On properties of C-normal operators II

For T∈ L(H) , an operator T is called C-normal if (CT) ^#CT= CT(CT) ^# for a conjugation C on H. In this paper, we continue our study, begun in Ko et al. (Banach J. Math. Anal. 14:1711–1727, 2020), of various properties of C-normal operators. Especially, we prove that if T= U| T| is the polar decomposition of T, C is a conjugation on H with U^∗CU^∗= C, and T is C-normal, then T^∗ possess the property (β) , the single valued extension property, the property (C) if and only if T possess, respectively. In addition, if T is C-normal, then T is binormal if and only if | T| ⁿ and C| T| ^mC commute for everyl positive integers m, n.