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One of the most notorious open problems in functional analysis is the invariant subspace problem for Hilbert spaces, which I will state here as a conjecture:
Conjecture 1 (Invariant Subspace Problem, ISP0) Let be an infinite dimensional complex Hilbert space, and let be a bounded linear operator. Then contains a proper closed invariant subspace (thus ).
As stated this conjecture is quite infinitary in nature. Just for fun, I set myself the task of trying to find an equivalent reformulation of this conjecture that only involved finite-dimensional spaces and operators. This turned out to be somewhat difficult, but not entirely impossible, if one adopts a sufficiently generous version of “finitary” (cf. my discussion of how to finitise the infinitary pigeonhole principle). Unfortunately, the finitary formulation that I arrived at ended up being rather complicated (in particular, involving the concept of a “barrier”), and did not obviously suggest a path to resolving the conjecture; but it did at least provide some simpler finitary consequences of the conjecture which might be worth focusing on as subproblems.
I should point out that the arguments here are quite “soft” in nature and are not really addressing the heart of the invariant subspace problem; but I think it is still of interest to observe that this problem is not purely an infinitary problem, and does have some non-trivial finitary consequences.
I am indebted to Henry Towsner for many discussions on this topic.
This month I am at MSRI, for the programs of Ergodic Theory and Additive Combinatorics, and Analysis on Singular Spaces, that are currently ongoing here. This week I am giving three lectures on the correspondence principle, and on finitary versions of ergodic theory, for the introductory workshop in the former program. The article here is broadly describing the content of these talks (which are slightly different in theme from that announced in the abstract, due to some recent developments). [These lectures were also recorded on video and should be available from the MSRI web site within a few months.]