Abstract
In this chapter we introduce slice hyperholomorphic functions with values in a quaternionic Banach space. As in the complex case, there are two equivalent notions, namely weak and strong slice hyperholomorphicity. In order to properly define a multiplication between slice hyperholomorphic functions, we give a third characterization in terms of the Cauchy–Riemann system. Operator-valued functions can be obtained by using the so-called S-functional calculus. This calculus is associated with the notions of S-spectrum and S-resolvent, which are introduced and studied. We also present some hyperholomorphic extension results and, finally, we study the Hilbert-space-valued quaternionic Hardy space of the ball and its backward-shift invariant subspaces.
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© 2016 Springer International Publishing Switzerland
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Alpay, D., Colombo, F., Sabadini, I. (2016). Operator-valued Slice Hyperholomorphic Functions. In: Slice Hyperholomorphic Schur Analysis. Operator Theory: Advances and Applications, vol 256. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-42514-6_7
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DOI: https://doi.org/10.1007/978-3-319-42514-6_7
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-42513-9
Online ISBN: 978-3-319-42514-6
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