Over the past dozen years or so the Hamiltonian theory of integrable mechanical systems of a finite number of degrees of freedom has been generalized to “integrable” partial differential systems. Recursive structures are particularly important in this theory [see, for example, I. Ya. Dorfman
, Phys. Lett. A 140, 378-382 (1989) and references therein]. The present paper appears to be largely ignorant of these developments preferring to cite papers written by the authors themselves over the same period, and therefore their results should be carefully compared with the former references. With this proviso we summarize their results. Hamiltonian operators, linear in the dependent variables, are defined and a means of extending them to larger function spaces is given. Sufficient for this is an algebraic structure condition on the function spaces. In the simplest cases the authors present the general solution of this condition and proceed to give examples of other special cases.