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Lower semicontinuity and the theorem of Datko and Pazy. (English) Zbl 1040.47033
Summary: Let \(\{T(t)\}_{t\geq 0}\) be a \(C_0\)-semigroup on a real or complex Banach space \(X\) and let \(J:C^+[0,\infty)\to [0,\infty]\) be a lower semicontinuous and nondecreasing functional on \(C^+ [0, \infty)\), the positive cone of \(C[0,\infty)\), satisfying \(J(c\mathbf{1})=\infty\) for all \(c>0\). We prove the following result: if \(\{T(t)\}_{t\geq 0}\) is not uniformly exponentially stable, then the set \[ \{x\in X:J( \| T(\cdot)x\|)= \infty\} \] is residual in \(X\).

MSC:
47D06 One-parameter semigroups and linear evolution equations
47D03 Groups and semigroups of linear operators
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