The single equality A⁎nAn=(A⁎A)nA⁎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⁎nWn=(W⁎W)nW⁎nWn=(W⁎W)n for some integer n⩾2n⩾2 is quasinormal. For any integer n⩾2n⩾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⁎nAn=(A⁎A)nA⁎nAn=(A⁎A)n is constructed. It is also shown that such an example can be constructed in the class of composition operators in L2L2-spaces over σ-finite measure spaces.