Non-standard Analysis - Invariant Subspace Problem

Invariant Subspace Problem

Abraham Robinson and Allen Bernstein proved that every polynomially compact linear operator on a Hilbert space has an invariant subspace.

Given an operator on Hilbert space, consider the orbit of a point under the iterates of . Applying Gram-Schmidt one obtains an orthonormal basis for . Let be the corresponding nested sequence of "coordinate" subspaces of . The matrix expressing with respect to is almost upper triangular, in the sense that the coefficients are the only nonzero subdiagonal coefficients. Bernstein and Robinson show that if is polynomially compact, then there is a hyperfinite index such that the matrix coefficient is infinitesimal. Next, consider the subspace of . If has finite norm, then is infinitely close to .

Now let be the operator acting on, where is the orthogonal projection to . Denote by the polynomial such that is compact. The subspace is internal of hyperfinite dimension. By transferring upper triangularisation of operators of finite-dimensional complex vector space, there is an internal orthonormal Hilbert space basis for where runs from to, such that each of the corresponding -dimensional subspaces is -invariant. Denote by the projection to the subspace . For a nonzero vector of finite norm in, one can assume that is nonzero, or to fix ideas. Since is a compact operator, is infinitely close to and therefore one has also . Now let be the greatest index such that . Then the space of all standard elements infinitely close to is the desired invariant subspace.

Upon reading a preprint of the Bernstein-Robinson paper, Paul Halmos reinterpreted their proof using standard techniques. Both papers appeared back-to-back in the same issue of the Pacific Journal of Mathematics. Some of the ideas used in Halmos' proof reappeared many years later in Halmos' own work on quasi-triangular operators.