On Paszkiewicz's conjecture about a product of positive contractions

The Paszkiewicz conjecture about a product of positive contractions asserts that given a decreasing sequence T₁≥ T₂≥ of positive contractions on a separable infinite-dimensional Hilbert space, the product Sₙ=Tₙ T₁ converges strongly. Recently, the first named author verified the conjecture for certain classes of sequences. In this paper, we take a new approach by analysing the limit of the positive part A=limn→ ∞|Sₙ| (which always exists, regardless of the existence of limn→ ∞Sₙ) and extend the class of examples for which the conjecture holds. We also show how an operator algebraic viewpoint can be useful to study the Paszkiewicz conjecture. Finally, we also show that the Paszkiewicz conjecture is true for all spectrally ordered sequences, i.e., those sequences for which T₁ᵏ≥ T₂ᵏ≥ holds for every k∈ N.