Operator Theory

This is the first part of the exposition which appears in this handbook under the common title “The Reproducing Kernel Property and Its Space.”

This is a follow-up of the chapter The Reproducing Kernel Property and Its Space: The Basics, which is the first part of the two-chapter project by the present author appearing in this handbook under the common title “The Reproducing Kernel Property and Its Space.”

The original Pick interpolation problem asks when an analytic function from the disk to the half-plane can interpolate certain prescribed values. This was solved by G. Pick in 1916. This chapter discusses this theorem and generalizations of it to other domains.

In this survey a brief review of results on the Bergman kernelBergman kernel and Bergman distance concentrating on those fields of complex analysis which remain in the focus of the research interest of the authors is presented. The topics discussed contain general discussion of ℒh2$$\mathcal{L}_{\mathrm{h}}^{2}$$ spaces, behavior of the Bergman distance, regularity of extension of proper holomorphic mappings, and recent development in the theory of Bergman distance stemming from the pluripotential theory and very short discussion of the Lu Qi Keng problem.

This work intends to serve as an introduction to sampling theory. Basically, sampling theory deals with the reconstruction of functions through their values on an appropriate sequence of points by means of sampling expansions involving these values. Reproducing kernel Hilbert spaces are suitable spaces for sampling purposes since evaluation functionals are continuous. As a consequence, the recovery of any function from a sequence of its samples depends on the basis properties of the reproducing kernel at the sampling points.

Reproducing kernels are a mathematical tool that is ubiquitous in many areas of theoretical and mathematical physics. Here we specialize the notion to show its applications to the theory of coherent states, coherent state and Berezin-Toeplitz quantization, and to the related theory of the continuous wavelet transform. The aim is to demonstrate the unifying mathematical aspect, given by the reproducing kernel, of these different theories.

It is shown how reproducing kernels, in a wide class, define in a very natural manner differential geometric objects like linear connections, covariant derivatives, and curvatures. The correspondence from kernels to connections is achieved through a pullback operation from the tautological universal bundle, using a suitable classifying morphism for the given kernel. The theory is illustrated by several examples including classical kernels in function spaces, kernels occurring in dilation theory for completely positive maps, and kernels on homogeneous vector bundles.

Here an introduction to linear spaces with an indefinite inner product, the so-called Kreĭn spaces, and to multi-valued operators between them is given. More precisely, the basic properties of these indefinite inner product spaces, which are generalizations of Hilbert spaces, are described, where the similarities to and differences compared with Hilbert spaces are emphasized. Secondly, the basic properties of multi-valued operators, which are generalizations of (linear) operators, are presented; such operators appear naturally, for instance, in the treatment of differential equations.

Here the basic properties of two classes of (multi-valued) operators between Kreĭn spaces are presented: the symmetric and isometric relations. Both types of multi-valued operators (relations) naturally appear for instance when studying differential equation; for example Sturm–Liouville equations with an indefinite weight.

This chapter contains a short review of the theory of boundary triplets, and the corresponding Weyl functions, of symmetric operators in Hilbert and Kreĭn spaces. The theory of generalized resolvents of such operators is exposed from the point of view of boundary triplets approach. Applications to different continuation problems related to the extension theory of Kreĭn space symmetric operators are discussed.

A brief survey of the commutant lifting theorem is presented. This is initially done in the Hilbert space context in which the commutant lifting problem was initially considered, both in Sarason’s original form and that of the later generalization due to Sz.-Nagy and Foias. A discussion then follows of the connection with contraction operator matrix completion problems, as well as with the Sz.-Nagy and Andô dilation theorems. Recent work in abstract dilation theory is outlined, and the application of this to various generalizations of the commutant lifting theorem are indicated. There is a short survey of the relevant Kreĭn space operator theory, focusing in particular on contraction operators and highlighting the fundamental differences between such operators on Kreĭn spaces and Hilbert spaces. The commutant lifting theorem is formulated in the Kreĭn space context, and two proofs are sketched, the first using a multistep extension procedure with a Kreĭn space version of the contraction operator matrix completion theorem, and the second diagrammatic approach which is a variation on a method due to Arocena. Finally, the problem of lifting intertwining operators which are not necessarily contractive is mentioned, as well as some open problems.

