Towards a general \(\mathcal{F}\)-resolvent equation and Riesz projectors
In Section 1, Introduction, the authors start explaining the steps in which the Fueter-Sce-Qian extension (or construction) consists (step A consists in the extension of holomorphic functions to the class of slice hyperholomorphic functions and step B consists in the extension of slice hyperholomorphic functions to monogenic functions). Step A has been helpful in the foundation of the quaternionic spectral theory on the \(S\)-spectrum. Step B generates the monogenic functional calculus based on the monogenic spectrum. The main goal of this paper is to study some properties of the \(\mathcal F\)-functional calculus, which is a bridge between the spectral theory on the \(S\)-spectrum and the monogenic spectral theory. Also in the Introduction there were made some considerations on the resolvent equations in various settings. It is recalled that in the slice hyperholomorphic setting there exist two different resolvent operators (according to left of right slice hyperholomorphicity) and they both are used to obtain the \(S\)-resolvent equation. Some of their properties are recalled. There are considered paravector operators \(T=T_0+e_1 T_1 + \cdots + e_n T_n\) with commuting components and the \(\mathcal F\)-spectrum of \(T\) is defined. It turns out that the \(\mathcal F\)-spectrum is the commutative version of the \(S\)-spectrum. At the end of the Introduction there are described the main results of the paper. ``Preliminary material'' \, is the he title of Section 2. The authors start this section recalling the story of how to extend a holomorphic function to another one that belongs to a wider class. Fueter started with holomorphic functions defined on open sets of the complex upper half plane and that satisfy few more conditions and gave a method to generate quaternionic Cauchy-Fueter regular functions. For his part, Sce started with the same kind of functions and gave a method to generate monogenic functions defined on a subset of \(\mathbb R^{n+1}\) and with values in a Clifford algebra (using an appropriate power of a Laplacian operator). Tao Qian considered the case when the power of the Laplacian is a fraction. These are the reason why the author refers to the extension process the Fueter-Sce-Qian mapping theorem. It is recalled what is an axially symmetric set \(U \subseteq \mathbb R^{n+1}\), what the left and right slice hyperholomorphic functions are (\(SH_L(U)\) and \(SH_R(U)\)). The (slice) left and right hyperholomorphic Cauchy kernels (\(S_L^{-1}(s,x)\) and \(S_R^{-1}(s,x)\)) and the Cauchy formulas are recalled also together with some of their properties. The \(\mathcal F\)-functional calculus is defined for \(n\)-tuples of commuting operators, \(n\) odd. It generates a monogenic functional calculus and it is based in the strategy of applying the Fueter-Sce operator \(T_{FS2} := \Delta^{(n-1)/2}\), with \(\Delta \) the Laplace operator in \(n+1\) variables, to the Slice Cauchy Formula, arriving to an integral formula in the setting of axially monogenic functions (AM(U)) and from here surges the \(\mathcal F\)-functional calculus. In the rest of Section 2 there are recalled some properties of the image of the slice Cauchy kernels under the operator \(T_{FS2}\). The \(\mathcal F\)-resolvent operators and their properties are presented also. Section 3 is devoted to new series expansions for the \(\mathcal F\)-resolvent operators. In Section 4 there are given the \(\mathcal F\)-resolvent equations for \(n=5 \) and for \(n=7\). In Section 5 the authors study the Riesz projectors for the \(\mathcal F\)-functional calculus for \(n=5\). In Section 6 it is studied the \(\mathcal F\)-resolvent equation and the Riesz projectors for the \(\mathcal F\)-functional calculus for the general case of \(n\) an odd number.
slice regular weighted composition operator
scientific article; zbMATH DE number 7595154
scientific article