The transformations f and F(different) appeared associated to the mean-square stability problem for infinite dimensional discrete bilinear systems evolving in a separable Hilbert space, being originally defined as infinite series in the Banach algebra of bounded linear operators on the Hilbert space where the system evolves. The present work starts with a previously defined sufficient stability condition, expressed by assumptions on the spectral radiuses of the mentioned transformations - both strictly less than one- and from the already known fact that the condition being partially fulfilled, that is, one of the spectral radiuses less than one, does not imply that it be so completely. Thus one poses a first question: in which cases does one have such an implication? The study is then developed on a simplification of the conditions from which it arose: both F and F(different) are taken as sums of only terms, and the initial question becomes the search for cases in which the equality betweem the spectral radiuses of F and F(different) occurs. More precisely, the terms that compose F and F(different) are products of operators in the above mentioned algebra, so that the behaviour of the spectral radiuses of F and F (different) is analysed by placing those operators in specific classes in that algebra. Under these assumptions, results related to the classes of self-adjoint, unitary, normal, isometries and subnormal operators are presented, as well as result referring to weighted shifts. Besides, a general result related to finite-dimensional spaces is also presented.