Invariant subspace problem

The Invariant Subspace Problem is a significant open question in functional analysis, a branch of mathematics. It concerns the existence of invariant subspaces for bounded linear operators on a Hilbert space. Specifically, the problem asks whether every bounded linear operator on an infinite-dimensional separable Hilbert space has a non-trivial closed invariant subspace. An invariant subspace for an operator \( T \) is a subspace \( M \) such that \( T(M) \subseteq M \).