Aronszajn-Smith theorem

In functional analysis, the Aronszajn-Smith theorem resolves the invariant subspace problem for compact operators on a Banach space. It was proved by Nachman Aronszajn and K. T. Smith.

The theorem states that a compact operator on a complex Banach space, of dimension greater than 1, has a nontrivial closed invariant subspace. The claim is trivial when the given compact operator has some non-zero complex number in its spectrum; however, this latter condition is not true in general. See, for instance, the Volterra operator.

References

*N. Aronszajn, and K.T. Smith, Invariant subspaces of completely continuous operators, "An. Math.", 60, 1954, 345-350.