Geodesic
Clifford-Hermite-monogenic operators
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1301-1322

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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.
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.
Classification : 30G35, 47B99, 47F05
Keywords: differential operators; Clifford analysis 
Brackx, Fred; de Schepper, Nele; Sommen, Frank. Clifford-Hermite-monogenic operators. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1301-1322. http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a15/
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     title = {Clifford-Hermite-monogenic operators},
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     zbl = {1164.47336},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a15/}
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