Lower semicontinuity and the theorem of Datko and Pazy

Abstract

Let {T(t)} _t≥0 be aC ₀-semigroup on a real or complex Banach spaceX and letJ:C ⁺[0,∞)→[0,∞] be a lower semicontinuous and nondecreasing functional onC ⁺[0,∞), the positive cone ofC[0,∞), satisfyingJ(c 1)=∞ for allc>0. We prove the following result: if {T(t)} _t≥0 is not uniformly exponentially stable, then the set

$$\{ x \in X: J(||T( \cdot )x||) = \infty \}$$

is residual inX.