Spectra of anticommutator for two orthogonal projections

ABSTRACT

In this note, for any two orthogonal projections P,Q on a Hilbert space, the characterization of spectrum of anticommutator PQ+QP has been obtained. As a corollary, an alternative proof of the norm formula $\parallel PQ+QP\parallel =\parallel PQ\parallel +\parallel PQ{\parallel }^2$ has been obtained (see Walters S. Anticommutator norm formula for projection operators, arXiv:1604.00699vl [math.FA] 3 Apr 2016).

KEYWORDS:

• Orthogonal projection
• anticommutator
• norm
• spectrum

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 47A05

Acknowledgements

The authors thank the referees for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was partially supported by the National Natural Science Foundation of China [grant numbers 11571211 and 11471200] and by the Fundamental Research Funds for the Central Universities [grant numbers GK201803009 and GK201801011].