\(C_ 0\)-semigroups on a locally convex space
Let X be a sequentially complete locally convex space with topology induced by a family \(\Gamma\) (called a calibration for X) of seminorms. For a linear operator T on X define \(\tilde p(\)T):\(=\sup \{p(Tx)\); \(p(x)\leq 1\}\), \(p\in \Gamma\), and \(\| T\|_{\Gamma}:=\sup \{\tilde p(T)\); \(p\in \Gamma \}\). The set \(B_{\Gamma}(X)\) of all T such that \(\| T\|_{\Gamma}<\infty\) is a Banach algebra with norm \(\|.\|_{\Gamma}\) and is a subspace of the space L(X) of all continuous linear operators on X. The Yosida theorem states that a linear operator A is the infinitesimal generator of a strongly continuous semigroup (C\({}_ 0\)-semigroup) \(\{T(t);\) \(t\geq 0\}\subset L(X)\) satisfying (*): \(\{e^{-\beta t}T(t)\); \(t\geq 0\}\) is equicontinuous for some \(\beta\geq 0\), if and only if (a) A is densely defined and \((\lambda -A)^{-1}\) exists for all \(\lambda >\beta\), and (b) \(\{(\lambda -\beta)^ k(\lambda -A)^{-k}\); \(\lambda >\beta\), \(k=0,1,2,...\}\) is equicontinuous. This paper presents a formulation in which (*) is replaced by (**): \(m_ 1(p)=\sup \{\tilde p(e^{-\beta t}T(t))\); \(t\geq 0\}<\infty\) for all \(p\in \Gamma,\) and (b) is replaced by (b'): \(m_ 2(p)=\sup \{\tilde p((\lambda -\beta)^ k(\lambda -A)^{- k})\); \(\lambda >\beta\), \(k=0,1,2,...\}<\infty\) for all \(p\in \Gamma.\) Moreover, \(m_ 1(p)=p_ 2(p)\) for each \(p\in \Gamma\). As a consequence, the Yosida theorem is reformulated with (b) replaced by (b''): \(\exists \Gamma \to \| (\lambda -A)^{-1}\|_{\Gamma}\leq (\lambda -\beta)^{-1}\) for \(\lambda >\beta.\) The author also proves, in the present setting, a stability theorem of Kato for generators, which includes as a special case a perturbation theorem for the generator of a quasi-equicontinuous \(C_ 0\)-semigroup by an operator in \(B_{\Gamma}(X)\).
infinitesimal generator of a strongly continuous semigroup
quasi-equicontinuous \(C_ 0\)- semigroup
scientific article; zbMATH DE number 3958049
scientific article