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Some perturbation results for analytic semigroups. (English) Zbl 0627.47020

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 \(ab^ *(x)=b^ *(x)a)\).

MSC:

47D03 Groups and semigroups of linear operators
47A55 Perturbation theory of linear operators
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References:

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