Estimates for the Norm of the n th Indefinite Integral

Let T be the Volterra operator on L 2 [0, 1] T f ( x ) = ∫ 0 x f ( t ) d t ,where f ∈ L 2 [0, 1], 0 ⩽ x ⩽ 1. It is well known that ‖ n ! T n ‖ = O (1/ n !). In a recent paper [ 1 ], D. Kershaw has proved thatlim n → ∞‖ n ! T n ‖ = 1 / 2 ,a result which was first conjectured by Lao and Whitley in [ 2 ]. It is easy to prove thatlim n → ∞   sup ‖ n ! T n ‖ ⩽ 1 / 2 ,For completeness, we give the proof using the familiar Schmidt norm estimate for the norm of an integral operator (see Section 2 below). Kershaw proves thatlim   inf n → ∞‖ n ! T n ‖ ⩾ 1 / 2by analysing the special positivity preserving properties of T * T . He uses one of the many abstract theorems on eigenvalues and eigenfunctions of compact operators which preserve a cone. In this paper we shall reprove (1), giving a short and direct proof of (2). 1991 Mathematics Subject Classification 47G10, 45‐04.