Grahame Bennett

On uncomplemented subspaces ofL p , 1 <p <2

It is shown tat, for 1 <p < 2, there is an uncomplemented subspace ofLp [0,1] that is isomorphic to Hilbert space.

Lower bounds for matrices

Abstract In a recent paper [3], Lyons has discovered an interesting lower bound for the Cesaro matrix C . His result says that ∥Cx∥⩾(π 6 )∥x∥ for every xϵl 2 satisfying x 1 ⩾ x 2 ⩾⋯⩾0. The purpose of this note is to establish analogous lower bounds for arbitrary matrices (with nonnegative entries) acting on arbitrary l p spaces, 1⩽ p ⩽∞.

Double dipping: the case of the missing binomial coefficient identities

A game of chance leads to the study of real sequences x and y with the remarkable property that the products of their differences are majorized by the differences of their product. Such sequences are said to form a double-dipping pair. The simplest examples: xn=(1 − p)n, yn=(1−q)n(n=0,1,2,…) with 0 ⩽p,q⩽1, arise when the game is played by tossing coins. The double-dipping property for these is the intriguing assertion (see Bennett, 1990) that the sum of any N terms from the set {pmqn:m,n=0,1,2,…} does not exceed 1+(p+q−pq)+⋯+(p+q−pq)N −1(N=1,2,…). Our purpose here is to prove the analogous inequalities that arise when the game is governed by sampling balls from urns, with various replacement schemes. This leads to the conjecture that x and y form a double-dipping pair whenever xN=(B−nb) and yn= (C−nc) (n=0,1,2,…), where B, b, C, c are non-negative integers with B⩾b and C⩾c.

From coin tossing to the Jacobi polynomials

A simple coin-tossing game leads to the study of real sequences, x and y, with the remarkable property that the products of their differences are majorized by the differences of their product. Such sequences are said to form a double-dipping pair . The following conjecture arises when the game is governed by sampling balls from urns: if A, a, B, b are non-negative integers with A ≥ a and B ≥ b , then x and y form a double-dipping pair, where x k = ( A − K a ) , y k = ( B − K b ) , k = 0 , 1 , 2 , … . The conjecture is proved here under the additional restriction - b ≤ A − B ≤ a . The proof is based, in part, upon the observation that the polynomials, x → ∑ k ( a k ) ( b n − k ) ( 1 − x ) k , n ≤ a + b , have reciprocals, all of whose Taylor coefficients (about x = 0) are non-negative.