Interpolation theorems for conjugations and applications

Let H be a separable complex Hilbert space. A conjugate-linear map C:H→ H is a conjugation if it is an involutive isometry. In this paper, we consider the following interpolation problems: Let ᵢ∈ I and ᵢ∈ I be two orthogonal sets of vectors in H, and let N and ₖ∈ K be normal operators such that the Nₖ's mutually commute. Then, under which conditions does there exist a conjugation C on H such that (a) Cxᵢ=yᵢ and CNₖC=Nₖ^* for all i∈ I and k∈ K; or (b) Cxᵢ=yᵢ, for every i∈ I, and CNC=-N^*. We provide complete answers to problems (a) and (b). As a consequence of our results, we give necessary and sufficient conditions for the existence of solutions of some equations in L∞(μ).