In this paper we consider operators acting on a subspace $\mathcal M$ of the space $L_2(\mathbb{R}^m;\mathbb{C}_m)$ of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace ${\mathcal M}$ is defined as the orthogonal sum of spaces ${\mathcal M}_{s,k}$ of specific Clifford basis functions of $L_2(\mathbb{R}^m;\mathbb{C}_m)$. Every Clifford endomorphism of ${\mathcal M}$ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they transform a space ${\mathcal M}_{s,k}$ into a similar space ${\mathcal M}_{s^{\prime }\!,k^{\prime }}$. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ${\mathcal M}$ is known. Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is studied in detail.
The paper is devoted to a careful analysis of the shape-preserving properties of the strongly continuous semigroup generated by a particular second-order differential operator, with particular emphasis on the preservation of higher order convexity and Lipschitz classes. In addition, the asymptotic behaviour of the semigroup is investigated as well. The operator considered is of interest, since it is a unidimensional Black-Scholes operator so that our results provide qualitative information on the solutions of classical problems in option pricing theory in Mathematical Finance.