TY - JOUR
T1 - A nonconstant coefficients differential operator associated to slice monogenic functions
AU - Colombo, Fabrizio
AU - Oscar González-Cervantes, J.
AU - Sabadini, Irene
PY - 2012
Y1 - 2012
N2 - 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 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.
AB - 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 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.
UR - http://www.scopus.com/inward/record.url?scp=84867758748&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-2012-05689-3
DO - 10.1090/S0002-9947-2012-05689-3
M3 - Artículo
SN - 0002-9947
VL - 365
SP - 303
EP - 318
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 1
ER -