# Questions tagged [semigroup-of-operators]

For questions related to theory of semigroups of linear operators and its applications to partial differential equations, stochastic processes such as Markov processes and other branches of mathematics.

21 questions
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### If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute

Let $A,B$ be complex valued square matrices. If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute. The converse of this statement can be an easy application of the Cauchy ...
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### Why does the semigroup commute with integration?

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 ...
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 ...