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Rank-1 perturbations of cosine functions and semigroups. (English) Zbl 1104.47046

Let \(X\) be a Banach space. Two of the main results of the paper may be stated as follows.
1. Suppose that either \(A\) is the generator of a cosine function and \(1/2 < \gamma \leq 1\), or \(A\) is the generator of a \(k\)-times integrated cosine function for some \(k \in {\mathbb N}_0\) and \(\gamma=1\). Let \(A_\gamma = (\omega-A)^\gamma\) with \(\omega\) large enough and let \(\varepsilon >0\). If for each \(a \in X\), \(b^* \in X^*\) satisfying \(\| a\| \leq \varepsilon\), \(\| b^*\| \leq \varepsilon\), there exists \(\ell \in {\mathbb N}\) such that \(A +ab^*A_\gamma\) generates an \(\ell\)-times integrated cosine function, then \(A\) is bounded.
2. Let \(A\) be a densely defined operator on \(X\) and let \(\varepsilon >0\). If for each \(a \in X\), \(b^* \in X^*\) satisfying \(\| a\| \leq \varepsilon\), \(\| b^*\| \leq \varepsilon\), the operator \(A + ab^*A\) generates a distribution semigroup, then \(A\) generates a holomorphic \(C_0\)-semigroup.
The paper is nicely written and the proofs are accurately presented.

MSC:

47D09 Operator sine and cosine functions and higher-order Cauchy problems
47D03 Groups and semigroups of linear operators
47A55 Perturbation theory of linear operators
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