  More than fifty years ago, in a couple of seminal works Kubo introduced the
important idea of generalized cumulants, extending to stochastic operators this
concept, implicitly introduced by Laplace in 1810. Kubo's idea has been applied
in several branches of physics, where the result of the average process is a
Lioville operator or an effective time evolution operator for the density
matrix of spin systems or the reduced density matrix for boson-fermions etc.
Despite this success, the theoretical developments in these Kubo works pose
problems that were highlighted many years ago by Fox and van Kampen and never
solved. These weaknesses and errors, in particular concerning the factorization
property of exponentials of cumulants and the explicit expressions that give
generalized cumulants in terms of generalized moments and vice-versa, caused
some perplexity (and confusion) about the possible application of this
procedure, limiting its use, in practice. In the present paper, we give a sound
ground to the approach to cumulant operators, working in a general framework
that shows the potentiality of the old Kubo's idea, today not yet fully
exploited. It results that for the same moment operators, different definitions
of generalized cumulants can be adopted. A general Kubo-Meeron closed-form
formula giving cumulant operators in terms of moment operators cannot be
obtained, but the reverse one, cumulants in terms of operators, is given and,
noticeably, formally it {\em does not} depend on the specific nature of the
moments, but just on the definition of the generalized cumulants.
