Invariant subspace problem

The invariant subspace problem asks whether every bounded linear operator on an infinite-dimensional complex Hilbert space has a closed invariant subspace other than zero and the whole space.

Per Enflo's invariant subspace counterexample: A construction of a bounded operator on a Banach space with no nontrivial closed invariant subspace. It settled the broader Banach-space version, but not the Hilbert-space question.
Invariant subspace: A linear subspace mapped into itself by a specified operator. This is the structure whose existence the problem asks about.
John von Neumann: A mathematician whose work helped establish the foundations of modern functional analysis and quantum mechanics. His work on operators helped shape the setting in which the problem emerged.
Invariant subspace theorem for compact operators: The theorem that every nonzero compact operator on an infinite-dimensional complex Banach space has a nontrivial closed invariant subspace. Compact operators satisfy a positive result, unlike the unrestricted class in the problem.
Read's invariant subspace counterexample: A bounded operator on a Banach space constructed to have no nontrivial closed invariant subspaces. It gave a striking later solution for a particular Banach space.
Bounded linear operator: A linear map between normed spaces whose output norm is bounded by a constant times the input norm. The problem quantifies over this class of transformations.
Paul Halmos: An influential twentieth-century mathematician known for contributions to operator theory and exposition. His writing helped frame the invariant subspace question as a central operator-theory problem.
Invariant subspace theorem for normal operators: The result that normal operators on complex Hilbert spaces have nontrivial closed invariant subspaces unless they are scalar multiples of the identity. Normality gives a broad Hilbert-space case where the sought subspace is guaranteed.
Enflo's Hilbert-space invariant subspace result: A result establishing invariant subspaces for certain operators on Hilbert spaces under additional conditions. It illustrates progress that falls short of resolving the unrestricted problem.
Closed subspace: A subspace containing the limits of all its convergent sequences in the ambient normed space. Closedness distinguishes the sought subspaces from merely algebraic invariant ones.

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