Let \((\mathcal {M}, \mathcal {N})\) be a pair of closed subspaces of a separable complex Hilbert space. A concrete characterization of all the conjugations C such that \(C(\mathcal {M})=\mathcal {N}\) is established which improves Liu et al.’s result (Liu et al. in J Math Anal Appl 500:125118, 2021). As applications, four classes of conjugations for the interpolation properties of idempotents are described.
