Let $T(t)$ be a semigroup of bounded linear operator on a Banach space $X$. When does the following holds $$ \int_0^t T(s)x ds = (\int_0^t T(s) ds)x, x \in X \, , $$ where $ t \in (0,1)$.
the bochner integral about a semigroup of bounded linear operator on a Banach space
Y Chen
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