Remarks on the structure of C-normal operators

We study a new class NC of Hilbert space operators, named C-normal operators for C a conjugation on a complex Hilbert space H, which were introduced by M. Ptak, K. Simik and A. Wicher. We provide a refined polar decomposition of C-normal operators. It is proved that those invertible ones are norm dense in NC and each contraction in NC is a mean of two unitary ones. For a dense class of operators, we prove that their C-normality coincides with C-symmetry. Also some illustrating examples are provided to show that NC is not closed under several natural operations.