The problem seems to have been stated in the mid-1900s after work by Beurling and von Neumann, who found a positive solution for the case of compact operators. It was then posed by Paul Halmos for the case of operators such that is compact. This was resolved affirmatively, for the more general class of polynomially compact operators, by Allen R. Bernstein and Abraham Robinson in 1966. For Banach spaces, the first example of an operator without an invariant subspace was constructed by Per Enflo. He 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. 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.
Precise statement
Formally, the invariant subspace problem for a complex Banach space of dimension > 1 is the question whether every bounded linear operator has a non-trivial closed -invariant subspace: a closed linear subspace of, which is different from and from, such that. A negative answer to the problem is closely related to properties of the orbits. If is an element of the Banach space, the orbit of under the action of, denoted by, is the subspace generated by the sequence. This is also called the -cyclic subspace generated by. From the definition it follows that is a -invariant subspace. Moreover, it is the minimal -invariant subspace containing : if is another invariant subspace containing, then necessarily for all , and so. If is non-zero, then is not equal to, so its closure is either the whole space or it is a non-trivial -invariant subspace. Therefore, a counterexample to the invariant subspace problem would be a Banach space and a bounded operator for which every non-zero vector is a cyclic vector for.
Known special cases
While the case of the invariant subspace problem for separable Hilbert spaces is still open, several other cases have been settled for topological vector spaces :
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 is not separable. In fact, if is a non-zero vector in, the norm closure of the linear orbit 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.
proved that every compact operator on any Banach space of dimension at least 2 has an invariant subspace.
proved using non-standard analysis that if the operator on a Hilbert space is polynomially compact then has an invariant subspace. Their proof uses the original idea of embedding the infinite-dimensional Hilbert space in a hyperfinite-dimensional Hilbert space.
, 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.
gave a very short proof using the Schauder fixed point theorem that if the operator on a Banach space commutes with a non-zero compact operator then 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 commutes with a non-scalar operator that commutes with a non-zero compact operator, then has an invariant subspace.
The first example of an operator on a Banach space with no non-trivial invariant subspaces was found by , and his example was simplified by.
The first counterexample on a "classical" Banach space was found by, who described an operator on the classical Banach space with no invariant subspaces.
Later constructed an operator on without even a non-trivial closed invariant subset, that is that for every vector the set is dense, in which case the vector is called hypercyclic.
gave an example of an operator without invariant subspaces on a nuclearFréchet space.
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.
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.
