Open AccessA nonconstant coefficients differential operator associated to slice monogenic functions
Author(s) -
Fabrizio Colombo,
J. Oscar González-Cervantes,
Irene Sabadini
Publication year - 2012
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/s0002-9947-2012-05689-3
Subject(s) - mathematics , differential operator , type (biology) , algorithm , operator (biology) , pure mathematics , algebra over a field , chemistry , biochemistry , repressor , gene , transcription factor , ecology , biology
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.\udIn 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.\udBoth 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 a global operator G.\udDespite the fact that G has non constant coefficients, we are able to prove that a subclass of functions in the kernel of G have a Cauchy formula.\udMoreover, 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