Editing Nonstandard analysis (section)

== Invariant subspace problem ==
Abraham Robinson and Allen Bernstein used nonstandard analysis to prove that every polynomially compact [[linear operator]] on a [[Hilbert space]] has an [[invariant subspace]].<ref>Allen Bernstein and Abraham Robinson, ''[https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-16/issue-3/Solution-of-an-invariant-subspace-problem-of-K-T-Smith/pjm/1102994835.full Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos]'', Pacific Journal of Mathematics 16:3 (1966) 421-431</ref>

Given an operator {{mvar|T}} on Hilbert space {{mvar|''H''}}, consider the orbit of a point {{mvar|v}} in {{mvar|H}} under the iterates of {{mvar|T}}. Applying Gram–Schmidt one obtains an orthonormal basis {{math|(''e<sub>i</sub>'')}} for {{mvar|H}}. Let {{math|(''H<sub>i</sub>'')}} be the corresponding nested sequence of "coordinate" subspaces of {{mvar|H}}. The matrix {{math|''a<sub>i,j</sub>''}} expressing {{mvar|T}} with respect to {{math|(''e<sub>i</sub>'')}} is almost upper triangular, in the sense that the coefficients {{math|''a''<sub>''i''+1,''i''</sub>}} are the only nonzero sub-diagonal coefficients. Bernstein and Robinson show that if {{mvar|T}} is polynomially compact, then there is a hyperfinite index {{mvar|w}} such that the matrix coefficient {{math|''a''<sub>''w''+1,''w''</sub>}} is infinitesimal. Next, consider the subspace {{math|''H<sub>w</sub>''}} of {{math|*''H''}}. If {{mvar|y}} in {{math|''H<sub>w</sub>''}} has finite norm, then {{math|''T''(''y'')}} is infinitely close to {{math|''H<sub>w</sub>''}}.

Now let {{math|''T<sub>w</sub>''}} be the operator <math>P_w \circ T</math> acting on {{math|''H<sub>w</sub>''}}, where {{math|''P<sub>w</sub>''}} is the orthogonal projection to {{math|''H<sub>w</sub>''}}. Denote by {{mvar|q}} the polynomial such that {{math|''q''(''T'')}} is compact. The subspace {{math|''H<sub>w</sub>''}} 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 {{math|(''e<sub>k</sub>'')}} for {{math|''H<sub>w</sub>''}} where {{mvar|k}} runs from {{math|1}} to {{mvar|w}}, such that each of the corresponding {{mvar|k}}-dimensional subspaces {{math|''E<sub>k</sub>''}} is {{mvar|T}}-invariant. Denote by {{math|Π<sub>''k''</sub>}} the projection to the subspace {{math|''E<sub>k</sub>''}}. For a nonzero vector {{mvar|x}} of finite norm in {{mvar|H}}, one can assume that {{math|''q''(''T'')(''x'')}} is nonzero, or {{math|{{!}}''q''(''T'')(''x''){{!}} > 1}} to fix ideas. Since {{math|''q''(''T'')}} is a compact operator, {{math|(''q''(''T<sub>w</sub>''))(''x'')}} is infinitely close to {{math|''q''(''T'')(''x'')}} and therefore one has also {{math|{{!}}''q''(''T<sub>w</sub>'')(''x''){{!}} > 1}}. Now let {{mvar|j}} be the greatest index such that <math display=inline>\left|q(T_w) \left (\prod_j(x) \right)\right|<\tfrac{1}{2}</math>. Then the space of all standard elements infinitely close to {{math|''E<sub>j</sub>''}} is the desired invariant subspace.

Upon reading a preprint of the Bernstein and Robinson paper, [[Paul Halmos]] reinterpreted their proof using standard techniques.<ref>P. Halmos, ''[https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-16/issue-3/Invariant-subspaces-of-polynomially-compact-operators/pjm/1102994836.full Invariant subspaces for Polynomially Compact Operators]'', Pacific Journal of Mathematics, 16:3 (1966) 433-437.</ref> 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.