Lebesgue type decompositions for linear relations and Ando's uniqueness criterion

TL;DR

This paper develops Lebesgue type decompositions for linear relations between Hilbert spaces, characterizes their uniqueness, and explores applications to range space relations, positive operators, and measures.

Contribution

It introduces a canonical Lebesgue decomposition for linear relations, characterizes when such decompositions are unique, and extends results to weak decompositions and applications.

Findings

Canonical Lebesgue decomposition characterized by maximal closable part

Necessary and sufficient conditions for decomposition uniqueness

Applications to range space relations and positive measures

Abstract

A linear relation, i.e., a multivalued operator $T$ from a Hilbert space ${\mathfrak H}$ to a Hilbert space ${\mathfrak K}$ has Lebesgue type decompositions $T=T_{1}+T_{2}$, where $T_{1}$ is a closable operator and $T_{2}$ is an operator or relation which is singular. There is one canonical decomposition, called the Lebesgue decomposition of $T$, whose closable part is characterized by its maximality among all closable parts in the sense of domination. All Lebesgue type decompositions are parametrized, which also leads to necessary and sufficient conditions for the uniqueness of such decompositions. Similar results are given for weak Lebesgue type decompositions, where $T_1$ is just an operator without being necessarily closable. Moreover, closability is characterized in different useful ways. In the special case of range space relations the above decompositions may be applied when dealing with pairs of (nonnegative) bounded operators and nonnegative forms as well as in the classical framework of positive measures.