Products of Projections, Polar Decompositions and Norms of Differences of Two Projections

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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Correspondence to Qingxiang Xu.

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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