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Transactions of the American Mathematical Society

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The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Geometric aspects of multiparameter spectral theory
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by Luzius Grunenfelder and Tomaž Košir PDF
Trans. Amer. Math. Soc. 350 (1998), 2525-2546 Request permission

Abstract:

The paper contains a geometric description of the dimension of the total root subspace of a regular multiparameter system in terms of the intersection multiplicities of its determinantal hypersurfaces. The new definition of regularity used here is proved to restrict to the familiar definition in the linear case. A decomposability problem is also considered. It is shown that the joint root subspace of a multiparameter system may be decomposable even when the root subspace of each member is indecomposable.
References
  • M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
  • F. V. Atkinson, Multiparameter eigenvalue problems, Mathematics in Science and Engineering, Vol. 82, Academic Press, New York-London, 1972. Volume I: Matrices and compact operators. MR 0451001
  • Paul Binding and Patrick J. Browne, Two parameter eigenvalue problems for matrices, Linear Algebra Appl. 113 (1989), 139–157. MR 978589, DOI 10.1016/0024-3795(89)90293-0
  • Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
  • R. J. Cook and A. D. Thomas, Line bundles and homogeneous matrices, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 120, 423–429. MR 559048, DOI 10.1093/qmath/30.4.423
  • A.C. Dixon, Note on the Reduction of Ternary Quartic to a Symmetrical Determinant, Proc. Cambridge Phil. Soc., 2:350–351, 1900–1902.
  • M. Faierman, Two-parameter eigenvalue problems in ordinary differential equations, Pitman Research Notes in Mathematics Series, vol. 205, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. MR 1124402
  • William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
  • G.A. Gadzhiev, Introduction to Multiparameter Spectral Theory (in Russian), Azerbaijan State University, Baku, 1987.
  • Luzius Grunenfelder and Tomaž Košir, An algebraic approach to multiparameter spectral theory, Trans. Amer. Math. Soc. 348 (1996), no. 8, 2983–2998. MR 1361640, DOI 10.1090/S0002-9947-96-01679-0
  • L. Grunenfelder and T. Košir, Coalgebras and Spectral Theory in One and Several Parameters, Recent Developments in Operator Theory and Its Applications (Manitoba, 1994), Operator Theory: Advances and Applications, vol. 87, Birkhäuser Verlag, 1996, pp. 177–192.
  • L. Grunenfelder and T. Košir, Koszul Cohomology for Finite Families of Comodule Maps and Applications, Comm. in Alg 25: 459–479, 1997.
  • Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
  • H.(G.A.) Isaev, Lectures on Multiparameter Spectral Theory, Dept. of Math. and Stats., University of Calgary, 1985.
  • Tomaž Košir, Kronecker bases for linear matrix equations, with application to two-parameter eigenvalue problems, Linear Algebra Appl. 249 (1996), 259–288. MR 1417422, DOI 10.1016/0024-3795(95)00361-4
  • Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
  • Jean-Pierre Serre, Algèbre locale. Multiplicités, Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965 (French). Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel; Seconde édition, 1965. MR 0201468
  • I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917
  • B. D. Sleeman, Multiparameter spectral theory in Hilbert space, Research Notes in Mathematics, vol. 22, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1978. MR 533152
  • Victor Vinnikov, Complete description of determinantal representations of smooth irreducible curves, Linear Algebra Appl. 125 (1989), 103–140. MR 1024486, DOI 10.1016/0024-3795(89)90035-9
  • Hans Volkmer, Multiparameter eigenvalue problems and expansion theorems, Lecture Notes in Mathematics, vol. 1356, Springer-Verlag, Berlin, 1988. MR 973644, DOI 10.1007/BFb0089295
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Additional Information
  • Luzius Grunenfelder
  • Affiliation: Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 3J5
  • Email: luzius@cs.dal.ca
  • Tomaž Košir
  • Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
  • Email: tomaz.kosir@fmf.uni-lj.si
  • Received by editor(s): September 26, 1996
  • Additional Notes: Research supported in part by the NSERC of Canada and by the Ministry of Science and Technology of Slovenia.
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 2525-2546
  • MSC (1991): Primary 13H15, 14C17, 15A54
  • DOI: https://doi.org/10.1090/S0002-9947-98-02078-9
  • MathSciNet review: 1451601