A Note on Some Non-Local Boundary Conditions and Their Use in Connection with Beltrami Fields

We consider two operators \(A_{0} and B_{0}\) between two Hilbert spaces satisfying \(A_{0}\subseteq -B_{0}^{\ast }\) and \(B_{0}\subseteq -A_{0}^{\ast }\) and inspect extensions \(A^{\#}\) and \(B^{\#}\) of \(A_{0}\) and \(B_{0}\) , respectively, whose domain consists of those elements satisfying an abstract periodic boundary condition. The motivating example is the derivative on some interval, where the so-defined realisation gives the classical derivative with periodic boundary conditions. We derive necessary and sufficient conditions for the operator equality \(A^{\#}=-\left (B^{\#}\right )^{\ast }\) and illustrate our findings by applications to the classical vector analytic operators \(\operatorname {grad},\operatorname {div}\) and \(\operatorname {curl}\) . In particular, the realisation \(\operatorname {curl}^{\#}\) naturally arises in the study of so-called Beltrami fields.