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

TL;DR

This paper discusses Paszkiewicz's conjecture on the strong convergence of products of decreasing sequences of positive contractions on infinite-dimensional Hilbert spaces, providing new formulations and partial proofs.

Contribution

It offers an equivalent formulation of the conjecture and proves its validity under specific spectral and algebraic conditions.

Findings

Conjecture holds if 1 is not in the essential spectrum of some T_n.

Conjecture holds if the generated von Neumann algebra admits a faithful normal tracial state.

Weak convergence version of the conjecture is confirmed.

Abstract

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.