Formulations of the -functional calculus and some consequences

Abstract

In this paper we introduce the two possible formulations of the F-functional calculus which are based on the Fueter-Sce mapping theorem in integral form and we introduce the pseudo F-resolvent equation. In the case of dimension 3 we prove the F-resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the F-functional calculus.AMS Classification: 47A10, 47A60. Key words: Fueter-Sce mapping theorem in integral form, the F-spectrum, formulations of the Ffunctional calculus for n-tuples of operators, projectors, quaternionic version of the F -functional calculus, quaternionic F-resolvent equation.

Schatten class and Berezin transform of quaternionic linear operators

“…We recall the spectral theorem for compact quaternionic operators ; note that this is a special case of the spectral theorem for arbitrary normal operators, which was recently established in . Theorem Givc with spherical point spectrum σ p S ( T ), there exists a Hilbert basis made of eigenvectors of T such that where is an eigenvalue relative to the eigenvector z , and if λ z ≠0, then there are only a finite number of distinct such that .…”

“…The Hille‐Phillips‐functional calculus for generators of strongly continuous groups was generalized to the quaternionic setting in . Moreover, the Fueter mapping theorem revealed a deep relation between slice hyperholomorphic and monogenic functions, which allowed to defined the ℱ‐functional calculus , a functional calculus for monogenic functions that is also based on the S‐spectrum. The spectral theorem for quaternionic bounded or unbounded normal operators based on the S‐spectrum has been proved recently in the papers .…”

“…The spectral theorem for quaternionic bounded or unbounded normal operators based on the S ‐spectrum has been proved recently in the papers . The case of compact operators can be found in .…”

“…The S ‐resolvent set is defined as The notion of S ‐spectrum for quaternionic operators arises naturally in the slice hyperholomorphic functional calculus, called S ‐functional calculus or quaternionic functional calculus, which is the quaternionic analogue of the Riesz–Dunford functional calculus for complex operators on a complex Banach space. Recently, it turned out that also the spectral theorem for quaternionic operators (bounded or unbounded) is based on the S ‐spectrum, see .…”

“…Additional information on how to orient these equivalence classes in accordance with the operator is needed and it is provided by the operator J . This phenomenon occurs, for instance, also for the Jordan normal form of quaternionic matrices in and the spectral theorem in . The definition of a Schatten class of operators in the quaternionic setting is only possible, if one restricts to operators that have a common orientation given by a fixed operator J .…”