LocallyLocally definitizable operators definitizableSpectrum operators have locally the same spectral properties as definitizable operators in Kreĭn spaces. It is shown in this note how to define spectral points of positive/negative type and spectral points of type π+∕π−$$\pi _{+}/\pi _{-}$$ via approximative eigensequences. This approach has the advantage that it does not make use of a local spectral function. Moreover, perturbation results for locally definitizable operators are discussed. Spectral points of type π₊ and π₋ are stable under compact perturbations. For real spectral points of type π₊ and type π₋ which are not in the interior of the spectrum the growth of the resolvent in an open neighborhood of these spectral points is of finite order. This can be utilized to show the existence of a local spectral function with singularities. With the help of this local spectral function one can also characterize spectral points of positive/negative type and spectral points of type π₊ and type π₋: It turns out that all spectral subspaces corresponding to sufficiently small neighborhoods of spectral points of positive/negative type are Hilbert or anti-Hilbert spaces and spectral subspaces corresponding to spectral points of type π₊ or type π₋ are Pontryagin spaces.Locally definitizable operators are used in the study of indefinite Sturm–Liouville problems, λ-dependent boundary value problems, ????$$\mathcal{P}\mathcal{T}$$-symmetric operators, and partial differential equations and in the study of problems of Klein–Gordon type.

Schur analysis comprises topics like: the Schur transformation on the class of Schur functions (by definition, the functions which are holomorphic and bounded by 1 on the open unit disk) and the Schur algorithm, Schur parameters and approximation, interpolation problems for Schur functions, factorization of rational 2 × 2 matrix polynomials, which are 100−1$$\left [\begin{array}{*{10}c} 1& 0\\ 0 &-1 \end{array} \right ]$$-unitary on the unit circle, and a related inverse scattering problem. This note contains a survey of indefinite versions of these topics related to the class of scalar generalized Schur functions. These are the meromorphic functions s(z) on the open unit disk for which the kernel 1−s(z)s(w)∗1−zw∗$$\frac{1-s(z)s(w)^{{\ast}}} {1-zw^{{\ast}}}$$ has finitely many negative squares. We also review a generalization of the Schur transformation to classes of functions on a general domain one of which is the class of scalar generalized Nevanlinna functions. These are the meromorphic functions n(z) on the open upper half plane for which the kernel n(z)−n(w)∗z−w∗$$\frac{n(z)-n(w)^{{\ast}}} {z-w^{{\ast}}}$$ has finitely many negative squares.

This chapter is an introduction to reproducing kernel Kreĭn spaces and their interplay with operator valued Hermitian kernels. Existence and uniqueness properties are carefully reviewed. The approach used in this survey involves the more abstract, but very useful, concept of linearization or Kolmogorov decomposition, as well as the underlying concepts of Kreĭn space induced by a selfadjoint operator and that of Kreĭn space continuously embedded. The operator range feature of reproducing kernel spaces is emphasized. A careful presentation of Hermitian kernels on complex regions that point out a universality property of the Szegö kernels with respect to reproducing kernel Kreĭn spaces of holomorphic functions is included.

This article gives an introduction and short overview on generalized Nevanlinna functions, with special focus on asymptotic behavior and its relation to the operator representation.

It is the aim of the present survey to provide an introduction into the theory of indefinite Hamiltonians and to give an overview over the most important results. Indefinite Hamiltonians can be seen as a distributional generalization of the classical theory of canonical Hamiltonian differential equations as studied among many others by M.G. Kreĭn and Louis de Branges. The spaces in the background of this theory are no longer Hilbert spaces as in the classical situation, but Pontryagin spaces. This type of spaces can be seen as a Hilbert where the Hilbert space scalar product is replaced by a finite dimensional perturbation. In a similar sense indefinite Hamiltonians can be seen as a certain perturbation of classical Hamiltonians. The theory of indefinite Hamiltonians involves certain reproducing kernel Pontryagin spaces consisting of entire function which constitutes a generalization of the theory of Louis de Branges on Hilbert spaces of entire functions.

