I am trying to incorporate the definition of dual operator in a Markovian setting. Say I have a Markov kernel $K:S\times S\to[0,1]$ i.e., a mapping such that
for every $A\in S$ $x\to K(x,A)$ is a measurable mapping, and
for every $x \in S$, $A\to K(x,A)$ is a probability measure.
I can see it as an operator acting on measures $(\mu K)(A)= \int K(x,A) d\mu(x)$ and as an operator acting on functions $Kf(x) = \int dK(x,y) f(y)$.
Now, given a reference measure $\nu$, I want to study the dual operator $K^\star_\nu$. Consider $K$ as a mapping form $L^p(\nu K)$ to $L^p(\nu)$, then $K^\star_\nu$ is a mapping from $(L^p(\nu))^\star$ to $(L^p(\nu K))^\star$ i.e., $L^q(\nu)$ to $L^q(\nu K)$ with $q=p/(p-1)$.
By definition I should have that $(K^\star_\nu f)(h) = f(Kh)$ where $h\in L^p(\nu K)$.
Now I should show that $(\nu K)(gK^\star_\nu(f))= \nu(K(g)f)$ but I am not sure how to derive this. Passing to integrals: $$ (\nu K)(gK^\star_\nu(f)) = \int d\nu K(y) g(y) (K^\star_\nu f)(y) = \int d\nu K(y) g(y) f(K?(y)) $$
my question is, what does "?" represent there?
I do know that $K_\nu^\star f\in L^q(\nu K)\approx (L^p(\nu K))^\star$ so the first piece makes sense formally if I see it as an operator acting on $L^q(\cdot)$ however, when I use the definition of dual operator, then $(K_\nu^\star f)(y)=f(K?y)$ and I don't know how to fill the question mark and how to bridge the duality between the operators and the duality between the spaces as $K$ should act on $L^p(\nu K)$ and not on the field.