Componentwise and Cartesian decompositions of linear relations

TL;DR

This paper explores decompositions of linear relations in Hilbert spaces, extending operator closability concepts, and establishes criteria for componentwise and Cartesian decompositions, including orthogonal cases and their connections to real and imaginary parts.

Contribution

It introduces new criteria for the existence and uniqueness of operator parts in linear relations and links Cartesian decompositions to the real and imaginary components of relations.

Findings

Criteria for existence of operator parts are established.

Orthogonal decompositions are valid for several classes of relations.

Connections between Cartesian decompositions and real/imaginary parts are analyzed.

Abstract

Let $A$ be a, not necessarily closed, linear relation in a Hilbert space $\sH$ with a multivalued part $\mul A$. An operator $B$ in $\sH$ with $\ran B\perp\mul A^{**}$ is said to be an operator part of $A$ when $A=B \hplus (\{0\}\times \mul A)$, where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for the existence of an operator part are established via the so-called canonical decomposition of $A$. In addition, conditions are developed for the decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation $A$ is said to have a Cartesian decomposition if $A=U+\I V$, where $U$ and $V$ are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of $A$ and the real and imaginary parts of $A$ is investigated.