Square root for backward operator weighted shifts with multiplicity $2$

Abstract

As is well-known, each positive operator $T$ acting on a Hilbert space has a positive square root which is realized by means of functional calculus. However, it is not always true that an operator have a square root. In this paper, by means of Schauder basis theory we obtain that if a backward operator weighted shift $T$ with multiplicity $2$ is not strongly irreducible, then there exists a backward shift operator $B$ (maybe unbounded) such that $T=B^2$. Furthermore, the backward operator weighted shifts in the sense of Cowen-Douglas are also considered.