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

TL;DR

This paper investigates properties of unbounded operators, including polynomial relations, spectral identities, and roots, providing new conditions, results, and counterexamples for these operator classes.

Contribution

It introduces new conditions for polynomial and spectral identities involving unbounded operators and explores nth roots of normal and nonnormal unbounded operators.

Findings

Conditions for $[p(T)]^*=ar{p}(T^*)$ established

Spectral identity $\sigma(AB)=\sigma(BA)$ analyzed for unbounded operators

Results on nth roots of unbounded normal and nonnormal operators

Abstract

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 with the validity of the identities $\sigma(AB)=\sigma(BA)$, where $A$ and $B$ are two unbounded operators. The equations $(TT^*)^*=TT^*$ and $(T^*T)^*=T^*T$, where $T$ is a densely defined closable operator, are also studied. A particular 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 and nonnormal (unbounded) operators. Some further consequences and counterexamples accompany our results.