Notes on a conjecture by Paszkiewicz on an ordered product of positive contractions

Paszkiewicz's conjecture asserts that given a decreasing sequence $T_1\ge T_2\ge \dots$ of positive contractions on a separable infinite-dimensional Hilbert space $H$, the product $S_n=T_nT_{n-1}\cdots T_1$ converges in the strong operator topology. In these notes, we give an equivalent, more precise formulation of his conjecture. Moreover, we show that the conjecture is true for the following two cases: (1) $1$ is not in the essential spectrum of $T_n$ for some $n\in \mathbb{N}$. (2) The von Neumann algebra generated by $\{T_n\mid n\in \mathbb{N}\}$ admits a faithful normal tracial state. We also remark that the analogous conjecture for the weak convergence is true.