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An operator equation, KdV equation and invariant subspaces


Authors: R. V. Garimella, V. Hrynkiv and A. R. Sourour
Journal: Proc. Amer. Math. Soc. 138 (2010), 717-724
MSC (2000): Primary 47A62, 46N20, 47N20
DOI: https://doi.org/10.1090/S0002-9939-09-10118-1
Published electronically: September 29, 2009
MathSciNet review: 2557188
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A$ be a bounded linear operator on a complex Banach space $ X$. A problem, motivated by the operator method used to solve integrable systems such as the Korteweg-deVries (KdV), modified KdV, sine-Gordon, and Kadomtsev-Petviashvili (KP) equations, is whether there exists a bounded linear operator $ B$ such that (i) $ AB+BA$ is of rank one, and (ii) $ (I+f(A)B)$ is invertible for every function $ f$ analytic in a neighborhood of the spectrum of $ A$. We investigate solutions to this problem and discover an intriguing connection to the invariant subspace problem. Under the assumption that the convex hull of the spectrum of $ A$ does not contain 0, we show that there exists a solution $ B$ to (i) and (ii) if and only if $ A$ has a non-trivial invariant subspace.


References [Enhancements On Off] (What's this?)

  • 1. H. Aden and B. Carl, On realizations of solutions of the KdV equation by determinants on operator ideals, J. Math. Phys. 37 (1996), 1833-1857. MR 1380877 (96m:58099)
  • 2. W. Arendt, F. Räbiger, and A. Sourour, Spectral properties of the operator equation $ AX+XB=Y$, Quart. J. Math, Oxford Series (2) 45 (1994), no. 178, 133-149. MR 1280689 (95g:47060)
  • 3. R. Bhatia, Matrix analysis, Springer-Verlag, New York, 1997. MR 1477662 (98i:15003)
  • 4. H. Blohm, Solution of nonlinear equations by trace methods, Nonlinearity 13 (2000), 1925-1964. MR 1794840 (2001i:37113)
  • 5. B. Carl and C. Schiebold, Nonlinear equations in soliton physics and operator ideals, Nonlinearity 12 (1999), 333-364. MR 1677783 (2000e:37129)
  • 6. B. Carl and S.-Z. Huang, On realizations of solutions of the KdV equation by the $ C_0$-semigroup method, Amer. J. Math. 122 (2000), 403-438. MR 1749054 (2001k:37107)
  • 7. J. Conway, A course in functional analysis, 2nd ed., Springer-Verlag, New York, 1990. MR 1070713 (91e:46001)
  • 8. R. Garimella, V. Hrynkiv, and A. R. Sourour, A solution of an operator equation related to the KdV equation, Linear Algebra Appl. 418 (2006), 788-792. MR 2260229 (2007e:47025)
  • 9. S.-Z. Huang, An operator method for finding exact solutions to vector Korteweg-de Vries equations, J. Math. Phys. 44 (2003), 1357-1388. MR 1958271 (2003m:37101)
  • 10. C. Kubrusly, Elements of operator theory, Birkhäuser, Boston, 2001. MR 1839920 (2002c:47001)
  • 11. G. Lumer and M. Rosenblum, Linear operator equations, Proc. Amer. Math. Soc. 10 (1959), 32-41. MR 0104167 (21:2927)
  • 12. H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, New York, 1973. MR 0367682 (51:3924)
  • 13. C. Schiebold, Explicit solution formula for the matrix-KP, Glasgow Math. J. 51 (2009), 147-155. MR 2481233

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Additional Information

R. V. Garimella
Affiliation: Department of Mathematics, University of Central Arkansas, Conway, Arkansas 72035
Email: rameshg@uca.edu

V. Hrynkiv
Affiliation: Department of Computer and Mathematical Sciences, University of Houston- Downtown, Houston, Texas 77002
Email: HrynkivV@uhd.edu

A. R. Sourour
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3P4 Canada
Email: sourour@math.uvic.ca

DOI: https://doi.org/10.1090/S0002-9939-09-10118-1
Keywords: Operator equation, invariant subspaces, rank one operators, spectrum
Received by editor(s): December 2, 2008
Received by editor(s) in revised form: July 10, 2009
Published electronically: September 29, 2009
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society

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