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In this paper, we are concerned with conditions under which$[p(T)]^*=\bar{p}(T^*)$, where $p(z)$ is a one-variable complex polynomial, and$T$ is an unbounded, densely defined, and linear operator. Then, we deal withthe validity of the identities $\sigma(AB)=\sigma(BA)$, where $A$ and $B$ aretwo unbounded operators. The equations $(TT^*)^*=TT^*$ and $(T^*T)^*=T^*T$,where $T$ is a densely defined closable operator, are also studied. Aparticular interest will be paid to the equation $T^*T=p(T)$ and its variants.Then, we have certain results concerning $nth$ roots of classes of normal andnonnormal (unbounded) operators. Some further consequences and counterexamplesaccompany our results.

Certain properties involving the unbounded operators $p(T)$, $TT^*$, and  $T^*T$; and some applications to powers and $nth$ roots of unbounded  operators