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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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References:
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