  A conjugation $C$ is an anti-linear isometric involution on a complex Hilbert
space $\clh$, and $T\in \clb(\clh)$ 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].
