A new proof and generalization of the Bogolyubov–Ruelle theorem

A new proof of the Bogolyubov–Ruelte theorem based on the general properties of a dual pair of Banach spaces $\langle'E_\xi, E_\xi\rangle$ is proposed. The new proof makes it possible to Lake into account readily the case of nonempty boundary conditions and extend the assertion of the theorem to a domain which includes the standard disk of analyticity with respect to the activity.