An operator equation, KdV equation and invariant subspaces

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.