Invariant subspace problem

The vector x is an eigenvector of the matrix A. Every operator on a non-trivial complex finite dimensional vector space has an eigenvector, solving the invariant subspace problem for these spaces.

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. The original form of the problem as posed by Paul Halmos was in the special case of polynomials with compact square. This was resolved affirmatively, for a more general class of polynomially compact operators, by Allen R. Bernstein and Abraham Robinson in 1966 (see Non-standard analysis#Invariant subspace problem for a summary of the proof).

Precise statement

More formally, 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.

History

The problem seems to have been stated in the mid-1900s after work by Beurling and von Neumann.[1]

For Banach spaces, the first example of an operator without an invariant subspace was constructed by Enflo. (For Hilbert spaces, the invariant subspace problem remains open.)

Per Enflo 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[2] Enflo's long "manuscript had a world-wide circulation among mathematicians"[1] and some of its ideas were described in publications besides Enflo (1976).[3][4] Enflo's works inspired a similar construction of an operator without an invariant subspace for example by Beauzamy, who acknowledged Enflo's ideas.[2]

In the 1990s, Enflo developed a "constructive" approach to the invariant subspace problem on Hilbert spaces.[5]

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):

Notes

  1. 1 2 Yadav, page 292.
  2. 1 2 Beauzamy 1988; Yadav.
  3. ↑ For example, Radjavi and Rosenthal (1982).
  4. ↑ Heydar Radjavi and Peter Rosenthal (March 1982). "The invariant subspace problem". The Mathematical Intelligencer 4 (1): 33–37. doi:10.1007/BF03022994.
  5. ↑ Page 401 in Foiaş, Ciprian; Jung, Il Bong; Ko, Eungil; Pearcy, Carl (2005). "On quasinilpotent operators. III". Journal of Operator Theory 54 (2): 401–414.. Enflo's method of ("forward") "minimal vectors" is also noted in the review of this research article by Gilles Cassier in Mathematical Reviews: MR 2186363
  6. ↑ Von Neumann's proof was never published, as relayed in a private communication to the authors of Aronszajn & Smith (1954). A version of that proof, independently discovered by Aronszajn, is included at the end of that paper.
  7. ↑ See Pearcy & Shields (1974) for a review.

References


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