Invariant subspace problem

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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 also easily seen to be true 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}.

While the general case of the conjecture is still open, several special cases have been settled:

• It is relatively easy to see that the conjecture is true if the Hilbert space H is not separable (i.e. if it has an uncountable orthonormal basis).
• 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 that if T ⁿ is compact for some positive integer n, then T has an invariant subspace.
• V. I. Lomonosov proved in 1973 that if T commutes with a non-zero compact operator then T 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 subspace have been exhibited by P. Enflo in 1975 and by Charles Read in 1984. 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[1] on his website; so far this proof has not been subjected to peer review.

[edit] 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. 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.