### Abstract

We establish Fredholm criteria and index formulas for one-dimensional zero-order pseudodifferential operators with piecewise continuous generating functions on L^{p} spaces with Muckenhoupt weights. The Fredholm symbol of such operators is shown to be a matrix function defined on a set which, roughly speaking, is a cylinder with a certain collection of horn shaped handles. The presence of these horns implies that, unlike the case of L^{p} spaces without weight or with so-called power weights, the spectrum may contain heavy parts, i. e. the set of the interior points of the spectrum need not be empty. Our proof makes essential use of recent results by Finck, Roch, Silbermann, Gohberg, and Krupnik on the inverse closedness of certain Banach algebras.

Original language | English (US) |
---|---|

Pages (from-to) | 251-269 |

Number of pages | 19 |

Journal | Integral Equations and Operator Theory |

Volume | 19 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 1994 |

### Fingerprint

### Keywords

- AMS subject classification: 45E05, 45E10, 47A53, 47G30

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory

### Cite this

*Integral Equations and Operator Theory*,

*19*(3), 251-269. https://doi.org/10.1007/BF01203665

**Pseudodifferential operators with heavy spectrum.** / Böttcher, A.; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Integral Equations and Operator Theory*, vol. 19, no. 3, pp. 251-269. https://doi.org/10.1007/BF01203665

}

TY - JOUR

T1 - Pseudodifferential operators with heavy spectrum

AU - Böttcher, A.

AU - Spitkovsky, Ilya

PY - 1994/9/1

Y1 - 1994/9/1

N2 - We establish Fredholm criteria and index formulas for one-dimensional zero-order pseudodifferential operators with piecewise continuous generating functions on Lp spaces with Muckenhoupt weights. The Fredholm symbol of such operators is shown to be a matrix function defined on a set which, roughly speaking, is a cylinder with a certain collection of horn shaped handles. The presence of these horns implies that, unlike the case of Lp spaces without weight or with so-called power weights, the spectrum may contain heavy parts, i. e. the set of the interior points of the spectrum need not be empty. Our proof makes essential use of recent results by Finck, Roch, Silbermann, Gohberg, and Krupnik on the inverse closedness of certain Banach algebras.

AB - We establish Fredholm criteria and index formulas for one-dimensional zero-order pseudodifferential operators with piecewise continuous generating functions on Lp spaces with Muckenhoupt weights. The Fredholm symbol of such operators is shown to be a matrix function defined on a set which, roughly speaking, is a cylinder with a certain collection of horn shaped handles. The presence of these horns implies that, unlike the case of Lp spaces without weight or with so-called power weights, the spectrum may contain heavy parts, i. e. the set of the interior points of the spectrum need not be empty. Our proof makes essential use of recent results by Finck, Roch, Silbermann, Gohberg, and Krupnik on the inverse closedness of certain Banach algebras.

KW - AMS subject classification: 45E05, 45E10, 47A53, 47G30

UR - http://www.scopus.com/inward/record.url?scp=0040112775&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040112775&partnerID=8YFLogxK

U2 - 10.1007/BF01203665

DO - 10.1007/BF01203665

M3 - Article

AN - SCOPUS:0040112775

VL - 19

SP - 251

EP - 269

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 3

ER -