An algebraic approach to multiparameter spectral theory
HTML articles powered by AMS MathViewer
- by Luzius Grunenfelder and Tomaz Kosir
- Trans. Amer. Math. Soc. 348 (1996), 2983-2998
- DOI: https://doi.org/10.1090/S0002-9947-96-01679-0
- PDF | Request permission
Abstract:
Root subspaces for multiprameter eigenvalue problems are described using coalgebraic techniques. An algorithm is given to construct bases for the root subspaces.References
- Eiichi Abe, Hopf algebras, Cambridge Tracts in Mathematics, vol. 74, Cambridge University Press, Cambridge-New York, 1980. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka. MR 594432
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- A. Akižanov, Heat flow in a piecewise non-homogeneous two-dimensional medium. Proof of the existence theorem, Izv. Akad. Nauk Kazah. SSR Ser. Fiz.-Mat. Nauk 1964 (1964), no. 2, 43–47 (Russian, with Kazakh summary). MR 0173858
- F. V. Atkinson, Multiparameter spectral theory, Bull. Amer. Math. Soc. 74 (1968), 1–27. MR 220078, DOI 10.1090/S0002-9904-1968-11866-X
- 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, Multiparameter root vectors, Proc. Edinburgh Math. Soc. (2) 32 (1989), no. 1, 19–29. MR 981988, DOI 10.1017/S0013091500006866
- 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
- P.A. Binding and T. Košir. Second Root Vectors for Multiparameter Eigenvalue Problems of Fredholm Type, Trans. Amer. Math. Soc. 348: 229–249, 1996.
- P. A. Binding and K. Seddighi, Elliptic multiparameter eigenvalue problems, Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 2, 215–228. MR 892692, DOI 10.1017/S0013091500028297
- 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
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- G.A. Gadzhiev. Introduction to Multiparameter Spectral Theory. Azerbaijan State University, Baku, 1987. (Russian)
- L. Grunenfelder and T. Košir. Representation of Commuting Maps by Tensor Products, preprint.
- L. Grünenfelder and M. Omladič, Linearly recursive sequences and operator polynomials, Linear Algebra Appl. 182 (1993), 127–145. MR 1207079, DOI 10.1016/0024-3795(93)90496-B
- Luzius Grünenfelder and Robert Paré, Families parametrized by coalgebras, J. Algebra 107 (1987), no. 2, 316–375. MR 885801, DOI 10.1016/0021-8693(87)90093-7
- Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
- H.(G.A.) Isaev. Lectures on Multiparameter Spectral Theory. Dept. of Math. and Stats., University of Calgary, 1985.
- A. Källström and B. D. Sleeman, Solvability of a linear operator system, J. Math. Anal. Appl. 55 (1976), no. 3, 785–793. MR 417833, DOI 10.1016/0022-247X(76)90081-0
- Tomaž Košir, Finite-dimensional multiparameter spectral theory: the nonderogatory case, Proceedings of the 3rd ILAS Conference (Pensacola, FL, 1993), 1994, pp. 45–70. MR 1306972, DOI 10.1016/0024-3795(94)90396-4
- T. Košir. Commuting Matrices and Multiparameter Eigenvalue Problems. PhD thesis, Dept. of Math. and Stats., University of Calgary, 1993.
- John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
- Brian Peterson and Earl J. Taft, The Hopf algebra of linearly recursive sequences, Aequationes Math. 20 (1980), no. 1, 1–17. MR 569945, DOI 10.1007/BF02190488
- 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
- B.D. Sleeman. Review of [10]. SIAM Review , 34:684–687, 1992.
- Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
- Hans Volkmer, On the completeness of eigenvectors of right definite multiparameter problems, Proc. Roy. Soc. Edinburgh Sect. A 96 (1984), no. 1-2, 69–78. MR 741647, DOI 10.1017/S0308210500020473
- G. J. O. Jameson, Mixed summing norms and finite-dimensional Lorentz spaces, Geometric aspects of Banach spaces, London Math. Soc. Lecture Note Ser., vol. 140, Cambridge Univ. Press, Cambridge, 1989, pp. 112–124. MR 1035508, DOI 10.1017/CBO9780511662300.010
Bibliographic Information
- Luzius Grunenfelder
- Affiliation: Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
- Email: luzius@cs.dal.ca
- Tomaz Kosir
- Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, 61000 Ljubljana, Sloevenia
- Email: tkosir@fmf.uni-lj.si
- Received by editor(s): May 2, 1994
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 2983-2998
- MSC (1991): Primary 16W30, 35P10, 47A13
- DOI: https://doi.org/10.1090/S0002-9947-96-01679-0
- MathSciNet review: 1361640