An introduction to hyperholomorphic spectral theories and fractional powers of vector operators (Q2047638)

The authors exhibit an overview dedicated to the role of holomorphicity in the spectral theories for \(n\)-tuples of operators, with \(n>1\). A first discussion concerns the extension of the Cauchy formula from the theory of holomorphic functions in several variables to \(n\)-tuples of operators. A second discussion is related to hyperholomorphic functions of quaternionic or paravector variables. They note that the Fueter-Sce-Qian mapping theorem leads to two different types of hyperholomorphic functions: slice hyperholomorphic and monogenic functions. These notions are used in the spectral theory via the \(S\)-spectrum or the monogenic spectrum, respectively, somehow related by the \(F\)-functional calculus, yet having different and complementary applications. The authors finally discuss an approach to fractional diffusion problems via the \(S\)-spectrum. The reviewer takes the opportunity to add to the references of this expository paper two recent works of himself, related to holomorphic functional calculus. 1) [\textit{F.-H. Vasilescu}, Concr. Oper. 8, 90--113 (2021; Zbl 1489.30050)] 2) [\textit{F.-H. Vasilescu}, Pure Appl. Funct. Anal. 7, No. 1, 389--407 (2022; Zbl 1501.47124)]