I am interested in representation theorems for bilinear forms, that go beyond treatment of bounded or even coercive bilinear forms.
Whilst I am thankful for any references regarding the topic mentioned above I am specifically looking for a treatment of positive semi-definite symmetric bilinear forms. The bilinear form I am currently concerned with is not bounded. The reason I'd like to know about that is that I try to check the preliminaries for the existence of an adjoint operator with respect to that certain bilinear form.
So written in it's abstract form, let $B(.,.)$ be a positive semi-definite symmetric but unbounded bilinear form. Let further be $H$ a Hilbert space, $f,g \in H$ and $A$ be a bounded linear operator on $H$. So I am interested in the answer to the following question: When does a linear operator (or any other kind of function on $H$) $A^*$ exist such that $$ B(f,Ag) = B(A^* f,g) $$
I know about basic representation theory for bounded bilinear forms and the Lax-Milgram Theorem. But of course they are of no help here. I'd appreciate any sort of hint into the right direction or to literature for further research on representation theory for bilinear forms and or the existence of adjoint operators.