Overview

This book introduces an innovative method to address two significant challenges in functional calculus theory: developing a general functional calculus for non-commuting n-tuples of operators and creating a functional calculus for quaternionic operators Our approach leverages recent advancements in Clifford analysis and the theory of quaternion-valued functions, providing a fresh perspective on these complex mathematical problems.

The classical Riesz–Dunford functional calculus highlighted the need for a functional calculus applicable to multiple operators This necessity was initially identified by Weyl in the 1930s Anderson later addressed this issue by employing the Fourier transform and n-tuples of self-adjoint operators that meet specific Paley–Wiener estimates.

In his foundational work, Taylor presents a novel method effective for n-tuples of commuting operators, while later exploring the Weyl calculus for noncommuting, self-adjoint operators These pioneering studies have laid the groundwork for various future developments in this field.

A successful approach to tackling noncommutativity involves utilizing Clifford algebra-valued functions This innovative concept has been effectively explored in the research of Jeffries, McIntosh, and their collaborators.

For a comprehensive understanding of this topic, refer to sources [60], [61], [62], [65], and [77] While the noncommutative framework is beneficial for analyzing multiple operators, it is important to recognize that there may still be limitations concerning the n-tuples of operators and their associated spectrum.

To broaden the scope of functional calculus, it is essential to move beyond current restrictions This can be achieved by exploring the development of a functional calculus that utilizes slice monogenic functions.

Hyperholomorphic Functions, Progress in Mathematics 289, DOI 10.1007/978-3-0348-0110-2_1, 1 © Springer Basel AG 2011

F Colombo et al., Noncommutative Functional Calculus: Theory and Applications of Slice

The functions initially introduced by the authors in [26] have since undergone significant theoretical development, as demonstrated by the extensive literature available on the topic, including works cited in [15], [18], [24], [26], [27], [28], [29], [30], [53], and [55].

To develop a functional calculus for a specific class of functions, establishing an appropriate integral formula is essential In the case of slice monogenic functions, this integral formula is presented and proven in Chapter 2 The foundational proof of this formula was initially provided by Colombo and Sabadini.

The integral formula discussed is computed over a path in the complex plane, highlighting a distinction from classical monogenic functions In the context of slice monogenic functions, the Cauchy kernel is represented by functions that are either left or right slice monogenic in a specific variable Consequently, two distinct kernels are required for handling left and right slice monogenic functions The resulting Cauchy formula for slice monogenic functions is particularly well-suited for defining a functional calculus applicable to both bounded and unbounded n-tuples of operators, regardless of whether they commute.

In the initial section of this book, we will explore the key findings of the theory of slice monogenic functions and the related functional calculus for n-tuples of operators that may not commute This calculus was first introduced in paper [25] for a specific class of functions and subsequently expanded to encompass the general case in [18].

In the second part of the book, we explore a distinct yet related problem that has intrigued researchers for many years and remains challenging Our focus is on defining a function of a single quaternionic linear operator While there are similarities to previous discussions, such as the noncommutative nature of the setting and the quaternion space being a Clifford algebra, the specific problem we are addressing differs from earlier analyses.

When dealing with the functional calculus forn-tuples of operators, our ap- proach is to embed the n-tuple of linear operators (over the real field) into the

In the context of Clifford algebra, we examine an operator that is quaternionic linear Due to the noncommutative nature of this setting, the operator can be classified as either left or right linear Our approach will highlight the distinctions between these two cases Understanding such operators is essential for applications in quaternionic quantum mechanics.

To construct a functional calculus for quaternion functions, it is essential to define the appropriate function space The most well-established space is that of regular functions, as introduced by Fueter in his seminal works These functions are differentiable within the quaternion space and adhere to the Cauchy–Fueter system, a set of first-order linear partial differential equations Importantly, the Cauchy–Fueter system applies to functions in both \(\mathbb{R}^4\) and \(\mathbb{R}^3\), with the latter being introduced earlier by G Moisil and N Theodorescu.

