Slice Hyperholomorphic Functional Calculi and the Spectral Theorem on the S-Spectrum

Fabrizio Colombo () and Irene Sabadini ()
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Fabrizio Colombo: Politecnico di Milano, Dipartimento di Matematica
Irene Sabadini: Politecnico di Milano, Dipartimento di Matematica

Chapter 69 in Operator Theory, 2026, pp 2159-2204 from Springer

Abstract: Abstract This chapter provides a comprehensive survey of the material developed since 2006, when the new concept of S-spectrum was introduced, leading to the emergence of a spectral theory related to the S-spectrum and its various extensions. Since then, the material has grown enormously, and many authors have made important contributions. In this chapter, it is difficult to condense all the materials contained in a growing number papers and over 10 books on this topic, but we will describe the main lines connected with the S-functional calculus and its H ∞ $$H^\infty $$ versions, the spectral theorem, and the fine structures on the S-spectrum that have broadened impressively this new spectral theory.

Keywords: S-Spectrum; S-Functional calculus; Quaternionic and Clifford linear operators; Spectral theorem for quaternionic and Clifford operators; Fueter–Sce mapping theorem; Fine structures on the S-spectrum; Slice hyperholomorphic functions; 47A10; 47A60; 30G35 (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16356-1_21

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