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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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Generator of a generated $C_0$ semigroup
Consider a $C_0$-semigroup $S_t:\mathscr{B(H)} \to \mathscr{B(H)}$ with generator $U$. Now define $P_t:\mathscr{B_1(H)} \to \mathscr{B_1(H)}$ where $P_t(\rho)=S_t\rho S_t^*$. How can I prove $P_t$ t …