Some perturbation results for analytic semigroups (Q1092394)

Theorem 1: Let A be the infinitesimal generator of an analytic semigroup on X and let F be a compact linear operator from \(X_ A\) into X. Then \((A+F)\) generates an analytic semigroup too. (We denote the Banach space D(A) equipped with the graph norm by \(X_ A).\) Theorem 2: Let X be a (complex) Banach space. Suppose that A is a linear operator in X so that there exists some \(\epsilon >0\) such that for each \(a\in X\), \(b^ *\in X^ *\) with \(\| a\| \leq \epsilon\), \(\| b^ *\| \leq \epsilon\), \(A+ab^ *A\) is the infinitesimal generator of a \(C_ 0\)-semigroup in X. Then A generates an analytic semigroup. (Here \(ab^ *\) is the operator defined by \(ab^ *(x)=b^ *(x)a)\).