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It is proved that each bounded injective bilateral weighted shift $W$satisfying the equality $W^{*n}W^{n}=(W^{*}W)^{n}$ for some integer $n\geq 2$is quasinormal. For any integer $n\geq 2$, an example of a boundednon-quasinormal weighted shift $A$ on a rootless directed tree with onebranching vertex which satisfies the equality $A^{*n}A^{n}=(A^{*}A)^{n}$ isconstructed. It is also shown that such an example can be constructed in theclass of composition operators in $L^2$-spaces over $\sigma$-finite measurespaces.

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

The single equality $A^{n}A^n = (A^A)^n$ does not imply the quasinormality of weighted shifts on rootless directed trees