# Invariant subspace problem

\begin{problem} Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace? \end{problem}

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Let $H$ be a Hilbert space. The subspaces $\{0\}$ and $H$ are trivially invariant under any linear operator on $H$, and so these are referred to as the trivial invariant subspaces. The problem is concerned with determining whether bounded operators necessarily have non-trivial invariant subspaces.

This is one of the most famous open problems in functional analysis. Enflo [1] constructed Banach spaces for which the corresponding question has a negative answer, and recently Argyros and Haydon constructed a Banach space for which the corresponding question has a positive answer [4].

For a nice overview to the problem see [2], [3] or [5].

## Bibliography

[1] P. Enflo, On the invariant subspace problem for Banach spaces, Acta Math. 158 (1987), 213-313. \MRhref{0892591}

[2] B. S. Yadav, The Present State and Heritages of the Invariant Subspace Problem, Milan J. Math. 73 (2005), 289-316. \MRhref{2175046} \href[another link]{http://www.math.leidenuniv.nl/~naw/serie5/deel06/jun2005/pdf/yadav.pdf}

[3] \href[H. Radjavi]{http://www.math.uwaterloo.ca/PM_Dept/Homepages/Radjavi/radjavi.shtml} and \href[P. Rosenthal]{http://www.math.toronto.edu/rosent/}, The Invariant Subspace Problem, The Mathematics Intelligencer 4 (1982), no. 1, 33-37. \MRhref{0678734}

[4] S. A. Argyros and R. G. Haydon, A hereditarily indecomposable $L_\infty$-space that solves the scalar-plus-compact problem, \arxiv{0903.3921} (2009).

[5] \href[J. Noel]{http://www.math.mcgill.ca/jnoel}. The Invariant Subspace Problem. Honours Thesis, Thompson Rivers University. \href[Link to pdf]{http://www.math.mcgill.ca/jnoel/pdf/Honours.pdf}.

* indicates original appearance(s) of problem.

## closed

Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial

closedinvariant subspace?