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Reductions of operator pencils


Author: Olivier Verdier
Journal: Math. Comp. 83 (2014), 189-214
MSC (2010): Primary 15A21, 15A22, 34A30, 47A10, 65L80
DOI: https://doi.org/10.1090/S0025-5718-2013-02740-8
Published electronically: June 27, 2013
MathSciNet review: 3120586
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Abstract: We study problems associated with an operator pencil, i.e., a pair of operators on Banach spaces. Two natural problems to consider are linear constrained differential equations and the description of the generalized spectrum. The main tool to tackle either of those problems is the reduction of the pencil. There are two kinds of natural reduction operations associated to a pencil, which are conjugate to each other.

Our main result is that those two kinds of reductions commute, under some mild assumptions that we investigate thoroughly.

Each reduction exhibits moreover a pivot operator. The invertibility of all the pivot operators of all possible successive reductions corresponds to the notion of regular pencil in the finite dimensional case, and to the inf-sup condition for saddle point problems on Hilbert spaces.

Finally, we show how to use the reduction and the pivot operators to describe the generalized spectrum of the pencil.


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Additional Information

Olivier Verdier
Affiliation: Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway
Email: olivier.verdier@math.ntnu.no

DOI: https://doi.org/10.1090/S0025-5718-2013-02740-8
Received by editor(s): May 27, 2011
Received by editor(s) in revised form: May 22, 2012
Published electronically: June 27, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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