Let $\Omega\subset\mathbb{R}^2$ be an open and bounded domain and consider a positive continuous linear operator $B:L^2(\Omega)\to L^2(\Omega)$ with the property that if $f\in L^{\infty}(\Omega)\subset L^2(\Omega)$ then $Bf\in L^{\infty}(\Omega)$.
Since $L^2(\Omega)$ is a Hilbert space there is an adjoint of $B$, denoted $B^*:L^2(\Omega)\to L^2(\Omega)$ given by the following property:
$$\int_{\Omega} y(x)(Bz)(x)\ dx=\int_{\Omega} (B^*y)(x)z(x)\ dx,\ \forall\ y,z\in L^2(\Omega)$$.
Here is my question: Is it true that the adjoint of $B$ has the same property as $B$, i.e.: $B^*$ is a positive continuous linear operator and
$$\forall\ f\in L^{\infty}(\Omega),\ B^*f\in L^{\infty}(\Omega)$$
??
P.S. By positive linear operator I mean that $\forall\ y\in L^2(\Omega), y(x)\geq 0$, a.e. on $\Omega$ we have that $(By)(x)\geq 0$ a.e. on $\Omega$.
What I have done?
I used the Riesz representation theorem to find that there is some positive function $\phi\in L^2(\Omega)$ with $\displaystyle\int_{\Omega} (By)(x)\ dx=\int_{\Omega} \phi(x)y(x)\ dx,\ \forall\ y\in L^2(\Omega)$, but I did not succed to make a complete proof using that fact.
Maybe there is some counter example, but I could not find one yet.