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# Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$

### Abstract

We consider a spectral problem over $\mathbb{R}^n$ for a Douglis-Nirenberg system of differential operators under limited smoothness assumptions and under the assumption of parameter-ellipticity in a closed sector $\mathcal{L}$ in the complex plane with vertex at the origin. We pose the problem in an $L_p$ Sobolev-Bessel potential space setting, $1 < p < \infty$, and denote by $A_p$ the operator induced in this setting by the spectral problem. We then derive results pertaining to the Fredholm theory for $A_p$ for values of the spectral parameter $\lambda$ lying in $\mathcal{L}$ as well as results pertaining to the invariance of the Fredholm domain of $A_p$ with $p$.

Key words: Parameter-ellipticity, Douglis-Nirenberg system, Fredholm properties.

### Article Information

 Title Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$ Source Methods Funct. Anal. Topology, Vol. 22 (2016), no. 4, 330-345 Milestones Received 27/04/2016 Copyright The Author(s) 2016 (CC BY-SA)

### Authors Information

M. Faierman
School of Mathematics and Statistics, The University of New South Wales, UNSW Sydney, NSW 2052, Australia

### Citation Example

M. Faierman, Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$, Methods Funct. Anal. Topology 22 (2016), no. 4, 330-345.

### BibTex

@article {MFAT913,
AUTHOR = {Faierman, M.},
TITLE = {Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {22},
YEAR = {2016},
NUMBER = {4},
PAGES = {330-345},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=913},
}

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