# invariant subspace problem

Let $X$ be a complex Banach space. Then every bounded operator   $T$ in $X$ has a non-trivial closed (http://planetmath.org/ClosedSet) invariant subspace  , i.e. there exists a closed vector subspace $S\subset X$ such that $S\neq 0$, $S\neq X$ and $T(S)\subseteq S$.

This conjecture was proven to be false when P. Enflo (1975) and . Read (1984) gave examples of bounded operators which did not have the above property.

However, if one considers only Hilbert spaces  , this is still an open problem. Today the invariant subspace conjecture is formulated as follows:

Let $H$ be a complex Hilbert space. Then every bounded operator $T$ in $H$ has a non-trivial invariant subspace, i.e. there exists a closed vector subspace $S\subset H$ such that $S\neq 0$, $S\neq H$ and $T(S)\subseteq S$.

Title invariant subspace problem InvariantSubspaceProblem 2013-03-22 17:24:02 2013-03-22 17:24:02 asteroid (17536) asteroid (17536) 5 asteroid (17536) Conjecture msc 47A15 msc 46-00 invariant subspace conjecture