[FOM] replies to two nonstandard postings

David Ross ross at math.hawaii.edu
Wed Nov 7 01:35:40 EST 2007

A J Franco de Oliveira <francoli at kqnet.pt> wrote:

> The answer is yes, in E. Nelson's IST

Joe's question does not even make sense in IST.  He is asking about
unique factorization of the ring as viewed _externally_ (in NSA
terminology), it is obviously true internally (as he has explained in
another posting).  IST supresses the internal/external distinction.

Gabriel Stolzenberg <gstolzen at math.bu.edu> asked:

>    Question.  What is the name of the famous conjecture in analysis
> whose proof by Abraham Robinson is sometimes offered as a
> of the power of nonstandard analysis?

This was the Bernstein-Robinson proof of the Smith-Halmos Invariant
Subspace Problem:  if T is a linear operator on a (complex) Hilbert
space H, and T^n is compact for some n, then T has a nontrivial
invariant closed subspace.  (For n=1, this was due to Aronszajn and
Smith, and the conjecture for larger n was asked by Halmos.)  A few
years later Lomonosov gave a much shorter, wonderfully clever standard
proof under the much weaker hypothesis that T commutes with _some_
compact operator.  I think that the existence of such an invariant
subspace for a general bounded linear operator is still open.

Prof. David A. Ross
Dept. of Mathematics, University of Hawaii
Group in Logic and Lattices:

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