This paper is a comment on the
solution of an invariant subspace problem by A. R. Bernstein and A. Robinson [2].
The theorem they prove can be stated as follows: if A is an operator on a Hilbert
space H of dimension greater than 1, and if p is a nonzero polynomial such that p(A)
is compact, then there exists a nontrivial subspace of H invariant under A.
(“Operator” means bounded linear transformation; “Hilbert space” means complete
complex inner product space; “compact” means completely continuous; “subspace”
means closed linear manifold; “nontrivial”, for subspaces, means distinct from
{0} and from H.) The Bernstein-Robinson proof has two aspects: it is an
ingenious adaptation of the proof by N. Aronszajn and K. T. Smith of the
corresponding theorem for compact operators [1], and it makes strong use of
metamathematical concepts such as nonstandard models of higher order
predicate languages. The purpose of this paper is to show that by appropriate
small modifications the Bernstein-Robinson proof can be converted (and
shortened) into one that is expressible in the standard framework of classical
analysis.
