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

Full-text PDF

Abstract | References | Similar Articles | 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 Korteweg-deVries (KdV), modified KdV, sine-Gordon, and Kadomtsev-Petviashvili (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 non-trivial invariant subspace.

**1.**H. Aden and B. Carl,*On realizations of solutions of the KdV equation by determinants on operator ideals*, J. Math. Phys.**37**(1996), no. 4, 1833–1857. MR**1380877**, https://doi.org/10.1063/1.531482**2.**Wolfgang Arendt, Frank Räbiger, and Ahmed Sourour,*Spectral properties of the operator equation 𝐴𝑋+𝑋𝐵=𝑌*, Quart. J. Math. Oxford Ser. (2)**45**(1994), no. 178, 133–149. MR**1280689**, https://doi.org/10.1093/qmath/45.2.133**3.**Rajendra Bhatia,*Matrix analysis*, Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997. MR**1477662****4.**Henning Blohm,*Solution of nonlinear equations by trace methods*, Nonlinearity**13**(2000), no. 6, 1925–1964. MR**1794840**, https://doi.org/10.1088/0951-7715/13/6/304**5.**B. Carl and C. Schiebold,*Nonlinear equations in soliton physics and operator ideals*, Nonlinearity**12**(1999), no. 2, 333–364. MR**1677783**, https://doi.org/10.1088/0951-7715/12/2/012**6.**Bernd Carl and Sen-Zhong Huang,*On realizations of solutions of the KdV equation by the 𝐶₀-semigroup method*, Amer. J. Math.**122**(2000), no. 2, 403–438. MR**1749054****7.**John B. Conway,*A course in functional analysis*, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR**1070713****8.**Ramesh V. Garimella, Volodymyr Hrynkiv, and A. R. Sourour,*A solution of an operator equation related to the KdV equation*, Linear Algebra Appl.**418**(2006), no. 2-3, 788–792. MR**2260229**, https://doi.org/10.1016/j.laa.2006.03.008**9.**Sen-Zhong Huang,*An operator method for finding exact solutions to vector Korteweg-de Vries equations*, J. Math. Phys.**44**(2003), no. 3, 1357–1388. MR**1958271**, https://doi.org/10.1063/1.1544414**10.**Carlos S. Kubrusly,*Elements of operator theory*, Birkhäuser Boston, Inc., Boston, MA, 2001. MR**1839920****11.**Gunter Lumer and Marvin Rosenblum,*Linear operator equations*, Proc. Amer. Math. Soc.**10**(1959), 32–41. MR**0104167**, https://doi.org/10.1090/S0002-9939-1959-0104167-0**12.**Heydar Radjavi and Peter Rosenthal,*Invariant subspaces*, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77. MR**0367682****13.**Cornelia Schiebold,*Explicit solution formulas for the matrix-KP*, Glasg. Math. J.**51**(2009), no. A, 147–155. MR**2481233**, https://doi.org/10.1017/S0017089508004862

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47A62,
46N20,
47N20

Retrieve articles in all journals with MSC (2000): 47A62, 46N20, 47N20

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