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**Quaternionic closed operators, fractional powers and fractional diffusion processes.**
*(English)*
Zbl 1458.47001

Operator Theory: Advances and Applications 274. Cham: Birkhäuser (ISBN 978-3-030-16408-9/hbk; 978-3-030-16411-9/pbk; 978-3-030-16409-6/ebook). viii, 322 p. (2019).

The book can be seen as a continuation of F. Colombo et al. [Spectral theory on the S-spectrum for quaternionic operators. Cham: Birkhäuser (2018; Zbl 1422.47002)]. It provides an up-to-date collection of material on the quaternionic spectral theory and functional analysis, based largely on publications by the authors and their coworkers. After a detailed preview of the content and a recollection of results on slice hyperholomorphic functions and the \(S\)-functional calculus for bounded operators, in particular those with commuting components, the \(S\)-functional calculus is developed in detail following the direct approach of J. Gantner [J. Oper. Theory 77, No. 2, 287–331 (2017; Zbl 1424.47040)], which does not require the previously used transformations reducing the case of unbounded operators to that of bounded ones. In particular, the assumption that the resolvent set must contain a real point is removed, so that previously excluded essential operators such as, e.g., the gradient are now included. The operator merely need be closed and have a nonempty resolvent set, but the admissible functions must be slice hyperholomorphic on the \(S\)-spectrum of the operator and at infinity. If the operator is the infinitesimal generator of a strongly continuous semigroup, the function need not be slice hyperholomorphic at infinity any more. This leads to the generalization of the functional calculus of R. S. Phillips [Trans. Am. Math. Soc. 71, 393–415 (1951; Zbl 0045.21502)], the results here being taken from D. Alpay et al. [Anal. Appl., Singap. 15, No. 2, 279–311 (2017; Zbl 1370.47015)]. Strongly continuous quaternionic groups as well as uniformly continuous and strongly continuous quaternionic semigroups are studied, culminating in an analogue of the Hille-Phillips-Yosida theorem. For closed right linear quaternionic operators, it is investigated under what condition the perturbation of a semigroup generator results in a generator again.

The second half of the book is devoted to the extension of the \(H^\infty\)-functional calculus of A. McIntosh [Proc. Cent. Math. Anal. Austral. Nat. Univ. 14, 210–231 (1986; Zbl 0634.47016)] to quaternionic sectorial operators. The approach here is that of F. Colombo and J. Gantner [Milan J. Math. 86, No. 2, 225–303 (2018; Zbl 1447.47063)] following the strategy of M. Haase [The functional calculus for sectorial operators. Basel: Birkhäuser (2006; Zbl 1101.47010)]. It gets along without the assumption that the operators be injective and have dense range, which was needed in the earlier work of D. Alpay et al. [J. Funct. Anal. 271, No. 6, 1544–1584 (2016; Zbl 1350.47017)]. As part of the generalizations is rather straightforward, the authors concentrate on the technically more involved proofs such as those for the chain rule and the spectral mapping theorem. The theory developed provides the tools for handling fractional powers. First, in direct approach, fractional powers of invertible sectorial operators and of operators with negative real part are considered . This is followed by an extension of the indirect approach of T. Kato [Proc. Japan Acad. 36, 94–96 (1960; Zbl 0097.31802)], leading to an \(S\)-analogue of Kato’s resolvent formula. As applications, the authors study the fractional heat equation and fractional diffusion.

The book ends with historical notes and an Appendix recalling some basic results of functional analysis which can with minor changes be carried over to the quaternionic setting.

The second half of the book is devoted to the extension of the \(H^\infty\)-functional calculus of A. McIntosh [Proc. Cent. Math. Anal. Austral. Nat. Univ. 14, 210–231 (1986; Zbl 0634.47016)] to quaternionic sectorial operators. The approach here is that of F. Colombo and J. Gantner [Milan J. Math. 86, No. 2, 225–303 (2018; Zbl 1447.47063)] following the strategy of M. Haase [The functional calculus for sectorial operators. Basel: Birkhäuser (2006; Zbl 1101.47010)]. It gets along without the assumption that the operators be injective and have dense range, which was needed in the earlier work of D. Alpay et al. [J. Funct. Anal. 271, No. 6, 1544–1584 (2016; Zbl 1350.47017)]. As part of the generalizations is rather straightforward, the authors concentrate on the technically more involved proofs such as those for the chain rule and the spectral mapping theorem. The theory developed provides the tools for handling fractional powers. First, in direct approach, fractional powers of invertible sectorial operators and of operators with negative real part are considered . This is followed by an extension of the indirect approach of T. Kato [Proc. Japan Acad. 36, 94–96 (1960; Zbl 0097.31802)], leading to an \(S\)-analogue of Kato’s resolvent formula. As applications, the authors study the fractional heat equation and fractional diffusion.

The book ends with historical notes and an Appendix recalling some basic results of functional analysis which can with minor changes be carried over to the quaternionic setting.

Reviewer: Heinrich Hering (Rockenberg)

### MSC:

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47S05 | Quaternionic operator theory |

47A60 | Functional calculus for linear operators |

26A33 | Fractional derivatives and integrals |

60J60 | Diffusion processes |