鲁宾逊不变子空间存在定理
(2019-08-30 20:11:36)鲁宾逊不变子空间存在定理
袁萌
附件:
Invariant subspace problem
Abraham Robinson and Allen Bernstein used non-standard analysis to prove that every polynomially compact linear operator on a Hilbert space has an invariant subspace.[16]
Given an operator T on Hilbert space H, consider the orbit of a point v in H under the iterates of T. Applying Gram-Schmidt one obtains an orthonormal basis (ei) for H. Let (Hi) be the corresponding nested sequence of "coordinate" subspaces of H. The matrix ai,j expressing T with respect to (ei) is almost upper triangular, in the sense that the coefficients ai+1,i are the only nonzero sub-diagonal coefficients. Bernstein and Robinson show that if T is polynomially compact, then there is a hyperfinite index w such that the matrix coefficient aw+1,w is infinitesimal. Next, consider the subspace Hw of *H. If y in Hw has finite norm, then T(y) is infinitely close to Hw.
Now let Tw be the operator
P w
T
| q ( T w ) ( Π j ( x ) ) | < 1 2 {\displaystyle |q(T_{w})\left(\Pi _{j}(x)\right)|<{\tfrac {1}{2}}}
. Then the space of all standard elements infinitely close to Ej is the desired invariant subspace.
Upon reading a preprint of the Bernstein-Robinson paper, Paul Halmos reinterpreted their proof using standard techniques.[17] 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.