Some results on subordination, selfdecomposability and operator semi-stability

Two topics are studied in relation to subordination, selfdecomposability and operator semi-stability. One is that strict operator semi-stability of the subordinand process by the selfdecomposable subordinator gives operator semi-selfdecomposability of the subordinated process. Another is an answer to the question whether any subordinated process arising from a strictly operator semi-stable subordinand and a strictly semi-stable subordinator can be strictly operator semi-stable.