Plan of the book

The article discusses a specific type of calculus involving functions that adhere to Fueter's regularity conditions, as outlined by the authors in reference [11] However, it highlights that this functional calculus falls short of expectations due to several factors, which are detailed in [11] A key observation is that even a straightforward function like f(q) = q² fails to meet Fueter's regularity criteria.

Recent research has introduced a new concept called slice regularity, which serves as the foundation for slice monogenicity, differing from Fueter's original notion The second part of this book will explore how slice regular functions can be utilized to develop a functional calculus for quaternionic linear operators over the quaternions The quaternionic functional calculus, particularly for functions with power series expansions, was initially presented in prior studies.

Chapter 4 draws inspiration from recent research that utilizes a new Cauchy formula, which serves as a fundamental tool for defining quaternionic functional calculus applicable to both bounded and unbounded operators, even when their components do not commute This paper also explores the quaternionic evolution operator, focusing specifically on bounded linear operators The study references findings that extend the Hille–Phillips–Yosida theory to the quaternionic framework, marking a significant advancement in understanding the relevance of this novel functional calculus in the field of physics.

Although slice monogenic and slice regular functions share similar definitions and properties, significant differences necessitate their independent treatment These distinctions primarily arise from the contrasting algebraic characteristics of quaternions and Clifford numbers, particularly in higher dimensions where the number of imaginary units generating the Clifford algebra exceeds two.

This book primarily draws on the authors' recent research, with the exception of fundamental concepts related to Clifford algebras, a brief appendix on the classical Riesz–Dunford functional calculus, and select findings included in the notes To highlight the key outcomes, we offer a concise overview of the central results presented within the text.

Slice monogenic functions Consider the universal Clifford algebraR n generated by n imaginary units {e 1 , , e n } satisfyinge i e j +e j e i = −2δ ij and a function f defined on the Euclidean spaceR n +1 , identified with the set of paravectors in

A slice monogenic function in R^n is defined by the condition that all restrictions of the function f to appropriate complex planes are holomorphic To illustrate these complex planes, we will utilize the sphere of unit 1-vectors.

From a geometric point of view, S is an (n−1)-sphere in R n+1 Note that an element I ∈S is again an imaginary unit sinceI 2 =−1 If we take any element

In the context of complex analysis, we can create the plane \( C_I \) within \( \mathbb{R}^{n+1} \), which passes through the points 1 and \( I \) This two-dimensional real subspace is isomorphic to the complex plane, establishing an algebraic isomorphism Consequently, we refer to \( C_I \) as a "complex plane," where an element in this space is typically expressed as \( x = u + Iv \).

In the context of R n+1, every element belongs to a complex plane, making the Euclidean space R n+1 the union of all complex planes C I, where I varies in S When considering an open set U ⊆ R n+1 and a function f : U → R n+1 that is differentiable in the real sense, we can restrict f to the complex plane C I for a given I ∈ S A function f is classified as a left slice monogenic function if it satisfies specific criteria across these complex planes.

Because of the noncommutativity we also have the right version of this notion and we say thatf is a right slice monogenic function if, for everyI∈S, we have

A slice monogenic function differs from classical monogenic functions in that it is not inherently harmonic, although its restrictions to any complex plane \( C_I \) are harmonic This distinction is significant in the context of monogenicity theory Notably, all convergent power series of the form \( \sum_{n \geq 0} a_n x^n \) are classified as left slice monogenic within their convergence domain, a property that is essential for developing a functional calculus.

To comprehend slice monogenic functions, it is essential to examine them within axially symmetric slice domains, as these represent their natural domains of definition A domain U in R n +1 is classified as a slice domain (or s-domain) if the intersection of U with R is non-empty and if the intersection of U with C I forms a domain in C I for all instances.

I∈S We say thatU ⊆R n +1 is an axially symmetric domain if, for allu+Iv∈U, the whole (n−1)-sphereu+vSis contained inU.

The class of slice monogenic functions defined over axially symmetric s-domains is characterized by a specific Representation Formula, also known as the Structure Formula, as established in [15].

