An operator equation, KdV equation and invariant subspaces
| Autor: | V. Hrynkiv, A. R. Sourour, Ramesh V. Garimella |
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| Rok vydání: | 2009 |
| Předmět: |
Pure mathematics
Applied Mathematics General Mathematics Mathematical analysis Invariant subspace Spectrum (functional analysis) Finite-rank operator Reflexive operator algebra Compact operator Strictly singular operator Bounded operator Nonlinear Sciences::Exactly Solvable and Integrable Systems Nonlinear Sciences::Pattern Formation and Solitons Invariant subspace problem Mathematics |
| Zdroj: | Proceedings of the American Mathematical Society. 138:717-724 |
| ISSN: | 0002-9939 |
| DOI: | 10.1090/s0002-9939-09-10118-1 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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