Operator Theory ¹

The Invariant Subspace Problem

Few questions in functional analysis have attracted sustained attention across as many decades as this one. It sits at the confluence of operator theory, spectral theory, and complex analysis, and every partial result has opened new territory rather than narrowing the problem to a routine case. Problem (Invariant Subspace Problem) Does every bounded linear operator $T$ on an infinite-dimensional separable complex Hilbert space $\mathcal{H}$ have a non-trivial closed invariant subspace?