### 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) |
---|---|

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 |

### Fingerprint

### 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

### Cite this

*Linear Algebra and Its Applications*,

*432*(6), 1412-1459. https://doi.org/10.1016/j.laa.2009.11.002

**A gentle guide to the basics of two projections theory.** / Böttcher, A.; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 432, no. 6, pp. 1412-1459. https://doi.org/10.1016/j.laa.2009.11.002

}

TY - JOUR

T1 - A gentle guide to the basics of two projections theory

AU - Böttcher, A.

AU - Spitkovsky, Ilya

PY - 2010/3/1

Y1 - 2010/3/1

N2 - 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.

AB - 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.

KW - Angle and gap

KW - Idempotent

KW - Skew and oblique projection

KW - Symbol calculus

KW - Two projections

KW - Two subspaces

UR - http://www.scopus.com/inward/record.url?scp=72949098608&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=72949098608&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2009.11.002

DO - 10.1016/j.laa.2009.11.002

M3 - Article

VL - 432

SP - 1412

EP - 1459

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 6

ER -