going deeper

For a student

The theorem turns an abstract Cauchy problem $u' = A u$, $u(0) = u_0$, in a Banach space into a question about the resolvent of $A$. For a densely defined closed operator, the resolvent bound is checked by solving the stationary equation $\lambda u - A u = f$ with an a priori estimate $\lambda \norm{u} \le \norm{f}$. This is how one shows that the heat equation and many parabolic equations are well posed. The semigroup is then the abstract replacement for the matrix exponential $e^{tA}$ of a system of linear ordinary differential equations. The Lumer–Phillips theorem is the form used most often in Hilbert spaces.