Continuous slice functional calculus in quaternionic Hilbert spaces

By Riccardo Ghiloni, Valter Moretti, A. Perotti. 2012

235 citations recorded by the source corpus.

The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic $C^*$--algebras and to define, on each of these $C^*$--algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/58b53f569071517e478c1c9e1098e47c2d033a65

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Entire slice regular functions (2015, background): “The Continuous slice functional calculus in quaternionic Hilbert spaces is studied in [114].”
The algebra of slice functions (2015, background): “The effectiveness of working with slice functions without restricting to the slice regular ones can be appreciated by looking at some of its applications, for instance the Cauchy integral formula for slice functions of class C (1), [16, Theorem 27], or the definition of a continuous (slice) function”
A New Resolvent Equation for the $$S$$S-Functional Calculus (2013, methodology): “Using the S-spectrum in [23] the authors introduce the continuous functional calculus in a quaternionic Hilbert space.”