I am trying to understand the adjoint of a linear operator geometrically. Since the graph of the adjoint can be constructed as the orthogonal complement of a "rotated" copy of the graph of the operator itself (details follow) I am wondering how orthogonality of the graphs translates to algebraic or topological relations between two operators.

What I mean by the above introductory remarks is that using the "rotation" $$U\colon H\oplus H\to H\oplus H, \qquad (x,y)\mapsto (-y,x)$$ we can get the relation $$\mathrm{Graph}(A^*) = U(\mathrm{Graph}(A))^\perp.$$

This question is similar to this one but since I am also interested in the infinte dimensional case I would like to avoid the singular value decomposition which is used in the answers there.