We consider two operators A₀andB₀ between two Hilbert spaces satisfying (formula presented) and inspect extensions A^# and B^# of A₀ and B₀, 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^#=−B#^∗ and illustrate our findings by applications to the classical vector analytic operators grad,div and curl. In particular, the realisation curl^# naturally arises in the study of so-called Beltrami fields.