[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
demonstration
> 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:
http://www.math.hawaii.edu/~ross/LogicLattices.htm
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