# general representation theorem for bilinear forms

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.

It is known that for any dual system $$(X,Y)$$ (two spaces are dual systems if there exists a non degenerate bilinear form from Cartesian product of $$X$$ and $$Y$$ into reals or complex) and any two adjoint operators $$A:X \to Y$$ and $$B:Y \to X$$, $$A$$ and $$B$$ are linear. (we have also uniqueness of adjoint if it exists). Even for pre-Hilbert spaces like $$X=C(G)$$ ($$G$$ some compact measurable subset of $$R^n$$) and any unknown operator $$A:X \to X$$, it is not known that whether its adjoint exists or not. For example $$A:C[0,1] \to C[0,1]$$ defined by $$Ax:=x(1)$$, doesn't have adjoint.