2022/08/09 by Mortad, Mohammed Hichem
#FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2208.04812
In this paper, we are concerned with conditions under which [p(T)]^*=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 σ(AB)=σ(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.