Non-standard behaviour of semibounded forms and their representations

Abstract

A number of results is presented concerning the extreme behaviour of products of linear operators or relations from one Hilbert space ${\mathfrak H}$ to another Hilbert space ${\mathfrak K}$. It is shown that there exists a relation $T$ from ${\mathfrak H}$ to ${\mathfrak K}$ such that the closure of the nonnegative relation $T^*T$ in ${\mathfrak H}$ 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 ${\rm dom\,}T^*T=\{0\}$, while $T^*T^{**}$ is a densely defined selfadjoint operator. Likewise it is shown that there exist a linear relation $T$ from ${\mathfrak H}$ to ${\mathfrak K}$ and a nonnegative everywhere defined contraction $R$ in ${\mathfrak K}$ for which the product $RT$ is singular and the product $R^{1/2} T$ 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.