Holomorphic continuations of \(C_ 0\)-groups on Banach spaces
The author proves that if \(iA\) is a generator of a \(C_ 0\)-group \(\{T(it), t\in {\mathbf R}\}\) in a Banach space \(E\) and if the spectral bound \(\sup\{\text{Re }\lambda,\lambda\in \sigma(A)\}\) of the operator \(A\) is finite, then there exist two closed \(T\)-invariant subspaces \(E_ 0,E_ 1\subset E\) with \(D(A)\subset E_ 0\oplus E_ 1\) so that \(A|_{E_ 1}\) (respectively, \(-A|_{E_ 0}\)) generates an analytic semigroup of angle \({\pi\over 2}\) in \(E_ 1\) (respectively, in \(E_ 0\)), with \(\sigma(A|_{E_ 1})= \sigma(A)\), and \(\sigma(A|_{E_ 0})=\emptyset\). The problem considered in the present paper is converse to the following result: if the \(C_ 0\)-semigroup \(\{T(s), s\in {\mathbf R}\}\) with the generator \(A\) can be extended analytically to a family of bounded operators \(\{T(z), \text{Re}(z)>0\}\) satisfying growth condition \(\sup_{\text{Re}(z)>0} e^{-\omega| z|}\| T(z)\|< \infty\), \(\omega> 0\), then \(\{T(it), t\in {\mathbf R}\}\) is the \(C_ 0\)- group with the generator \(iA\), where \(T(it)\) is the strong limit of \(T(s+ it)\) as \(s\to 0\) [see \textit{E. Hille} and \textit{S. Phillips}, Functional analysis and semigroups, Am. Math. Soc. Colloq. Publ., Vol. 31, Providence, R.I. (1957; Zbl 0078.100), section 17.9]. The second result of the paper concerns the growth conditions of the group \(\{T(it), t\in {\mathbf R}\}\) at infinity which are sufficient for triviality of the subspace \(E_ 0\).
scientific article; zbMATH DE number 611118