invariant subspace problem

Initially formulated for Banach spaces, the invariant subspace conjecture stated the following:

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 $S\mathrm{\subset }X$ such that $S\mathrm{\ne }\mathrm{0}$, $S\mathrm{\ne }X$ and $T\mathit{}\mathrm{(}S\mathrm{)}\mathrm{\subseteq }S$.

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 $S\mathrm{\subset }H$ such that $S\mathrm{\ne }\mathrm{0}$, $S\mathrm{\ne }H$ and $T\mathit{}\mathrm{(}S\mathrm{)}\mathrm{\subseteq }S$.