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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
Published electronically: September 29, 2009
MathSciNet review: 2557188
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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.

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Additional Information

R. V. Garimella
Affiliation: Department of Mathematics, University of Central Arkansas, Conway, Arkansas 72035

V. Hrynkiv
Affiliation: Department of Computer and Mathematical Sciences, University of Houston- Downtown, Houston, Texas 77002

A. R. Sourour
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3P4 Canada

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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