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

We consider two operators $A_0,B_0$ between two Hilbert spaces satisfying $A_0\subseteq -B_0^{\ast }$ and $B_0\subseteq -A_0^{\ast }$ and inspect extensions $A^{\mathrm{\#}}$ and $B^{\mathrm{\#}}$ 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 $A^{\mathrm{\#}}=-{\left(B^{\mathrm{\#}}\right)}^{\ast }$ and illustrate our findings by applications to the classical vector analytic operators $\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d},\mathrm{d}\mathrm{i}\mathrm{v}$ and $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}$. In particular, the realisation ${\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}^{\mathrm{\#}}$ naturally arises in the study of so-called Beltrami fields.