Reductions of Operator Pencils
2014 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 83, no 285, 189-214 p.Article in journal (Refereed) Published
We study problems associated with an operator pencil, i.e., a pair of operators onBanach spaces. Two natural problems to consider are linear constrained dierentialequations and the description of the generalized spectrum. The main tool to tackleeither of those problems is the reduction of the pencil. There are two kinds of naturalreduction operations associated to a pencil, which are conjugate to each other.Our main result is that those two kinds of reductions commute, under some mildassumptions that we investigate thoroughly.Each reduction exhibits moreover a pivot operator. The invertibility of all thepivot operators of all possible successive reductions corresponds to the notion ofregular pencil in the nite dimensional case, and to the inf-sup condition for saddlepoint problems on Hilbert spaces.Finally, we show how to use the reduction and the pivot operators to describe thegeneralized spectrum of the pencil.
Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2014. Vol. 83, no 285, 189-214 p.
Research subject Mathematics
IdentifiersURN: urn:nbn:se:umu:diva-79400DOI: 10.1090/S0025-5718-2013-02740-8OAI: oai:DiVA.org:umu-79400DiVA: diva2:641398