An invariant subspace theorem of J. Feldman
T. Alastair Gillespie · Pacific Journal of Mathematics · 1968
Theorem.Let t be a quasi-nilpotent bounded linear operator on a complex normed space X of dimension greater than one.Suppose further that there is a sequence \p n (t)} of polynomials in t and a nonzero compact operator s on X such that Pn(t)-> s (in norm) as n-> oo.Then t has a proper closed invariant subspace.In [3], Feldman proves this theorem in the case when X is a Hubert space.By adapting the proof given by Bonsall [2, Theorem (20.1)] of the Bernstein-Robinson invariant subspace theorem [1], the result can be shown to hold when X is a normed space, the necessary changes in the proof given in [2] being suggested by [3].For the sake of completeness, the proof below repeats the relevant arguments in [2], We need the following notation and simple results.(i) If E is a nonempty subset of X and xe X, the distance from x to E, d(x, E), is defined by d(x, E) = mί{\\x -y\\:yeE} .