A Class of Invertible Bilateral Weighted Shifts

Abstract

Abstract. In this note we study a class of invertible weighted bilateral shifts on Hilbertspace introduced by Haskell Rosenthal recently. We show that every Rosenthal shift isunitarily equivalent to its inverse, not quasisimilar to its adjoint, and has a nontrivialhyperinvariant subspace. We write, as usual, Z for the set of integers and N(N 0 ) for the set of positive(nonnegative) integers. We also write l 2 (Z) for the separable, infinite dimensional,complex Hilbert space l 2 (Z) := ff n g n2 Z :  n 2 C ; ∑ n2 Z j n j 2 0, then the operator B w 2 L( l 2 (Z)) definedby(1) B w e n = w n e n +1 ; n 2 Z ; is a (invertible, forward) weighted bilateral shift. A trivial calculation gives thedefining equations(2)