I have a question about Theorem 7.4.2 in Evan’s PDEs book. If $S(t)$ is a contraction semigroup on a Banach space $X$. He uses $$S(r)\int_0^t S(s)u\,ds = \int_0^t S(r+s)u\,ds$$ and I don’t understand how $S(r)$ and integration commute. Could anyone explain why this is true?

Let $X$ be a Banach space and $\mathcal{L}(X)$ the Banach algebra of all bounded linear operators $L:X\to X$, where the norm is given by $$\|L\|_\mathcal{L}=\sup\{\|L(x)\|_X;\;\|x\|_X=1\}$$ and the product is the composition of operators. If $T,F:(0,\infty)\to\mathcal{L}(X)$ are differentiable functions, can we apply the product rule to derive $T(t)F(t)$? Particularly, I’m interested in the case that $\{T(t)\}_{t\geq […]

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