A solution of an operator equation related to the KdV equation

Abstract

For a given nonzero bounded linear operator A on a Banach space X, we show that if A or A ∗ has an eigenvalue then, except when the dimension of X is equal to two and the trace of A is zero, there exists a bounded linear operator B on X 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. This result was motivated by the operator method used by Carl et al. [H. Aden, B. Carl, On realizations of solutions of the KdV equation by determinants on operator ideals, J. Math. Phys. 37 (1996) 1833–1857; H. Blohm, Solution of nonlinear equations by trace methods, Nonlinearity 13 (2000) 1925–1964; B. Carl, C. Schiebold, Nonlinear equations in soliton physics and operator ideals, Nonlinearity 12 (1999) 333–364; B. Carl, S.-Z. Huang, On realizations of solutions of the KdV equation by the C 0-semigroup method, Amer. J. Math. 122 (2000) 403–438; S.-Z. Huang, An operator method for finding exact solutions to vector Korteweg–de Vries equations, J. Math. Phys. 44 (2003) 1357–1388] to solve nonlinear partial differential equations such as the Korteweg–deVries (KdV), modified KdV, and Kadomtsev–Petviashvili equations.