On properties of C-normal operators II

Abstract

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. © 2023, Tusi Mathematical Research Group (TMRG).