The hyperinvariant subspace problem asks whether an operator has a nonzero proper hyperinvariant subspace, with the class of operators specified as part of the question. In particular, one formulation asks whether every nonzero quasinilpotent operator on an infinite-dimensional separable complex Hilbert space has such a subspace. This requires invariance under the full commutant, a stronger condition than invariance under the operator alone.
Hyperinvariant Subspace Problem
See also
Commutant, Hyperinvariant Subspace, Invariant Subspace, Invariant Subspace Problem, Quasinilpotent OperatorExplore with Wolfram|Alpha
References
Chalendar, I. and Partington, J. R. "Recent Perspectives on the Invariant Subspace Problem." 29 Jul 2025. https://arxiv.org/abs/2507.21834.Kim, H. J. "The Hyperinvariant Subspace Problem for Quasinilpotent Operators." https://www.math.sci.hokudai.ac.jp/coe/sympo/070223/abst/H.J.Kim.pdf.Cite this as:
Weisstein, Eric W. "Hyperinvariant Subspace Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperinvariantSubspaceProblem.html