Given two self-adjoint, positive, compact operators A; B on a separable Hilbert space, we show that there exists a self-adjoint, positive, compact operator C commuting with B such that lim t-N jjðe Bt 2 e At e Bt 2 Þ 1 t À e C jj ¼ 0: r

Abstract. In this paper we consider a linear homogeneous system of m equations in n unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed k + 1 for some positive integer k. We show that if the system has a nontrivial solution then there exists a nontrivial solution x = (x 1 , . . . , xn) ⊤ such thatfor each i, j satisfying x i x j = 0. This inequality is sharp.We also prove a conjecture of A. Tyszka related to our results.

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