Invariant subspace problem

In the field of mathematics known as functional analysis, one of the most prominent open problems is the invariant subspace problem, sometimes optimistically known as the invariant subspace conjecture. It is the question whether the following statement is true::Given a complex Hilbert space "H" of dimension > 1 and a bounded linear operator "T" : "H" → "H", then "H" has a non-trivial closed "T"-invariant subspace, i.e. there exists a closed linear subspace "W" of "H" which is different from {0} and "H" such that "T"("W") ⊆ "W".

The statement is true for all finite-dimensional complex vector spaces of dimension at least 2: the eigenvalues of a linear operator (matrix) are the zeros of its characteristic polynomial; this polynomial has zeros because of the fundamental theorem of algebra; a corresponding eigenvector will span an invariant subspace. The statement is true in the infinite-dimensional case if "W" is not required to be closed: pick any non-zero vector "x" in "H" and consider the subspace "W" of "H" spanned by {"T" ^"n"("x") : "n" ≥ 0} . "W" is invariant. Moreover, "W" is a meager set in "H" and so by the Baire category theorem must be distinct from "H".

While the general case of the conjecture is still open, several special cases have been settled:
* 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.
* The spectral theorem shows that all normal operators admit invariant subspaces.
* Every compact operator has an invariant subspace, as proved by Aronszajn and Smith in 1954. The theory of compact operators is in many ways similar to the theory of operators on a finite-dimensional space, so this result is not too surprising.
* Bernstein and Robinson proved in 1966, using nonstandard analysis that if "T" ^"n" is compact for some positive integer "n", then "T" has an invariant subspace. Paul Halmos subsequently provided a proof which did not rely on nonstandard methods.
* V. I. Lomonosov proved in 1973 that if "T" commutes with a non-zero compact operator then "T" has an invariant subspace. 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.

In recent years, some mathematicians have attempted to construct counterexamples to the conjecture using the theory of random matrices.

If one considers Banach spaces instead of Hilbert spaces, the conjecture becomes false; explicit examples of bounded operators without invariant subspaces have been exhibited by P. Enflo (who in 1975 sketched out a construction, of which arevised version was produced in 1981 and eventually published in 1987),and by Charles Read in 1984, who laterproduced a further example on the classical Banach space "l¹". However, the statement is true for certain classes of operators.

In 1964, Louis de Branges published an alleged proof of the invariant subspace conjecture which was later found to be false. He recently published a new claimed proof [http://www.math.purdue.edu/~branges/invariantsubspaceconjecture.pdf] on his website; so far this proof has not been subjected to peer review.

References

* Paul Halmos. Invariant Subspaces. "American Mathematical Monthly", Vol. 85, No. 3 (March 1978), pages 182-183.
* B. S. Yadav. The present state and heritages of the invariant subspace problem. "Milan J. Math." 73 (2005), pages 289-316.
* Piotr Sniady. Generalized Cauchy identities, trees and multidimensional Brownian motions. Part I: bijective proof of generalized Cauchy identities. Section 1.5. [http://arxiv.org/abs/math.CO/0412043 Preprint 2004] .
*Enflo, P. On the invariant subspace problem in Banach spaces. Séminaire Maurey--Schwartz (1975-1976) Espaces "Lp" applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 14-15, Centre Math., École Polytech., Palaiseau, 1976.
*Enflo, Per On the invariant subspace problem for Banach spaces. Acta Math. 158 (1987), no. 3-4, pages 213-313.
*Read, C. J. A solution to the invariant subspace problem. Bull. London Math. Soc. 16 (1984), no. 4, pages 337-401.