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

Paweł Pietrzycki

doi:10.1016/j.jmaa.2015.09.062

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arXiv · 1502.06396

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

Paweł Pietrzycki

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Abstract

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 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.

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Paweł Pietrzycki. 2015-02-23. The single equality $A^{*n}A^n = (A^*A)^n$ does not imply the quasinormality of weighted shifts on rootless directed trees. https://doi.org/10.1016/j.jmaa.2015.09.062

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