## Abstract

Some characterizations of products of two projections on a Hilbert space are generalized to the case of products of a finite number of projections on a Hilbert \(C^*\)-module. An example is constructed to show that in the Hilbert \(C^*\)-module case, \(\mathfrak {X}\) and its subset \(\mathfrak {X}_\bot \) can be different, where \(\mathfrak {X}\) denotes the set of all products of two projections on a Hilbert \(C^*\)-module, and \(\mathfrak {X}_\bot \) consists of those elements in \(\mathfrak {X}\) that have the polar decomposition. Some new phenomena are revealed when an operator is taken in \(\mathfrak {X}\) instead of \(\mathfrak {X}_\bot \). Some refinements are made on norms of differences of two projections.

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## Acknowledgements

The authors are grateful to the referees for very helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (11671261, 11971136) and a grant from Science and Technology Commission of Shanghai Municipality (18590745200).

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Communicated by Mohammad S. Moslehian.

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Xu, Q., Yan, G. Products of Projections, Polar Decompositions and Norms of Differences of Two Projections.
*Bull. Iran. Math. Soc.* (2021). https://doi.org/10.1007/s41980-020-00518-y

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### Keywords

- Hilbert \(C^*\)-module
- Polar decomposition
- Product of projections
- Norm of the difference of two projections

### Mathematics Subject Classification

- 46L08
- 47A05