The invariant-subspace problem is probably an open problem for reflexive spaces which asks:



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> Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?
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> 
> 


This problem is closely related to properties of the orbits, which are
called $T$-cyclic subspaces.


On the other hand, Li–Yorke chaos is based on infinite-dimensional separable complex Banach spaces and bounded operators. My question here is: If the answer to the above question is *yes,* what are the consequences for Li–Yorke chaos?



Consequences of invariant-subspace problem to Li–Yorke chaos
Li Yorke chaos means there is an uncountable scrambled set. Linear operators can be Li-York chaotic (see the work of A. Peris and his collaborators, for example) and for the case of a linear operator $T:X \to X$, Li-York chaos is equivalent to there being an irregular vector: i.e., a vector $x\in X$ such that $\liminf\_n \| T^n x \|=0$ and $\limsup\_n \| T^n x \|=\infty$ (see Bermudez, Bonilla, Martinez-Gimenez and Peris, "Li-Yorke and distributionally chaotic operators", J. Math. Anal. Appl. 373 (2011) 83-93).


All that being said, I am aware of no relation of this to the ISP (but maybe there is one?). In general, I know of no really interesting result that depends on the outcome of the ISP. A lot of work has been put into answering this question: it is maddening that we do not know the answer for such a simple query! On the other hand, a lot of interesting techniques have been developed in the search for an answer. I recommend Radjavi and Rosenthal's classic book "Invariant Subspaces" (the second, Dover, edition, has an updated version of the development of the field).


