Square root for backward operator weighted shifts with multiplicity 2 (Q437737)

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 has a square root. In this paper, by means of Schauder basis theory the authors obtain that, if a backward operator weighted shift \(T\) with multiplicity 2 is not strongly irreducible, then there exists a backward shift operator \(B\) (possibly unbounded) such that \(T=B^2\). Furthermore, the backward operator weighted shifts in the sense of Cowen-Douglas are also considered.