On the Aluthge Transforms of Weighted Shifts with Moments of Fibonacci Type. Application to Subnormality

Abstract

In this paper, we investigate the properties of the Aluthge transforms of weighted shifts whose moment sequence is a generalized Fibonacci sequence of order r (or of Fibonacci type). We provide sufficient conditions under which the Aluthge transform has also a moment sequence of Fibonacci type. Notably, we detail the case when the moment sequence is a generalized Fibonacci sequence of order r ≤ 4. This allows us to recover the main result given by S. H. Lee et al. concerning the subnormality of Aluthge transforms of weighted shifts with two atomic Berger measure. We end this paper by considering the extended case of the subnormal weighted shift of a finite atomic Berger measure, with a view to whether its Aluthge transforms is again subnormal. As a result, we give a negative answer to a recent conjecture posed by S. H. Lee et al. in its general setting. Even so, under some mild assumptions we give a positive answer to this conjecture.