Consider the indefinite Sturm–Liouville problem −f′′=λrf$$-f^{{\prime\prime}} = \lambda rf$$ on [−1, 1] with Dirichlet boundary conditions and with a real weight function r ∈ L1[−1, 1] changing its sign. The question is studied whether or not the eigenfunctions form a Riesz basis of the Hilbert space L_{| r |}2[−1, 1] or, equivalently, ∞ is a regular critical point of the associated definitizable operator in the Kreĭn space L _r 2[−1, 1]. This question is also related to other subjects of mathematical analysis like half range completeness, interpolation spaces, HELP-type inequalities, regular variation, and Kato’s representation theorems for non-semibounded sesquilinear forms. The eigenvalue problem can be generalized to arbitrary self-adjoint boundary conditions, singular endpoints, higher order, higher dimension, and signed measures. The present paper tries to give an overview over the so far known results in this area.

The aim of this chapterFinite dimensional indefinite inner product spaces is to give a few examples for the fruitful interaction of the theory of finite-dimensional indefinite inner product spaces as a special theme in Operator Theory on the one hand and Numerical Linear Algebra as a special theme in Numerical Analysis on the other hand. Two particular topics are studied in detail. First, the theory of polar decompositions in indefinite inner product spaces is reviewed, and the connection between polar decompositions and normal matrices is highlighted. It is further shown that the adaption of existing algorithms from Numerical Linear Algebra allows the numerical computation of these polar decompositions. Second, two particular applications are presented that lead to the Hamiltonian eigenvalue problem. The first example deals with Algebraic Riccati Equations that can be solved via the numerical computation of the Hamiltonian Schur form of a corresponding Hamiltonian matrix. It is shown that the question of the existence of the Hamiltonian Schur form can only be completely answered with the help of a particular invariant discussed in the theory of indefinite inner products: the sign characteristic. The topic of the second example is the stability of gyroscopic systems, and it is again the sign characteristic that allows the complete understanding of the different effects that occur if the system is subject to either general or structure-preserving perturbations.

In this essay algebraic Riccati equations will be discussed. It turns out that Hermitian solutions of algebraic Riccati equations which originate from systems and control theory may be studied in terms of invariant Lagrangian subspaces of matrices which are selfadjoint in an indefinite inner product. The essay will describe briefly certain problems in systems and control theory where the algebraic Riccati equation plays a role. The focus in the main part of the essay will be on those aspects of the theory of matrices in indefinite inner product spaces that were motivated and largely influenced by the connection with the study of Hermitian solutions of algebraic Riccati equations. This includes the description of uniqueness and stability of invariant Lagrangian subspaces and of invariant maximal semidefinite subspaces of matrices that are selfadjoint in the indefinite inner product, which leads to the concept of the sign condition. Also, it is described how the inertia of solutions of a special type of algebraic Riccati equation may be described completely in terms of the invariant Lagrangian subspaces connected with the solutions.

The theory of Hilbert spaces of entire functions was conceived as a generalization of Fourier analysis by its founder, Louis de Branges. The Paley–Wiener spaces provided the motivating example. This chapter outlines the early development of the theory, showing how key steps were guided by the Hamburger moment problem, matrix differential equations, and eigenfunction expansions.

The subject of this survey is to review the basics of Louis de Branges’ theory of Hilbert spaces of entire functions, and to present results bringing together the notions of de Branges spaces on one hand and growth functions (proximate orders) on the other hand.After a few introductory words, the paper starts off with a short companion on de Branges theory (section “A Short Companion on Hilbert Spaces of Entire Functions”) where much of the terminology and cornerstones of the theory are presented. Then growth functions are very briefly introduced (section “Growth Functions”). The following two sections of the survey are devoted to growth properties. First (section “General Theorems Relating de Branges Spaces and Growth”), some general theorems, where the growth of elements of a de Branges space is discussed in relation with generating Hermite–Biehler functions and associated canonical systems, and results on growth of subspaces of a given space are presented. Second (section “Some Examples”), some more concrete examples which appear “in nature,” and where growth of different rates is exhibited.It should be said explicitly that this survey is of course far from being exhaustive. For example, since the main purpose is to study growth properties of spaces of entire functions, all what relates to spectral measures (inclusion in L2-spaces, etc.) is omitted from the presentation.

