Stability and spectra of \(C_ 0\)-semigroups
Since its publication \textit{L. Gearhart}'s [Trans. Am. Math. Soc. 236, 385-394 (1978; Zbl 0326.47038)] on the boundary of the spectrum of the generator of a contraction semigroup has not lost its fascination, perhaps because the deep reasons for its truth are not quite well understood. The present paper sheds new light on this topic. Let us describe at least one of the interesting results contained in it: Let A be the infinitesimal generator of the strongly continuous semigroup \(U=(U_ t)_{t\geq 0}\) on the Banach space X. Then the intersection \(D_{\infty}(A)=\cap_{n}D(A^ n)\) of the domains of \(A^ n\) is a Fréchet-space when equipped with its natural projective topology. Moreover \(D_{\infty}(A)\) is invariant under U. Denote by \(\sigma (U_ t,D_{\infty}(A))\) the spectrum in the sense of Allan, \(\rho (U_ t,D_{\infty}(A))={\mathbb{C}}\setminus \sigma (U_ t,D_{\infty}(A))\). Let \(\sigma\) (A) denote the ordinary spectrum of A, and set \(\rho (A)={\mathbb{C}}\setminus \sigma (A).\) Theorem: For fixed t the following assertions are equivalent: (1) \(\exp (tz)\in \rho (U_ t,D_{\infty}(A)).\) (2) \(z+2\pi i{\mathbb{Z}}/t\subset \rho (A)\) and \(k\to \| (z+2\pi ik/t- A)^{-1}\|\) is slowly increasing. (3) There exists m such that \(D(A^ m)\subset (\exp (tz)-U_ t)(X)\) and \(\{z+2\pi i{\mathbb{Z}}/t\}\cap (P\sigma (A))=0.\) (There is one further equivalence.) Various applications to stability theory are given and moreover interesting theorems e.g. of Slemrod or Schaefer are improved.
boundary of the spectrum of the generator of a contraction semigroup
scientific article; zbMATH DE number 4144778
scientific article