Operator theoretic properties of semigroups in terms of their generators (Q2717580)

Let \(A\) be the generator of a \(C_0\)-semigroup \(\{T_t\}_{t\geq 0}\) on a complex Banach space \(X\), \({\mathcal A}\subset {\mathcal L}(x)\) be some normed operator ideal. The authors establish the equivalent of the following conditions: NEWLINENEWLINENEWLINE(a) \(R(\lambda,A) \in{\mathcal A}(x)\) for one (all) \(\lambda\in \rho(A)\); NEWLINENEWLINENEWLINE(b) \(j_1(A):D(-A) \hookrightarrow X\) belongs \({\mathcal A}(D(A),x)\); NEWLINENEWLINENEWLINE(c) \(\widetilde S_t\in {\mathcal A}(x)\) for one all \(t>0\) where \(\widetilde S_t x:= \int^t_0e^{-bs} T_sx ds\) for some \(b>\omega (T_t)\).NEWLINENEWLINENEWLINEThen they prove the analogous results for the Phillips functional calculus NEWLINE\[NEWLINE\widehat g(-A) x: = \int^\infty_0 T_sxg(s)ds,\quad g\in L^+_1(a)NEWLINE\]NEWLINE with norm \(\|g\|_{1,a}: = \int^\infty_0 |g(s)|e^{as}ds <\infty\), \(a>\omega (T_t)\), including the \({\mathcal A}\)-generalization of Pazy's result on compact semigroups. They also prove that for all \(\alpha>0\) and \(\beta\in (0,1]\) one has:NEWLINENEWLINENEWLINE\(\|T_t \|_{\mathcal A} \leq ct^{\beta-\alpha}\) on \(\mathbb{R}_+ \Leftrightarrow\|R( \lambda, A)^\alpha \|_{\mathcal A}\leq D\lambda^{-\beta}\) on \(\mathbb{R}_+\).NEWLINENEWLINENEWLINEMany examples are also given. See also the authors' paper in Arch. Math. 79, 109-118 (2002; Zbl 1006.47036).