A note on the convergence of linear semi-groups of class (1,A) (Q911012)

A strongly continuous semigroup T(t), \(t>0\) is said to be of class (1,A) iff the following conditions are satisfied: \[ (a)\quad \int^{1}_{0}\| T(t)\| dt<\infty,\quad (b)\quad \lim_{\lambda \to \infty}\lambda \int^{\infty}_{0}e^{-\lambda t}T(t)x dt=x,\quad Re \lambda >\omega,\quad x\in X. \] It follows from (a) that the integral in (b) exists and defines a bounded linear operator R(\(\lambda)\) on X. Moreover the closure \(\bar A_ 0\) exists and \(Re(\lambda:\bar A_ 0)=R(\lambda),\) for Re \(\lambda\) \(>\Omega.\) The author assumes that the strongly continuous semigroup T(t), \(t>0\) satisfies the following two conditions: For each n in \({\mathbb{N}}\), let \(\{T(t;A_ n)\), \(t>0\}\) be a semigroup of linear operators on \(X_ n\) of class (1,A). Let there exist constants \(M,K>0\) and \(\omega >0\) such that \[ (1)\quad \int^{\infty}_{0}e^{- \omega t}\| T( ;A_ n)\|_ ndt\leq M,\quad n\in {\mathbb{N}}, \] and \[ (2)\quad \| R(\lambda;A_ n)\|_ n\leq K\quad \lambda \geq \omega,\quad n\in {\mathbb{N}}. \] For the above semigroup of operators, the author proves Trotter-Kato type convergence theorems.