The single equality A*nAn = (A*A)n does not imply the quasinormality of weighted shifts on rootless directed trees

It is proved that each bounded injective bilateral weighted shift W satisfying the equality W*W-n(n) = (W*W)(n) for some integer n >= 2 is quasinormal. For any integer n >= 2, an example of a bounded non-quasinormal weighted shift A on a rootless directed tree with one branching vertex which satisfies the equality A*(n)A(n) = (A*A)(n) is constructed. It is also shown that such an example can be constructed in the class of composition operators in L-2-spaces over sigma-finite measure spaces. (C) 2015 Elsevier Inc. All rights reserved.