Operators not positive with respect to any basis
Angela Spalsbury · Quaestiones Mathematicae · 2000
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 L 2 (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 T x ∊ M then x ∊ M.