An operator equation, KdV equation and invariant subspaces
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- by R. V. Garimella, V. Hrynkiv and A. R. Sourour PDF
- Proc. Amer. Math. Soc. 138 (2010), 717-724 Request permission
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
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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
- 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
- © Copyright 2009 American Mathematical Society
- 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
- MathSciNet review: 2557188