On square roots of isometries

Abstract

In various normed spaces we answer the question of when a given isometry is a square of some isometry. In particular, we consider (real and complex) matrix spaces equipped with unitarily invariant norms and unitary congruence invariant norms, as well as some infinite dimensional spaces illustrating the difference between finite and infinite dimensions.

Keywords:

• Isometry
• Frobenius norm
• unitarily invariant norm
• unitary congruence invariant norm

AMS Subject Classifications:

• Primary: 15A60
• Secondary: 46B04
• 15A86

Acknowledgements

The authors would like to thank C.K. Li for several fruitful discussions which led to the extension of the original draft and for bringing the paper [8] to the authors’ notice. The authors would also like to thank M. Mbekhta for making the preprint of the paper [22] available to them. The authors are also indebted to M. Gaál for informing them about the paper [15].

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Dijana Ilišević has been fully supported by the Croatian Science Foundation [project number IP-2016-06-1046]. Bojan Kuzma was partially supported by Slovenian Research Agency [research core fundings No. P1-0222].