# Commuting linear operators and algebraic decompositions

Archivum Mathematicum (2007)

• Volume: 043, Issue: 5, page 373-387
• ISSN: 0044-8753

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## Abstract

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For commuting linear operators ${P}_{0},{P}_{1},\cdots ,{P}_{\ell }$ we describe a range of conditions which are weaker than invertibility. When any of these conditions hold we may study the composition $P={P}_{0}{P}_{1}\cdots {P}_{\ell }$ in terms of the component operators or combinations thereof. In particular the general inhomogeneous problem $Pu=f$ reduces to a system of simpler problems. These problems capture the structure of the solution and range spaces and, if the operators involved are differential, then this gives an effective way of lowering the differential order of the problem to be studied. Suitable systems of operators may be treated analogously. For a class of decompositions the higher symmetries of a composition $P$ may be derived from generalised symmmetries of the component operators ${P}_{i}$ in the system.

## How to cite

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Gover, Rod A., and Šilhan, Josef. "Commuting linear operators and algebraic decompositions." Archivum Mathematicum 043.5 (2007): 373-387. <http://eudml.org/doc/250153>.

@article{Gover2007,
abstract = {For commuting linear operators $P_0,P_1,\dots ,P_\ell$ we describe a range of conditions which are weaker than invertibility. When any of these conditions hold we may study the composition $P=P_0P_1\cdots P_\ell$ in terms of the component operators or combinations thereof. In particular the general inhomogeneous problem $Pu=f$ reduces to a system of simpler problems. These problems capture the structure of the solution and range spaces and, if the operators involved are differential, then this gives an effective way of lowering the differential order of the problem to be studied. Suitable systems of operators may be treated analogously. For a class of decompositions the higher symmetries of a composition $P$ may be derived from generalised symmmetries of the component operators $P_i$ in the system.},
author = {Gover, Rod A., Šilhan, Josef},
journal = {Archivum Mathematicum},
keywords = {commuting linear operators; decompositions; relative invertibility; commuting linear operators; decompositions; relative invertibility},
language = {eng},
number = {5},
pages = {373-387},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Commuting linear operators and algebraic decompositions},
url = {http://eudml.org/doc/250153},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Gover, Rod A.
AU - Šilhan, Josef
TI - Commuting linear operators and algebraic decompositions
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 5
SP - 373
EP - 387
AB - For commuting linear operators $P_0,P_1,\dots ,P_\ell$ we describe a range of conditions which are weaker than invertibility. When any of these conditions hold we may study the composition $P=P_0P_1\cdots P_\ell$ in terms of the component operators or combinations thereof. In particular the general inhomogeneous problem $Pu=f$ reduces to a system of simpler problems. These problems capture the structure of the solution and range spaces and, if the operators involved are differential, then this gives an effective way of lowering the differential order of the problem to be studied. Suitable systems of operators may be treated analogously. For a class of decompositions the higher symmetries of a composition $P$ may be derived from generalised symmmetries of the component operators $P_i$ in the system.
LA - eng
KW - commuting linear operators; decompositions; relative invertibility; commuting linear operators; decompositions; relative invertibility
UR - http://eudml.org/doc/250153
ER -

## References

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6. Gover A. R., Laplacian operators and Q-curvature on conformally Einstein manifolds, Mathematische Annalen, 336 (2006), 311–334. Zbl1125.53032MR2244375
7. Gover A. R., Šilhan J., Commuting linear operators and decompositions; applications to Einstein manifolds, Preprint math/0701377 , www.arxiv.org. Zbl1195.47038MR2585804
8. Graham C. R., Jenne R., Mason J. V., Sparling G. A., Conformally invariant powers of the Laplacian, I: Existence, J. London Math. Soc. 46, (1992), 557–565. (1992) Zbl0726.53010MR1190438
9. Miller W., Jr., Symmetry and separation of variables, Encyclopedia of Mathematics and its Applications, Vol. 4. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977, xxx+285 pp. (1977) Zbl0368.35002MR0460751

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