On necessary and sufficient conditions relating the adjoint of a column to a row of linear relations

TL;DR

This paper investigates the conditions under which the adjoint of a column of linear relations in Hilbert spaces corresponds to a row, providing necessary and sufficient criteria and discussing various related results.

Contribution

It introduces new necessary and sufficient conditions linking the adjoint of a column to a row of linear relations in Hilbert spaces.

Findings

Derived criteria for the adjoint of a column to be a row

Presented conditions involving sums and intersections of linear relations

Discussed several related theoretical outcomes

Abstract

A row and a column of two linear relations in Hilbert spaces are presented respectively as a sum and an intersection of two linear relations. As an application, necessary and sufficient conditions for the adjoint of a column to be a row are examined. Several outcomes are discussed as well.