Point spectrum of an SOT-typical positive contraction on ℓ2

Determine whether the point spectrum of an SOT-typical positive contraction on ℓ2 equals the open unit disk.

Background

For general contractions on ℓ2, Eisner and Mátrai showed that a typical contraction is unitarily equivalent to a backward shift, yielding point spectrum equal to the open unit disk. However, for positive contractions the key co-isometry property fails typically, so the existing argument does not apply.

The paper asks whether the same spectral picture nevertheless holds in the positive-contraction setting.