Invariant subspace problem

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In the field of mathematics known as functional analysis, 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.

For the most important case of Hilbert spaces H the problem remains open (as of 2013), though Per Enflo found the first example of a Banach space operator with no invariant subspace.

Known special cases[edit]

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):

  • For non-zero finite-dimensional vector spaces every operator admits an eigenvector, so it has a 1-dimensional invariant subspace.
  • 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.
  • von Neumann showed that any compact operator on a Hilbert space of dimension at least 2 has a non-trivial invariant subspace.
  • The spectral theorem shows that all normal operators admit invariant subspaces.
  • Aronszajn & Smith (1954) proved that every compact operator on any Banach space of dimension at least 2 has an invariant subspace.
  • Bernstein & Robinson (1966) proved using non-standard analysis that if the operator T on a Hilbert space is polynomially compact (in other words P(T) is compact for some non-zero polynomial P) then T has an invariant subspace. Their proof uses the original idea of embedding the infinite-dimensional Hilbert space in a hyperfinite-dimensional Hilbert space (see Non-standard analysis#Invariant subspace problem).
  • Halmos (1966), 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.
  • Lomonosov (1973) gave a very short proof using the Schauder fixed point theorem that if the operator T on a Banach space commutes with a non-zero compact operator then T 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 S commutes with a non-scalar operator T that commutes with a non-zero compact operator, then S has an invariant subspace.[1]
  • The first example of an operator on a Banach space with no invariant subspaces was found by Per Enflo (1976, 1987), and his example was simplified by Beauzamy (1985).
  • The first counterexample on a "classical" Banach space was found by Charles Read (1984, 1985), who described an operator on the classical Banach space l1 with no invariant subspaces.
  • Later Charles Read (1988) constructed an operator on l1 without even a non-trivial closed invariant subset, that is, with every vector hypercyclic, solving in the negative the invariant subset problem for the class of Banach spaces.
  • Atzmon (1983) gave an example of an operator without invariant subspaces on a nuclear Fréchet space.
  • Śliwa (2008) 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.
  • Argyros & Haydon (2009) announced 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.

Notes[edit]

  1. ^ See Pearcy & Shields (1974) for a review.

References[edit]