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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

   数学守旧派认为,非标准数学能够证明的定理,标准数学也能够证明。这是数学守旧派的“挡箭牌”。

   实际情况是:1966年,;鲁宾逊关于希尔伯特空间的一类紧致线性算子存在不变子空间定理的证明比标准方法的证明要早了好几年。数学守旧派的“挡箭牌”,不攻自破。

   对此有兴趣的读者,请见本文附件文章。

袁萌  陈启清   830

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  

 acting on Hw, where Pw is the orthogonal projection to Hw. Denote by q the polynomial such that q(T) is compact. The subspace Hw 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 (ek) for Hw where k runs from 1 to w, such that each of the corresponding k-dimensional subspaces Ek is T-invariant. Denote by Πk the projection to the subspace Ek. For a nonzero vector x of finite norm in H, one can assume that q(T)(x) is nonzero, or |q(T)(x)| > 1 to fix ideas. Since q(T) is a compact operator, (q(Tw))(x) is infinitely close to q(T)(x) and therefore one has also |q(Tw)(x)| > 1. Now let j be the greatest index such that

| 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.

 


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