Invariant subspace problem (Hilbert space)

Invariant subspace problem. Let $H$ be an infinite-dimensional, separable, complex Hilbert space and let $T\colon H\to H$ be a bounded linear operator. Then there is a closed linear subspace $S\subseteq H$ such that

The source asks this as a yes/no question ("Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?"); this item states the affirmative answer. A proof of the statement or a proof of its negation both settle the problem. It is one of the most famous open problems in functional analysis.

Formalization Note Complex scalars are used, as is standard for this problem. Separability is assumed because on non-separable spaces the answer is easily positive; infinite dimension is assumed because the finite-dimensional case is elementary. The zero operator is not excluded (it trivially satisfies the conclusion).