Proof of the Paszkiewicz's conjecture about a product of positive contractions

TL;DR

This paper proves the Paszkiewicz conjecture that the product of a decreasing sequence of positive contractions on an infinite-dimensional Hilbert space converges strongly, confirming a long-standing open problem in operator theory.

Contribution

The paper establishes the conjecture in full generality and extends it to some generalized cases, advancing understanding of operator products in Hilbert spaces.

Findings

Confirmed the convergence of products of positive contractions in full generality.

Extended the conjecture to certain generalized cases.

Provided a comprehensive proof resolving the conjecture.

Abstract

The Paszkiewicz conjecture about a product of positive contractions asserts that given a decreasing sequence $T_1\ge T_2\ge \dots$ of positive contractions on a separable infinite-dimensional Hilbert space, the product $S_n=T_n\dots T_1$ converges strongly. Recently, the first named author verified the conjecture for certain classes of sequences. In this paper, we prove the Paszkiewicz conjecture in full generality. Moreover, we show that in some cases, a generalized version of the Paszkiewicz conjecture also holds.