Invariant subspace problem

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. Many variants of the problem have been solved, by restricting the class of bounded operators considered or by specifying a particular class of Banach spaces. The problem is still open for separable Hilbert spaces (in other words, all the examples found of operators with no non-trivial invariant subspaces act on Banach spaces which are not separable Hilbert spaces).
History.
The problem seems to have been stated in the mid-1900s after work by Beurling and von Neumann, who found (but never published) a positive solution for the case of compact operators. It was then posed by Paul Halmos for the case of operators formula_1 such that formula_2 is compact. This was resolved affirmatively, for the more general class of polynomially compact operators (operators formula_1 such that formula_4 is a compact operator for a suitably chosen non-zero polynomial formula_5), by Allen R. Bernstein and Abraham Robinson in 1966 (see for a summary of the proof).
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 (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.
Precise statement.
Formally, the invariant subspace problem for a complex Banach space formula_6 of dimension > 1 is the question whether every bounded linear operator formula_7 has a non-trivial closed formula_1-invariant subspace: a closed linear subspace formula_9 of formula_6, which is different from formula_11 and from formula_6, such that formula_13.
A negative answer to the problem is closely related to properties of the orbits formula_1. If formula_15 is an element of the Banach space formula_6, the orbit of formula_15 under the action of formula_1, denoted by formula_19, is the subspace generated by the sequence formula_20. This is also called the formula_1-cyclic subspace generated by formula_15. From the definition it follows that formula_19 is a formula_1-invariant subspace. Moreover, it is the "minimal" formula_1-invariant subspace containing formula_15: if formula_9 is another invariant subspace containing formula_15, then necessarily formula_29 for all formula_30 (since formula_9 is formula_1-invariant), and so formula_33. If formula_15 is non-zero, then formula_19 is not equal to formula_11, so its closure is either the whole space formula_6 (in which case formula_15 is said to be a cyclic vector for formula_1) or it is a non-trivial formula_1-invariant subspace. Therefore, a counterexample to the invariant subspace problem would be a Banach space formula_6 and a bounded operator formula_7 for which every non-zero vector formula_43 is a cyclic vector for formula_1. (Where a "cyclic vector" formula_15 for an operator formula_1 on a Banach space formula_6 means one for which the orbit formula_19 of formula_15 is dense in formula_6.)
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 (over the field of complex numbers):
References.
 