This survey article contains various aspects of the direct and inverse spectral problem for two-dimensional Hamiltonian systemsTwo-dimensional Hamiltonian system, that is, two-dimensional canonical systems of homogeneous differential equations of the form Jy′(x)=−zH(x)y(x),x∈[0,L],0<L≤∞,z∈ℂ,$$\displaystyle{Jy^{{\prime}}(x) = -zH(x)y(x),\ x \in [0,L],\ \ 0 < L \leq \infty,\ z \in \mathbb{C},}$$ with a real non-negative definite matrix function H ≥ 0 and a signature matrix J, and with a standard boundary condition of the form y₁(0+) = 0. Additionally it is assumed that Weyl’s limit point case prevails at L. In this case the spectrum of the canonical system is determined by its Titchmarsh–Weyl coefficient Q which is a Nevanlinna function, that is, a function which maps the upper complex half-plane analytically into itself. In this article an outline of the Titchmarsh–Weyl theory for Hamiltonian systems is given and the solution of the direct spectral problem is shown. Moreover, Hamiltonian systems comprehend the class of differential equations of vibrating strings with a non-homogeneous mass-distribution function as considered by M.G. Kreĭn. The inverse spectral problem for two-dimensional Hamiltonian systems was solved by L. de Branges by use of his theory of Hilbert spaces of entire functions, showing that each Nevanlinna function is the Titchmarsh–Weyl coefficient of a uniquely determined normed Hamiltonian. More detailed results of this connection for, e.g., systems with a semibounded or discrete or finite spectrum are presented, and also some results concerning spectral perturbation, which allow an explicit solution of the inverse spectral problem in many cases.

This work presents a contemporary treatment of Kreĭn’s entire operators with deficiency indices (1, 1) and de Branges’ Hilbert spaces of entire functions. Each of these theories played a central role in the research of both renown mathematicians. Remarkably, entire operators and de Branges spaces are intimately connected and the interplay between them has had an impact in both spectral theory and the theory of functions. This work exhibits the interrelation between Kreĭn’s and de Branges’ theories by means of a functional model and discusses recent developments, giving illustrations of the main objects and applications to the spectral theory of difference and differential operators.

Let ω be a non-negative function on ℝ$$\mathbb{R}$$. Is it true that there exists a non-zero f from a given space of entire functions X satisfying (a)|f|≤ωor (b)|f|≍ω?$$\displaystyle{\mbox{ (a)}\quad \vert f\vert \leq \omega \mbox{ or (b)}\quad \vert f\vert \asymp \omega?}$$ The classical Beurling–Malliavin Multiplier Theorem corresponds to (a) and the classical Paley–Wiener space as X. This is a survey of recent results for the case when X is a de Branges space ℋ(E)$$\mathcal{H}(E)$$. Numerous answers mainly depend on the behavior of the phase function of the generating function E. For example, if argE$$\arg E$$ is regular, then for any even positive ω non-increasing on [0, ∞) with logω∈L1((1+x2)−1dx)$$\log \omega \in L^{1}((1 + x^{2})^{-1}dx)$$ there exists a non-zero f∈ℋ(E)$$f \in \mathcal{H}(E)$$ such that | f | ≤ | E | ω. This is no longer true for the irregular case. The Toeplitz kernel approach to these problems is discussed. This method was recently developed by N. Makarov and A. Poltoratski.

This is a short survey on relationships between Jacobi matrices, de Branges spaces, and canonical systems.

This paper discusses the inverse spectral theory of Schrödinger equations from the point of view of canonical systems and de Branges’s theory of Hilbert spaces of entire functions. The basic idea is to view Schrödinger equations as special canonical systems. For canonical systems, a complete inverse spectral theory is available: there is a one-to-one correspondence between the coefficient functions, on the one hand, and suitable spectral data, on the other hand. The task then is to identify those subclasses that correspond to Schrödinger equations.

