# Invariant subspace problem

(Redirected from Invariant subspace conjecture)

In the field of mathematics known as functional analysis, 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.

For the most important case of Hilbert spaces H the problem remains open (as of 2013), though Per Enflo found the first example of a Banach space operator with no invariant subspace.

## Known special cases

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 non-zero finite-dimensional vector spaces 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.[1]
• 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) announced 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.

## Notes

1. ^ See Pearcy & Shields (1974) for a review.