It is shown that for every positive integer $n$ there exists a subnormalweighted shift on a directed tree (with or without root) whose $n$th power isdensely defined while its $(n+1)$th power is not. As a consequence, for everypositive integer $n$ there exists a non-symmetric subnormal compositionoperator $C$ in an $L^2$ space over a $\sigma$-finite measure space such that$C^n$ is densely defined and $C^{n+1}$ is not.
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机译：结果表明，对于每一个正整数$ n $，在有向树（有或没有根）上都有一个次正规加权移位，其有$ n $次幂被密集地定义，而其第（n + 1）$次幂则没有。结果，对于每个正整数$ n $，在$ \ sigma $有限度量空间的$ L ^ 2 $空间中存在一个不对称的次正规合成运算符$ C $，使得$ C ^ n $被密集定义并$ C ^ {n + 1} $不是。
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