For S a contractive analytic operator-valued function on the unit disk ??$$\mathbb{D}$$, de Branges and Rovnyak associate a Hilbert space of analytic functions ℋ(S)$$\mathcal{H}(S)$$ and related extension space ??(S)$$\mathcal{D}(S)$$ consisting of pairs of analytic functions on the unit disk ??$$\mathbb{D}$$. This survey describes three equivalent formulations (the original geometric de Branges–Rovnyak definition, the Toeplitz operator characterization, and the characterization as a reproducing kernel Hilbert space) of the de Branges–Rovnyak space ℋ(S)$$\mathcal{H}(S)$$,de Branges–Rovnyak spaces as well as its role as the underlying Hilbert space for the modeling of completely non-isometric Hilbert-space contraction operators. Also examined is the extension of these ideas to handle the modeling of the more general class of completely nonunitary contraction operators, where the more general two-component de Branges–Rovnyak model space ??(S)$$\mathcal{D}(S)$$ and associated overlapping spaces play key roles. Connections with other function theory problems and applications are also discussed. More recent applications to a variety of subsequent applications are given in a companion survey article.

For S a contractive analytic operator-valued function on the unit disk ??$$\mathbb{D}$$, de Branges and Rovnyak associate a Hilbert space of analytic functions ℋ(S)$$\mathcal{H}(S)$$. A companion survey provides equivalent definitions and basic properties of these spaces as well as applications to function theory and operator theory. The present survey brings to the fore more recent applications to a variety of more elaborate function theory problems, including H∞-norm constrained interpolation, connections with the Potapov method of Fundamental Matrix Inequalities, parametrization for the set of all solutions of an interpolation problem, variants of the Abstract Interpolation Problem of Katsnelson, Kheifets, and Yuditskii, boundary behavior and boundary interpolation in de Branges–Rovnyak spaces themselves, and extensions to multivariable and Kreĭn-space settings.

This article is the first of a two-part series on de Branges spaces of vector valued functions and some of their applications to direct and inverse problems for canonical differential systems and Dirac–Kreĭn systems. This first part is intended as an introduction to the spaces; the second, which is written as a continuation of the first, will focus on applications. The exposition is divided into a number of short sections, each of which deals with one topic. The list of topics covered in this part includes: reproducing kernel Hilbert spaces, J-inner matrix valued functions and some of their important subclasses, the Potapov–Ginzburg transform, linear fractional transformations, the de Branges spaces ℋ(U)$$\mathcal{H}(U)$$, the de Branges space ℬ(??)$$\mathcal{B}(\mathfrak{E})$$.

This article is the second of a two part series on de Branges spaces of vector valued functions and their applications. This part focuses on applications to direct and inverse problems for canonical differential systems and Dirac–Kreĭn systems. The exposition is again divided into a number of short sections, each of which focuses on one topic. The list of topics covered includes: spectral functions for the spaces ℋ(U)$$\mathcal{H}(U)$$ and ℬ(??)$$\mathcal{B}(\mathfrak{E})$$, generalized Fourier transforms, direct and inverse spectral problems for regular canonical differential systems, the Kreĭn accelerant extension problem, the inverse monodromy problem, other directions. The notation is the same as in the first part.

This chapter deals with the realization theory of different classes of Herglotz–Nevanlinna operator-valued functions as impedance functions of linear conservative L-systems. Nowadays realizations of various classes of operator-valued functions play an important role in modern spectral and system theories. An overview of comprehensive analysis of the abovementioned L-systems with, generally speaking, unbounded operators that satisfy the metric conservation law is provided. The treatment of realization problems for Herglotz–Nevanlinna functions and their various subclasses when members of these subclass are realized as impedance functions of L-systems is presented. In particular, the conservative realizations of Stieltjes, inverse Stieltjes, and general Herglotz–Nevanlinna functions and their connections to L-systems of different types with accretive, sectorial, and accumulative state-space operators are considered. The detailed study of the subject is based upon a new method involving extension theory of linear operators with the exit into rigged Hilbert spaces. A one-to-one correspondence between the impedance of L-systems and related extensions of unbounded operators with the exit into rigged Hilbert spaces is established. This material can be of interest to researchers in the field of operator theory, spectral analysis of differential operators, and system theory.

