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An introduction to the fine structures on the $S$-spectrum

Fabrizio Colombo, Antonino De Martino, Stefano Pinton, Irene Sabadini, Peter Schlosser

Abstract

Holomorphic functions are fundamental in operator theory and their Cauchy formula is a crucial tool for defining functions of operators. The Fueter-Sce extension theorem (often called Fueter-Sce mapping theorem) provides a two-step procedure for extending holomorphic functions to hyperholomorphic functions. In the first step, slice hyperholomorphic functions are obtained, and their associated Cauchy formula establishes the $S$-functional calculus for noncommuting operators on the $S$-spectrum. The second step produces axially monogenic functions, which lead to the development of the monogenic functional calculus. In this review paper we discuss the second operator in the Fueter-Sce mapping theorem that takes slice hyperholomorphic to axially monogenic functions. This operator admits several factorizations which generate various function spaces and their corresponding functional calculi, thereby forming the so-called fine structures of spectral theories on the $S$-spectrum.

An introduction to the fine structures on the $S$-spectrum

Abstract

Holomorphic functions are fundamental in operator theory and their Cauchy formula is a crucial tool for defining functions of operators. The Fueter-Sce extension theorem (often called Fueter-Sce mapping theorem) provides a two-step procedure for extending holomorphic functions to hyperholomorphic functions. In the first step, slice hyperholomorphic functions are obtained, and their associated Cauchy formula establishes the -functional calculus for noncommuting operators on the -spectrum. The second step produces axially monogenic functions, which lead to the development of the monogenic functional calculus. In this review paper we discuss the second operator in the Fueter-Sce mapping theorem that takes slice hyperholomorphic to axially monogenic functions. This operator admits several factorizations which generate various function spaces and their corresponding functional calculi, thereby forming the so-called fine structures of spectral theories on the -spectrum.

Paper Structure

This paper contains 6 sections, 1 theorem, 29 equations.

Table of Contents

  1. Introduction
  2. The Fueter-Sce mapping theorem and spectral theories
  3. What are the fine structures on the $S$-spectrum
  4. A unified way to see some function spaces
  5. The quaternionic setting
  6. The five dimensional case

Key Result

Theorem 2.1

Let $f$ be a holomorphic function in an open set of the upper half complex plane and let $f(x+iy)=u(x,y)+iv(x,y)$, for $x,y\in \mathbb{R}$, where $u$ and $v$ are real differentiable functions with values in $\mathbb{R}$. Consider the Euclidean space $\mathbb{R}^{n+1}$ whose points $(x_0,....,x_n)$ a

Theorems & Definitions (13)

  • Definition 1.1: Slice Cauchy domain
  • Definition 1.2: Slice hyperholomorphic functions (or slice monogenic functions)
  • Definition 1.3
  • Definition 1.5
  • Definition 1.6: Axially monogenic function
  • Theorem 2.1: Sce, 1957
  • Definition 3.1
  • Definition 3.2: holomorphic Cliffordian of order $k$
  • Definition 3.4: anti-holomorphic Cliffordian of order $k$
  • Definition 3.5: polyharmonic of degree $k$
  • ...and 3 more