Abstract. In a sequence of four recent papers (cf. below), it was eventually shown that the hyperinvariant-subspace lattice $\mathop{\rm Hlat}(T)$ of an arbitrary nonalgebraic operator $T$ on Hilbert space is lattice-isomorphic to $\mathop{\rm Hlat}(A)$ for some $A$ in a special class $({\cal A}_{\theta })$ of operators defined below. In this note, which might be regarded as a first step in an attempt to better understand the structure of the class $({\cal A}_{\theta })$, we construct and study a certain subclass ${\cal(S}_{\theta }{\cal )}$ of this collection consisting of some operator--weighted bilateral shifts.

AMS Subject Classification (1991): 47A15, 47A45

Received May 15, 2006, and in revised form May 24, 2006. (Registered under 5942/2009.)