The Lidskii trace property and the nest approximation property in Banach spaces

Abstract

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 [2, Theorem 3.1] that an atomic Boolean subspace lattice that has only two atoms must have the strong rank one density property.