invariant subspace problem

Initially formulated for Banach spacesMathworldPlanetmath, the invariant subspace conjecture stated the following:

Let X be a complex Banach space. Then every bounded operatorMathworldPlanetmathPlanetmath T in X has a non-trivial closed ( invariant subspacePlanetmathPlanetmath, i.e. there exists a closed vector subspace SX such that S0, SX and T(S)S.

This conjecture was proven to be false when P. Enflo (1975) and . Read (1984) gave examples of bounded operators which did not have the above property.

However, if one considers only Hilbert spacesMathworldPlanetmath, this is still an open problem. Today the invariant subspace conjecture is formulated as follows:

Let H be a complex Hilbert space. Then every bounded operator T in H has a non-trivial invariant subspace, i.e. there exists a closed vector subspace SH such that S0, SH and T(S)S.

Title invariant subspace problem
Canonical name InvariantSubspaceProblem
Date of creation 2013-03-22 17:24:02
Last modified on 2013-03-22 17:24:02
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 5
Author asteroid (17536)
Entry type Conjecture
Classification msc 47A15
Classification msc 46-00
Synonym invariant subspace conjecture