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A Purity Criterion for Pairs of Linear Transformations

Published online by Cambridge University Press:  20 November 2018

Uri Fixman  and
Frank A. Zorzitto
Uri Fixman
Affiliation:
Queen's University, Kingston, Ontario
Frank A. Zorzitto
Affiliation:
Queen's University, Kingston, Ontario
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In connection with the study of perturbation methods for differential eigenvalue problems, Aronszajn put forth a theory of systems (X, Y; A, B) consisting of a pair of linear transformations A, B:X → Y (see [1]; cf. also [2]). Here X and Y are complex vector spaces, possibly of infinite dimension. The algebraic aspects of this theory, where no restrictions of topological nature are imposed, where developed in [3] and [5]. We hasten to point out that the category of C2-systems (definition in § 1) in which this algebraic investigation takes place is equivalent to the category of all right modules over the ring of matrices of the form

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 3 , June 1974 , pp. 734 - 745
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Aronszajn, N., Quadratic forms on vector spaces, Proc. Intern. Symposium on Linear Spaces 1960, Jerusalem, 1961.Google Scholar
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3. Aronszajn, N. and Fixman, U., Algebraic spectral problems, Studia Math. 30 (1968), 273–338.Google Scholar
4. Dickson, S. E., A torsion theory for abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223–235.Google Scholar
5. Fixman, U., On algebraic equivalence between pairs of linear transformations, Trans. Amer. Math. Soc. 113 (1964), 424–453.Google Scholar
6. Zorzitto, F. A., Topological decompositions in systems of linear transformations, Ph.D. Thesis, Queen's University, 1972.Google Scholar
7. Zorzitto, F. A., Purity and copurity in systems of linear transformations (to appear).Google Scholar