In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. The original form of the problem as posed by Paul Halmos was in the special case of polynomials with compact square. This was resolved affirmatively, for a more general class of polynomially compact operators, by Allen R. Bernstein and Abraham Robinson in 1966 (see Non-standard analysis#Invariant subspace problem for a summary of the proof).
More formally, the invariant subspace problem for a complex Banach space H of dimension > 1 is the question whether every bounded linear operator T : H → H has a non-trivial closed T-invariant subspace (a closed linear subspace W of H which is different from {0} and H such that T(W) ⊆ W).
To find a counterexample to the invariant subspace problem means to answer affirmatively the following equivalent question: Does there exist a bounded linear operator T : H → H such that for every non-zero vector x, the vector space generated by the sequence {T n(x) : n ≥ 0} is norm dense in H? Such operators are called cyclic.
The problem seems to have been stated in the mid-1900s after work by Beurling and von Neumann.
For Banach spaces, the first example of an operator without an invariant subspace was constructed by Enflo. (For Hilbert spaces, the invariant subspace problem remains open.)
Per Enflo proposed a counterexample to the invariant subspace problem in 1975, publishing an outline in 1976. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987 Enflo's long "manuscript had a world-wide circulation among mathematicians" and some of its ideas were described in publications besides Enflo (1976). Enflo's works inspired a similar construction of an operator without an invariant subspace for example by Beauzamy, who acknowledged Enflo's ideas.
In the 1990s, Enflo developed a "constructive" approach to the invariant subspace problem on Hilbert spaces.
While the general case of the invariant subspace problem is still open, several special cases have been settled for topological vector spaces (over the field of complex numbers):
For finite-dimensional complex vector spaces of dimension greater than two every operator admits an eigenvector, so it has a 1-dimensional invariant subspace.
The conjecture is true if the Hilbert space H is not separable (i.e. if it has an uncountable orthonormal basis). In fact, if x is a non-zero vector in H, the norm closure of the vector space generated by the infinite sequence {T n(x) : n ≥ 0} is separable and hence a proper subspace and also invariant.
von Neumann showed that any compact operator on a Hilbert space of dimension at least 2 has a non-trivial invariant subspace.
The spectral theorem shows that all normal operators admit invariant subspaces.
Aronszajn & Smith (1954) proved that every compact operator on any Banach space of dimension at least 2 has an invariant subspace.
Bernstein & Robinson (1966) proved using non-standard analysis that if the operator T on a Hilbert space is polynomially compact (in other words P(T) is compact for some non-zero polynomial P) then T has an invariant subspace. Their proof uses the original idea of embedding the infinite-dimensional Hilbert space in a hyperfinite-dimensional Hilbert space (see Non-standard analysis#Invariant subspace problem).
Halmos (1966), after having seen Robinson's preprint, eliminated the non-standard analysis from it and provided a shorter proof in the same issue of the same journal.
Lomonosov (1973) gave a very short proof using the Schauder fixed point theorem that if the operator T on a Banach space commutes with a non-zero compact operator then T has a non-trivial invariant subspace. This includes the case of polynomially compact operators because an operator commutes with any polynomial in itself. More generally, he showed that if S commutes with a non-scalar operator T that commutes with a non-zero compact operator, then S has an invariant subspace.
The first example of an operator on a Banach space with no invariant subspaces was found by Per Enflo (1976, 1987), and his example was simplified by Beauzamy (1985).
The first counterexample on a "classical" Banach space was found by Charles Read (1984, 1985), who described an operator on the classical Banach space l1 with no invariant subspaces.
Later Charles Read (1988) constructed an operator on l1 without even a non-trivial closed invariant subset, that is, with every vector hypercyclic, solving in the negative the invariant subset problem for the class of Banach spaces.
Atzmon (1983) gave an example of an operator without invariant subspaces on a nuclear Fréchet space.
Śliwa (2008) proved that any infinite dimensional Banach space of countable type over a non-Archimedean field admits a bounded linear operator without a non-trivial closed invariant subspace. This completely solves the non-Archimedean version of this problem, posed by van Rooij and Shikhof in 1992.
Argyros & Haydon (2009) gave the construction of an infinite-dimensional Banach space such that every continuous operator is the sum of a compact operator and a scalar operator, so in particular every operator has an invariant subspace.
