Operators not positive with respect to any basis

Abstract

In this paper we use an equivalent form of Titchmarsh's Convolution theorem to show that the Volterra operator is an example of a quasi-nilpotent operator on a Hilbert space that is not positive with respect to any basis of L²(0, 1). In proving this, it then becomes clear how to generalize this to any operator T with the property that if M is an invariant subspace of T and Tx ∈ M then x ∈ M.

Keywords: invariant subspaces, positive operators

Quaestiones Mathematicae 23(2000), 489–494