Vol. 16, No. 3, 1966

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Invariant subspaces of polynomially compact operators

P. R. Halmos

Vol. 16 (1966), No. 3, 433–437

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.

Mathematical Subject Classification
Primary: 47.35
Received: 10 October 1964
Published: 1 March 1966
P. R. Halmos