Urbanik introduces in  and  the classes Lm
of distributions on R1
and finds relations with stable distributions. Kumar-Schreiber  and Thu  extend some of the results to distributions on Banach spaces. Sato  gives alternative definitions of the classes Lm
and studies their properties on Rd
. Earlier Sharpe  began investigation of operator-stable distributions and, subsequently, Urbanik  considered the operator version of the class L on Rd
. Jurek  generalizes some of Sato’s results  to the classes associated with one-parameter groups of linear operators in Banach spaces. Analogues of Urbanik’s classes Lm
) in the operator case are called multiply (or completely) operator-selfdecomposable. They are studied in relation with processes of Ornstein-Uhlenbeck type or with stochastic integrals based on processes with homogeneous independent increments (Wolfe , , Jurek-Vervaat , Jurek , , and Sato-Yamazato , ). The purpose of the present paper is to continue the preceding papers, to give explicit characterizations of completely operator-selfdecomposable distributions and operator-stable distributions on Rd
, and to establish relations between the two classes. For this purpose we explore the connection of the structures of these classes with the Jordan decomposition of a basic operator Q.