On the uniqueness of the polar decomposition of bounded operators in Hilbert spaces (Q2859494)

It is known that the polar decomposition of a regular matrix exists and it is uniquely determined. The authors prove the following generalization of this result: Let \(A\) be a bounded linear operator defined on the Hilbert space \(H\) with values in the Hilbert space \(K\). The polar decomposition of \(A\) is unique if and only if either \(A\) is injective or it has dense range. They also include an example showing that a related statement in [\textit{A. Arai} and \textit{E. Hiroshi}, Mathematical structure of quantum mechanics. I. Tokyo: Asakura Publishing (1999; Zbl 1309.81001)], is not correct.