A hyperinvariant subspace of a bounded operator 
on a Hilbert space 
is a closed linear subspace
invariant under every bounded operator commuting
with 
.
Thus 
is hyperinvariant when 
for every 
in the commutant 
. Because 
belongs to its own commutant,
a hyperinvariant subspace is an invariant subspace
of 
.
The converse need not hold. A commutant is transitive
when it has no common nonzero proper closed invariant subspace; in that case 
has no nontrivial hyperinvariant subspace.
Hyperinvariant Subspace
See also
Commutant, Hyperinvariant Subspace Problem, Invariant SubspaceExplore 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." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperinvariantSubspace.html