A nonconstant coefficients differential operator associated to slice monogenic functions
By Fabrizio Colombo, And J. OSCAR GONZ´ALEZ-CERVANTES, Irene Sabadini. 2012
76 citations recorded by the source corpus.
. Slice monogenic functions have had a rapid development in the past few years. One of the main properties of such functions is that they allow the definition of a functional calculus, called S -functional calculus, for (bounded or unbounded) noncommuting operators. In the literature there exist two different definitions of slice monogenic functions that turn out to be equivalent under suitable conditions on the domains on which they are defined. Both the existing definitions are based on the validity of the Cauchy-Riemann equations in a suitable sense. The aim of this paper is to prove that slice monogenic functions belong to the kernel of the global operator defined by G ( x ) := | x | 2 ∂∂x 0 + x (cid:2) nj =1 x j ∂∂x j , where x is the 1-vector part of the paravector x = x 0 + x and n ∈ N . Despite the fact that G has nonconstant coefficients, we are able to prove that a subclass of functions in the kernel of G have a Cauchy formula. Moreover, we will study some relations among the three classes of functions and we show that the kernel of the operator G strictly contains the functions given by the other two definitions.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/bd33782dc424dd3d266037b07ceb20eac8502eef
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In Applied Mathematics
- Irene Sabadini
- Fabrizio Colombo
- Riccardo Ghiloni
- A. Perotti
- Daniele C. Struppa
- Caterina Stoppato
- J. González‐Cervantes
- Graziano Gentili
Citations in the corpus
- Entire slice regular functions (2015, background): “The generalization of slice regularity to real alternative algebra was developed in the papers [26, 116, 117, 118, 119] and, finally, while some results associated with the global operator associated to slice hyperholomorphicity are in the papers [53, 70, 80, 120].”
- The C-property for slice regular functions and applications to the Bergman space (2013, background): “The proof of the following result is given in [6].”
- Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions (2012, unknown): “We also point out that there exists a non-constant coefficient differential operator whose kernel contains slice hyperholomorphic functions defined on suitable domains; see [23].”
- Global differential equations for slice regular functions (2014, background): “See again [1] for the related operator G = n(Im(x))θ in the quaternionic case.”