On the adjoint of the unbounded operators $p(T)$, $TT^*$, and $T^*T$, and certain applications to $nth$ roots of unbounded operators

Mohammed Hichem Mortad

Published 2022-08-09Version 1

In this paper, we are mainly concerned with conditions under which $[p(T)]^*=\overline{p}(T^*)$, where $p(z)$ is a one-variable complex polynomial, and $T$ is an unbounded, densely defined, and linear operator. Then, we deal with the validity of the identities $(TT^*)^*=TT^*$ and $(T^*T)^*=T^*T$, where $T$ is a densely defined closable operator. A particular interest will be paid to the equation $T^*T=p(T)$ and its variants. Also, we have certain results concerning $nth$ roots of classes of normal and nonnormal (unbounded) operators. Some further consequences and counterexamples accompany our results.