Factorization of anti-linear and $C$-normal operators

A conjugation $C$ is an anti-linear isometric involution on a complex Hilbert space $\text{clh}$, and $T\in \text{clb}(\text{clh})$ is conjugate normal if $T^{\ast }T=CT{T}^{\ast }C$ holds for some conjugation ©. 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].