Representation Formula LetU ⊆R n +1 be an axially symmetric s-domain and let f be a left slice monogenic function on U For any vector x=x 0 +I x |x| ∈U and for allI∈S, we have f(x) = 1

The Representation Formula states that if we know the value of a slice mono- genic function on the intersection of an axially symmetric s-domainUwith a plane

C I , then we can reconstruct the function on all ofU.

An analogous formula exists for right slice monogenic functions, requiring appropriate modifications The initial step in developing a functional calculus involves demonstrating a Cauchy integral formula with a slice monogenic kernel While it is possible to establish an integral representation formula using the standard Cauchy kernel (x − x₀)⁻¹, this method is constrained because the kernel lacks slice monogenic properties Therefore, we will examine the Cauchy kernel series specifically for left slice monogenic functions, where x and s are elements of Rⁿ⁺¹, acknowledging that these elements generally do not commute.

The left noncommutative Cauchy kernel series, denoted as S −1 L (s, x), converges for |x| < |s| and is slice monogenic in x The sum of this series can be computed as n ≥0 x n s −1− n = −(x² − 2 Re[s]x + |s|²) −1 (x − s) This function, S −1 L (s, x), serves as a suitable Cauchy kernel for left slice monogenic functions, as it aligns with the conventional Cauchy kernel in complex analysis when restricted to the commutative plane C I Furthermore, S −1 (s, x) is left slice monogenic in x and right slice monogenic in s within its defined domain, with similar principles applicable to right slice monogenic functions.

S −1 R (s , x) : n ≥0 s − n −1 x n a right noncommutative Cauchy kernel series; it is convergent for | x | 0be such thatΔ I (y 0 , r) ={(u+Iv) : (u−y 0 ) 2 +v 2 ≤r 2 } is contained in

U∩C I If M I = max{|f(x)|:x ∈∂Δ I (y 0 , r)} and if M = inf{M I :I∈S}, then

Proof For anyI∈S, it is possible to write

Therefore, for anyI∈Swe can write the following sequence of inequalities:

By taking the infimum, forI∈S, of the right-hand side of the inequality we prove the assertion

Using the previous result it is immediate to show the following

Theorem 2.4.6 (Liouville) Letf : R n +1 →R n be an entire s-monogenic function.

If f is bounded, then f is constant onR n+1

Proof Suppose that|f| ≤M onR n +1 By the previous theorem we have:

≤ M r n , n≥0, and by lettingr→+∞we obtain

∂u n (0) = 0 for anyn >0, which impliesf(x) =c, withc∈R n

Corollary 2.4.7 Let f : R n +1 → R n be an entire s-monogenic function If lim x →∞ f exists, then f is constant onR n +1

Theorem 2.4.8 LetU be an open set inR n+1 If f : U →R n is an s-monogenic function, then

∂Δ d x f(x) = 0 for any disc Δ⊂U ∩C I with center in a point on the real axis.

Proof This result is an easy consequence of the analogous result for holomorphic functions of one complex variable and of the Splitting Lemma Conversely, we have the following result:

Theorem 2.4.9 states that if U is an axially symmetric s-domain and f: U → R^n is a real differentiable function, then f is an s-monogenic function provided that the integral of f over any closed, piecewise C^1 curve γ I within U ∩ C I, which is homotopic to a point, equals zero.

Proof This is a consequence of the classical Morera’s theorem and of the definition of s-monogenic function

Proposition 2.4.10 states that if \( f : B(0, R) \to \mathbb{R}^n \) is an s-monogenic function represented by the series \( x^m a_m \) converging on \( B \), then the composition of \( f \) and \( I_0 = x^{-1} \) results in an s-monogenic function defined on \( \mathbb{R}^{n+1} \setminus B(0, 1/R) \) This new function can also be expressed by the series \( x^{-m} a_m \) that converges on \( \mathbb{R}^{n+1} \setminus B(0, 1/R) \).

Proof Proposition 2.3.4 implies thatf◦ I 0is an s-monogenic function onR n +1 \ B(0,1/R) The statement follows from the analogous result for holomorphic func- tions in one complex variable

Theorem 2.4.11 (Laurent series) Letf be an s-monogenic function in a spherical shell A={ x ∈R n+1 | R 1