Factorization of anti-linear and C-normal operators

A conjugation C is an anti-linear isometric involution on a complex Hilbert space , and T∈ () is conjugate normal if T^*T = CTT^*C holds for some conjugation (C). In this paper, we provide a factorization and range inclusion theorem for anti-linear operators, and consequently, establish the polar decomposition for anti-linear operators by applying the Douglas theorem on majorization of Hilbert space operators. Moreover, we present a factorization of C-normal operators based on the polar decomposition. Lastly, we study the Cartesian decomposition of conjugate normal operators, thereby expanding the results in [18].