The Lidskii Trace Property and the Nest Approximation Property in Banach Spaces

For a Banach space X, the Lidskii trace property is equivalent to the nest approximation property; that is, for every nuclear operator on X that has summable eigenvalues, the trace of the operator is equal to the sum of the eigenvalues if and only if for every nest N of closed subspaces of X, there is a net of finite rank operators on X, each of which leaves invariant all subspaces in N , that converges uniformly to the identity on compact subsets of X. The Volterra nest in Lp(0, 1), 1 ≤ p < ∞, is shown to have the Lidskii trace property. A simpler duality argument gives an easy proof of the result [ALL, Theorem 3.1] that an atomic Boolean subspace lattice that has only two atoms must have the strong rank one density property.