# 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
1answer
696 views

### 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 ...
2answers
262 views

### 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 ...
1answer
355 views

### Is the product rule true in a Banach algebra?

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 ...
2answers
2k views