<p>The purpose of this paper is to revisit the polar decomposition theorem for closed everywhere defined linear relations and to illustrate some of its applications. In doing so, we extend Furuta’s results and those of Ito, Yamazaki, and Yanagida, which were previously established for operators, to the realm of closed linear relations. Additionally, we propose potential directions for future research, such as extending the theoretical framework to include <i>C</i>-polar decomposition for closed linear relations, further studying weak-type normal linear relations, and investigating inequalities for closed linear relations components. These developments significantly broaden the scope of polar decomposition theory and open new avenues for further exploration.</p>

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The polar decomposition of closed linear relations and its applications

  • Mohamed Amine Aouichaoui,
  • Zied Garbouj

摘要

The purpose of this paper is to revisit the polar decomposition theorem for closed everywhere defined linear relations and to illustrate some of its applications. In doing so, we extend Furuta’s results and those of Ito, Yamazaki, and Yanagida, which were previously established for operators, to the realm of closed linear relations. Additionally, we propose potential directions for future research, such as extending the theoretical framework to include C-polar decomposition for closed linear relations, further studying weak-type normal linear relations, and investigating inequalities for closed linear relations components. These developments significantly broaden the scope of polar decomposition theory and open new avenues for further exploration.