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Topic: Invariant subspace problem solved?
Replies: 9   Last Post: Mar 6, 2013 7:34 AM

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David Bernier

Posts: 3,187
Registered: 12/13/04
Re: Invariant subspace problem solved?
Posted: Mar 6, 2013 7:34 AM
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On 02/28/2013 08:12 PM, David Bernier wrote:
> On 02/28/2013 07:56 PM, David Bernier wrote:
[...]

>> I was reading on problems considered "modulo a compact
>> operator K".
>>
>> If T is a bounded operator on a separable, infinite
>> dimensional Hilbert space H,
>>
>> can one show that for some compact operator K
>> on H, T+K has a non-trivial invariant
>> subspace?

> [...]
>
> Ok. I think we can find a finite-rank K such that
> T+K has some non-zero eigenvector, i.e.
> one can construct K such that for some v in H,
>
> T+K maps v to a scalar multiple of v.

[...]

Alexey I. Popov and Adi Tcaciuc have an arxiv pre-print
implying that on a separable infinite-dimensional
Hilbert space, any operator T can be perturbed by
a rank-one operator K such that
T + K has an invariant subspace of infinite dimension and codimension.

They prove more than that.

Cf. the abstract here:

http://arxiv.org/abs/1208.5831

David Bernier
--
dracut:/# lvm vgcfgrestore
File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID
993: sh
Please specify a *single* volume group to restore.



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