Let $\Omega$ be a Polish space, denote $E = (C_b(\Omega),\|\cdot\|)$ be the Banach space of continuous and bounded functions with the usual supremum norm. A family $(P_t)_{t\geq 0} $ of linear operators on $E$ is called a semigroup iff
- $P_0 = I$, the identity map on $E$,
- The map $t\mapsto P_t$ is continuous in the sense: $\forall f \in E$, $t\mapsto P_tf$ is a continuous map from $\mathbb R^+$ to $E$,
- $P_{t+s} = P_tP_s, \forall t,s \geq 0$,
A semigroup is called Markov if in addition,
- $P_t1 = 1$, (1 is the function that takes only value 1),
- $P_t$ is positive, i.e, if $f\geq 0$ then $P_tf \geq 0$.
Questions:
A) Prove that if $P$ is Markov then it is contractive, i.e, $\| P_tf\| \leq \| f\| $?
B) Let $A$ be a positive bounded linear operator on $E$, and that $\| A \| \leq 1$ (in operator norm), $A1 =1$. Let $\lambda >0 $. Verify that $$ P_t := e^{t\lambda(A-I)} = \sum_{n=0}^\infty \frac{(\lambda t)^n}{ n!}(A-I)^n $$ is a Markov operator.
Context: I encountered A) in a lecture note, which I don't know how to prove and begin with. For B), I can only verify 1,3,4. I don't know how to verify the continuity (2.) and also, I suspect that it is differentiable as well. For 5., I have tried to prove the finite sum $H_N = \sum^N_{n=0}...$ has the property
$$\| H_Nf \| \leq \| f\|,$$
for $N=0,1,2,3$, but I don't know quite how to do it inductively for all $N$.
I hope to get a detailed treatment of 2. and its differentiability as well if possible (for me to learn this kind of calculus) and only some hints on A) and how to prove 5 inductively. Thank you in advance :D.