This paper presents an overview of my work with Bruce Francis on asymptotic behavior of linear systems of countably many kinematic points with “nearest neighbor” dynamics. Both first and second order systems are considered. The novelty of the results considered here is that, unlike previous work in this area where the state space was a Hilbert sequence (or function) space, the state space is the Banach sequence space of bounded doubly infinite scalar sequences with the standard supremum norm. The basic problem lying at the heart of the theory for first order systems is the “serial pursuit and rendezvous problem.” Unlike the case of finitely many points where the asymptotic behavior of the system is straightforward, for infinitely many points the asymptotic behavior of the system connects with the classical study of Borel summability of sequences. The symmetric synchronizations problems are dependent on determining the subspace of initial configurations which give convergence in the serial pursuit problem. The finite dimensional version of the infinite second order system we study arises in physics in the theory of phonons, in the simplest case of one-dimensional lattice dynamics. We compare the asymptotic behavior of the finite system case to the infinite system one, both for undamped and damped systems. The results are quite unexpected. Despite the fact that the system is unbounded there are many cases where, asymptotically, synchronization takes place both in the damped and undamped case.

In this survey article some linear transformations that play a fundamental role in signal processing and optical systems are reviewed. After a brief discussion of the general theory of linear systems, specific linear transformations are introduced. An important class of signals to which most of these linear transformations are applied is the class of bandlimited signals and some of its generalizations. The article begins by an introduction to this class of signals and some of its properties, in particular, the property that a bandlimited signal can be perfectly reconstructed from its samples on a discrete set of points. The main tool for the reconstruction is known as the sampling theorem. Some of the transformations presented, such as the windowed Fourier transform, the continuous wavelet transform, the Wigner distribution function, the radar ambiguity function, and the ambiguity transformation, fall into the category of time–frequency, scale-translation, or phase-space representations. Such transformations make it possible to study physical systems from two different perspectives simultaneously. Another group of transformations presented is closely related to the Fourier transform, such as the fractional Fourier transform. Generalizations of the fractional Fourier transform, including the special affine Fourier transformation, and their applications in optical systems are introduced, together with sampling theorems for signals bandlimited in the domains of the aforementioned transformations.

In Gohberg et al. (Classes of linear operators, Theorem 4.2, Chapter XVII. Birkhäuser, Basel, 2003), some sufficient conditions are given so that if A is an unbounded Fredholm linear operator and if B is another (possibly unbounded) linear operator, then their algebraic sum A + B is a Fredholm operator. The main objective here consists of extending the previous result to the case of three unbounded linear operators. Namely, some sufficient conditions are given so that if A, B, C are three unbounded linear operators with A being a Fredholm operator, then their algebraic sum A + B + C is also a Fredholm operator.

This chapter describes the frequency domain approach to the robust stabilization problem in linear control theory. The exposition is restricted to single-input single-output systems. After introducing the preliminaries on linear control systems, their transfer functions, stable and nonstable systems, the stabilization problem and its solution are discussed via the factorization approach, and finally an appropriate metric on the set of transfer functions making stabilizability a robust property is given along with some simple prototypical computational examples.

The main objects of this chapter are “semi-separable systems,” sometimes called “quasi-separable systems.” These are systems of equations, in which the operator has a special structure, called “semi-separable” in this chapter. By this is meant that the operator, although typically infinite dimensional, has a recursive structure determined by sequences of finite matrices, called transition matrices. This type of operator occurs commonly in Dynamical System Theory for systems with a finite dimensional state space and/or in systems that arise from discretization of continuous time and space. They form a natural generalization of finite matrices and a complete theory based on sequences of finite matrices is available for them. The chapter concentrates on the invertibility of such systems: either the computation of inverses when they exist, or the computation of approximate inverses of the Moore–Penrose type when not. Semi-separable systems depend on a single principal variable (often identified with time or a single dimension in space). Although there are several types of semi-separable systems depending on the continuity of that principal variable, the present chapter concentrates on indexed systems (so-called discrete-time systems). This is the most straightforward and most appealing type for an introductory text. The main workhorse is “inner–outer factorization,” a technique that goes back to Hardy space theory and generalizes to any context of nest algebras, as is the one considered here. It is based on the definition of appropriate invariant subspaces in the range and co-range of the operator. It translates to attractive numerical algorithms, such as the celebrated “square-root algorithm,” which uses proven numerically stable operations such as QR-factorization and singular value decomposition (SVD).

