Polar Decomposition for Non-Adjointable Maps

TL;DR

This paper establishes conditions under which non-adjointable operators between Hilbert modules can be decomposed into polar form using partial isometries that are not necessarily adjointable, introducing the concept of modular operators.

Contribution

It introduces the notion of modular operators for non-adjointable maps and extends polar decomposition theory to a broader class of operators between Hilbert modules.

Findings

Conditions for polar decomposition of non-adjointable operators

Introduction of modular operators for such operators

Extension of polar decomposition framework to Hilbert modules

Abstract

We give conditions when not necessarily adjointable operators between Hilbert modules allow for a polar decomposition involving not necessarily adjointable partial isometries. While the latter have been introduced and discussed by Shalit and Skeide [SS23,Ske25], here we are led, as a basic new ingredient, to the notion of not necessarily adjointable operators $a$ that admit a modulus $|a|$, so-called modular operators.