Proof of the Paszkiewicz conjecture about a product of positive contractions

Abstract The Paszkiewicz conjecture about a product of positive contractions asserts that given a decreasing sequence $$T_1\ge T_2\ge \dots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≥</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>≥</mml:mo> <mml:mo>⋯</mml:mo> </mml:mrow> </mml:math> of positive contractions on a separable infinite-dimensional Hilbert space, the product $$S_n=T_n\dots T_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>⋯</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> 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.