Linear system theory over finite fields has played a major role in unveiling the properties of linear error correction codes, thus providing essential insights into key design parameters and features, such as minimal realizations, distance spectra, trapping sets, and efficient decoder structures, among others. A more recent thrust in error correction coding (linear or otherwise) is in secrecy systems, in the form of physical layer security that can complement, and in certain cases even replace, classical cryptography in specific communication settings. This chapter reviews the basic principles of secrecy coding, focusing on the properties of linear codes that approach secrecy capacity, as a precursor to understanding design strategies that attain these properties, as offered in the references. Applications beyond secure communications of these same coding techniques, notably in watermarking and steganography, are also outlined.

Let {T₁, …, T _n } be a set of n commuting bounded linear operators on a Hilbert space ℋ$$\mathcal{H}$$. Then the n-tuple (T₁, …, T _n ) turns ℋ$$\mathcal{H}$$ into a module over ℂ[z1,…,zn]$$\mathbb{C}[z_{1},\ldots,z_{n}]$$ in the following sense: ℂ[z1,…,zn]×ℋ→ℋ,(p,h)↦p(T1,…,Tn)h,$$\displaystyle{\mathbb{C}[z_{1},\ldots,z_{n}] \times \mathcal{H}\rightarrow \mathcal{H},\quad \quad (p,h)\mapsto p(T_{1},\ldots,T_{n})h,}$$where p∈ℂ[z1,…,zn]$$p \in \mathbb{C}[z_{1},\ldots,z_{n}]$$ and h∈ℋ$$h \in \mathcal{H}$$. The above module is usually called the Hilbert module over ℂ[z1,…,zn]$$\mathbb{C}[z_{1},\ldots,z_{n}]$$. Hilbert modules over ℂ[z1,…,zn]$$\mathbb{C}[z_{1},\ldots,z_{n}]$$ (or natural function algebras) were first introduced by R.G. Douglas and C. Foias in 1976. The two main driving forces were the algebraic and complex geometric views to multivariable operator theory.This article gives an introduction of Hilbert modules over function algebras and surveys some recent developments. Here the theory of Hilbert modules is presented as combination of commutative algebra, complex geometry and the geometry of Hilbert spaces, and its applications to the theory of n-tuples (n ≥ 1) of commuting operators. The topics which are studied include: model theory from Hilbert module point of view, Hilbert modules of holomorphic functions, module tensor products, localizations, dilations, submodules and quotient modules, free resolutions, curvature, and Fredholm Hilbert modules. More developments in the study of Hilbert module approach to operator theory can be found in a companion paper, “Applications of Hilbert Module Approach to Multivariable Operator Theory.”

A commuting n-tuple (T1,…,Tn)$$(T_{1},\ldots,T_{n})$$ of bounded linear operators on a Hilbert space ℋ$$\mathcal{H}$$ associates a Hilbert module ℋ$$\mathcal{H}$$ over ℂ[z1,…,zn]$$\mathbb{C}[z_{1},\ldots,z_{n}]$$ in the following sense: ℂ[z1,…,zn]×ℋ→ℋ,(p,h)↦p(T1,…,Tn)h,$$\displaystyle{\mathbb{C}[z_{1},\ldots,z_{n}] \times \mathcal{H}\rightarrow \mathcal{H},\quad \quad (p,h)\mapsto p(T_{1},\ldots,T_{n})h,}$$ where p∈ℂ[z1,…,zn]$$p \in \mathbb{C}[z_{1},\ldots,z_{n}]$$ and h∈ℋ$$h \in \mathcal{H}$$. A companion survey provides an introduction to the theory of Hilbert modules and some (Hilbert) module point of view to multivariable operator theory. The purpose of this survey is to emphasize algebraic and geometric aspects of Hilbert module approach to operator theory and to survey several applications of the theory of Hilbert modules in multivariable operator theory. The topics which are studied include generalized canonical models and Cowen–Douglas class, dilations and factorization of reproducing kernel Hilbert spaces, a class of simple submodules and quotient modules of the Hardy modules over polydisk, commutant lifting theorem, similarity and free Hilbert modules, left invertible multipliers, inner resolutions, essentially normal Hilbert modules, localizations of free resolutions, and rigidity phenomenon.This article is a companion paper to “An Introduction to Hilbert Module Approach to Multivariable Operator Theory”.

