For a Markov process define its infinitesimal operator: [T(t) fleft(X_{s} ight)=int f(y) pleft(t, X_{s}, y ight) d

For a Markov process define its infinitesimal operator:

\[T(t) f\left(X_{s}\right)=\int f(y) p\left(t, X_{s}, y\right) d y=E\left[f\left(X_{t+s}\right) \mid X_{s}\right]\]

Show that this operator is a contraction with respect to:

(a) Norm 0: \(\|f(x)\|_{0}=\max f(x)\)

(b) Norm 1: \(\|f(x)\|_{1}=\left|\int^{x} f(x) d x\right|\)

(c) Norm 2: \(\|f(x)\|_{2}=\left(\int f(x)^{2} d x\right)^{\frac{1}{2}}\)

(d) Norm \(p\) : \(\|f(x)\|_{p}=\left(\int f(x)^{p} d x\right)^{\frac{1}{p}}\), for all \(p \geq 1\)