### Abstract

This paper is a survey of the basics of the theory of two projections. It contains in particular the theorem by Halmos on two orthogonal projections and Roch, Silbermann, Gohberg, and Krupnik's theorem on two idempotents in Banach algebras. These two theorems, which deliver the desired results usually very quickly and comfortably, are missing or wrongly cited in many recent publications on the topic, The paper is intended as a gentle guide to the field. The basic theorems are precisely stated, some of them are accompanied by full proofs, others not, but precise references are given in each case, and many examples illustrate how to work with the theorems.

Original language | English (US) |
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Pages (from-to) | 1412-1459 |

Number of pages | 48 |

Journal | Linear Algebra and Its Applications |

Volume | 432 |

Issue number | 6 |

DOIs | |

State | Published - Mar 1 2010 |

### Keywords

- Angle and gap
- Idempotent
- Skew and oblique projection
- Symbol calculus
- Two projections
- Two subspaces

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

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## Cite this

Böttcher, A., & Spitkovsky, I. M. (2010). A gentle guide to the basics of two projections theory.

*Linear Algebra and Its Applications*,*432*(6), 1412-1459. https://doi.org/10.1016/j.laa.2009.11.002