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), 717724
MSC (2000):
Primary 47A62, 46N20, 47N20
Published electronically:
September 29, 2009
MathSciNet review:
2557188
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Additional Information
Abstract: Let be a bounded linear operator on a complex Banach space . A problem, motivated by the operator method used to solve integrable systems such as the KortewegdeVries (KdV), modified KdV, sineGordon, and KadomtsevPetviashvili (KP) equations, is whether there exists a bounded linear operator such that (i) is of rank one, and (ii) is invertible for every function analytic in a neighborhood of the spectrum of . 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 does not contain 0, we show that there exists a solution to (i) and (ii) if and only if has a nontrivial invariant subspace.
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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:
http://dx.doi.org/10.1090/S0002993909101181
PII:
S 00029939(09)101181
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
