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(Usually one-parameter) semigroups of linear operators and their applications to partial differential equations, stochastic processes such as Markov processes and other branches of mathematics.

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The part of an operator as an analytic generator

This is wrong. Let us assume that $Y$ is a closed subspace of $X$ to clearify the problem. As Matthew Daws already said, you have to assume that the semigroup $(T(t))$ generated by $A$ leaves $Y$ inva …
Stephan Fackler's user avatar