Perturbation of non-exponentially-bounded \(\alpha\)-times integrated \(C\)-semigroups (Q1425469)

This paper is concerned with the perturbation of an \(\alpha\)-times integrated \(C\)-semigroup \(T(\cdot)\) with the generator \(A\) on a Banach space \(X\), where \(T(\cdot)\) may be degenerate and non-exponentially bounded. The main results are as follows: (1) Let \(B\) be the generator of a \(C_0\) group \(S(\cdot)\) commuting with \(T(\cdot)\) and \(C\). Then \(\overline{A+B}\) is the generator of an \(\alpha\)-times integrated \(C\)-semigroup under suitable conditions on \(T(\cdot)\) and \(S(\cdot)\). (2) If \(B\) is a bounded linear operator on \(X\), and \(B\) is commuting with \(T(\cdot)\) and \(C\), then \(A+B\) also is the generator of an \(\alpha\)-times integrated \(C\)-semigroup.