Conditions on the generator of a uniformly bounded \(C_0\)-semigroup
Suppose \(X\) is a complex Banach space with dual \(X^*\). Let \(A\) be a closed, densely defined operator on \(X\) with spectrum in the closed left half plane. The author proves that under the integral condition \[ \sup_{\delta> 0} \Biggl\{\delta \int^{\delta+ i\infty}_{\delta- i\infty}|\langle R^2(A, \lambda)x,y\rangle||d\lambda|\Biggr\}< \infty,\quad \forall x\in X,\;\forall y\in X^*, \] the operator \(A\) generates a uniformly bounded \(C_0\)-semigroup of operators. He also shows that this condition is not necessary in general Banach spaces, but for Hilbert spaces it is also necessary. The short and elegant proof is based on the Banach-Steinhaus theorem and the Miyadera-Feller-Phillips generation theorem. This result nicely complements the characterization of self-adjoint operators on a Hilbert space by \textit{J. A. van Casteren} [Pro. J. Math. 104, 241-255 (1983; Zbl 0501.47001)], see also Theorem 3.3 in the reviewer's paper, Collect. Math. 45, No. 3, 287-300 (1994; Zbl 0819.47018).
uniformly bounded \(C_0\)-semigroup of operators
Miyadera-Feller-Phillips generation theorem
scientific article; zbMATH DE number 1452678