Conjugate phase retrieval on general Hilbert spaces

A conjugation on a Hilbert space H means an antilinear bounded operator that the squares to the identity, which generalizes the traditional conjugation on complex Euclidean spaces. In this paper, with the help of normalized conjugation we introduce the notion of conjugate phase retrieval on general Hilbert spaces. We characterize frames that do conjugate phase retrieval; prove that every conjugate phase retrieval frame for H consisting of all real vectors has the complement property in H, and the converse is true if dim(H) <= 2; and also prove that a small perturbation of conjugate phase retrieval frame still gives a conjugate phase retrieval frame if dim(H) < infinity, but it is false if dim(H) = infinity.