Non-standard behaviour of semibounded forms and their representations

A number of results is presented concerning the extreme behaviour of products of linear operators or relations from one Hilbert space H to another Hilbert space K. It is shown that there exists a relation T from H to K such that the closure of the nonnegative relation T^*T in H is not equal to the nonnegative selfadjoint relation T^*T**; moreover, this may happen for a singular but also for a regular (i.e., closable) relation T. In fact, as an extreme situation the paper contains a construction of a densely defined closable operator T, such that dom\,T^*T=\0\, while T^*T** is a densely defined selfadjoint operator. Likewise it is shown that there exist a linear relation T from H to K and a nonnegative everywhere defined contraction R in K for which the product $RT$ is singular and the product R1/2 T is regular. Such results illustrate situations that can arise in the Lebesgue type decomposition of linear relations and the Lebesgue type decomposition of semibounded forms.