Brackx, Fred; de Schepper, Nele; Sommen, Frank Clifford–Hermite-monogenic operators. (English) Zbl 1164.47336 Czech. Math. J. 56, No. 4, 1301-1322 (2006). Summary: 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 \({\mathcal M}_{s'\!,k'}\). 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. Cited in 2 Documents MSC: 47B99 Special classes of linear operators 30G35 Functions of hypercomplex variables and generalized variables Keywords:differential operators; Clifford analysis PDFBibTeX XMLCite \textit{F. Brackx} et al., Czech. Math. J. 56, No. 4, 1301--1322 (2006; Zbl 1164.47336) Full Text: DOI EuDML Link References: [1] F. Brackx, R. Delanghe, F. Sommen: Clifford Analysis. Pitman Publ., Boston-London-Melbourne, 1982. [2] F. Brackx, N. De Schepper, K. I. Kou, and F. Sommen: The Mehler formula for the generalized Clifford-Hermite polynomials. Acta Mathematica Sinica. Accepted. · Zbl 1125.30042 [3] R. Delanghe, F. Sommen, and V. Souček: Clifford Algebra and Spinor-Valued Functions. Kluwer Acad. Publ., Dordrecht, 1992. · Zbl 0747.53001 [4] F. Sommen: Special functions in Clifford analysis and axial symmetry. J. Math. Anal. Appl. 130 (1988), 110–133. · Zbl 0634.30042 · doi:10.1016/0022-247X(88)90389-7 [5] F. Sommen, N. Van Acker: Monogenic differential operators. Results Math. 22 (1992), 781–798. · Zbl 0765.30033 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.