Zhu, Fang and Shi's question on invariant projections of weighted irrational rotations

Theorem 1.1: for every irrational $\theta\in(0,1)$ there is a continuous $f:\mathbb T\to[0,1]$ vanishing only at $1$, with $\int\log f\,dm=-\infty$, such that every projection $p\in R_\theta$ with $(1-p)Uf(V)p=0$ is $0$ or $1$; the operator is nonzero and norm-quasinilpotent. The angle is prescribed in advance with no Diophantine condition. Invariant subspaces of the Hilbert-space representation whose projections lie outside the factor are not excluded. An appendix gives a second realization on an infinite product of circles.

Theorem 1.1: for every irrational $\theta\in(0,1)$ there is a continuous $f:\mathbb T\to[0,1]$ vanishing only at $1$, with $\int\log f\,dm=-\infty$, such that every projection $p\in R_\theta$ with $(1-p)Uf(V)p=0$ is $0$ or $1$; the operator is nonzero and norm-quasinilpotent. The angle is prescribed in advance with no Diophantine condition. Invariant subspaces of the Hilbert-space representation whose projections lie outside the factor are not excluded. An appendix gives a second realization on an infinite product of circles.