A calculus on L\'evy exponents and selfdecomposability on Banach spaces

In infinite dimensional Banach spaces there is no complete characterization of the L'evy exponents of infinitely divisible probability measures. Here we propose \emph{a calculus on L'evy exponents} that is derived from some random integrals. As a consequence we prove that \emph{each} selfdecomposable measure can by factorized as another selfdecomposable measure and its background driving measure that is s-selfdecomposable. This complements a result from the paper of Iksanov-Jurek-Schreiber in the Annals of Probability $\mathbf{\text{32}}$, 2004.}