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Factorization, majorization, and domination for linear relations

Let ${\mathfrak H}_A$, ${\mathfrak H}_B$, and ${\mathfrak H}$ be Hilbert spaces. Let $A$ be a linear relation from ${\mathfrak H}$ to ${\mathfrak H}_A$ and let $B$ be a linear relation from ${\mathfrak H}$ to ${\mathfrak H}_B$. If there exists an operator $Z \in \mathbf{B}({\mathfrak H}_B,{\mathfrak H}_A)$ such that $ZB \subset A$, then $B$ is said to dominate $A$. This notion plays a major role in the theory of Lebesgue type decompositions of linear relations and operators. There is a strong connection to the majorization and factorization in the well-known lemma of Douglas, when put in the context of linear relations. In this note some aspects of the lemma of Douglas are discussed in the context of linear relations and the connections with the notion of domination will be treated.