Dilation theoryDilation theory of single Hilbert space contractions is an important and very useful part of operator theory. By the main result of the theory, every Hilbert space contraction has the uniquely determined minimal unitary dilation. In many situations this enables to study instead of a general contraction its unitary dilation, which has much nicer properties.The present paper gives a survey of dilation theory for commuting tuples of Hilbert space operators. The paper is organized as follows:1.Introduction2.Dilation theory of single contractions3.Regular dilations4.The Ando dilation and von Neumann inequality5.Spherical dilations6.Analytic models7.Further examples8.Concluding remarks

The Drury–Arveson space Hd2$$H_{d}^{2}$$ (also known as symmetric Fock space or the d-shift space), is the reproducing kernel Hilbert space on the unit ball of ℂd$$\mathbb{C}^{d}$$ with the kernel k(z,w)=(1−⟨z,w⟩)−1$$k(z,w) = (1 -\langle z,w\rangle )^{-1}$$. The operators Mzi:f(z)↦zif(z)$$M_{z_{i}}: f(z)\mapsto z_{i}f(z)$$, arising from multiplication by the coordinate functions z1,…,zd$$z_{1},\ldots,z_{d}$$, form a commuting d-tuple Mz=(Mz1,…,Mzd)$$M_{z} = (M_{z_{1}},\ldots,M_{z_{d}})$$. The d-tuple M _z —which is called the d-shift—gives the Drury–Arveson space the structure of a Hilbert module. This Hilbert module is arguably the correct multivariable generalization of the Hardy space on the unit disc H2(??)$$H^{2}(\mathbb{D})$$. It turns out that the Drury–Arveson space Hd2$$H_{d}^{2}$$ plays a universal role in operator theory (every pure, contractive Hilbert module is a quotient of an ampliation of Hd2$$H_{d}^{2}$$) as well as in function theory (every irreducible complete Pick space is essentially a restriction of Hd2$$H_{d}^{2}$$ to a subset of the ball). These universal properties resulted in the Drury–Arveson space being the subject of extensive studies, and the theory of the Drury–Arveson is today broad and deep. This survey aims to introduce the Drury–Arveson space, to give a panoramic view of the main operator theoretic and function theoretic aspects of this space, and to describe the universal role that it plays in multivariable operator theory and in Pick interpolation theory.

The notion of spectrum of an operator is one of the central concepts of operator theory. It is closely connected with the existence of a functional calculus which provides important information about the structure of Banach space operators. The situation for commuting n-tuples of Banach space operators is much more complicated. There are many possible definitions of joint spectra. However, the joint spectrum introduced by J.L. Taylor has a distinguished property—there exists a functional calculus for functions analytic on a neighborhood of this spectrum. The present paper gives a survey of basic properties of the Taylor spectrum and Taylor functional calculus.

A numberUnbounded operators of results on integrability and extendability of Lie algebras of unbounded skew-symmetric operators with common dense domain in Hilbert space are proved. By integrability for a Lie algebra ??$$\mathfrak{g}$$, it means that there is an associated unitary representation ??$$\mathcal{U}$$ of the corresponding simply connected Lie group such that ??$$\mathfrak{g}$$ is the differential of ??$$\mathcal{U}$$. The results extend earlier integrability results in the literature and are new even in the case of a single operator. Applications include a new invariant for certain Riemann surfaces.

The main purpose of this chapter is to introduce some new tools from harmonic analysis and the theory of operator algebras into the study of arithmetic functions, i.e., functions defined from the natural numbers ℕ$$\mathbb{N}$$ to the complex numbers ℂ$$\mathbb{C}$$. The cases are from number theory (for example, Dirichlet L-functions, etc.), from the theory of moments, and from probability theory (e.g., generating functions). Algebras of arithmetic functions and their representations are considered. In particular, direct decompositions and tensor-factorizations of arithmetic functions are studied. One can do this with a reduction over the primes; and with the use of free probability spaces, one for every prime. The algebras are represented in Kreĭn spaces. The notion of freeness here is analogous to independence in classical statistics. As an application, the study of certain representations of countable discrete groups is considered.

In this chapter, relations between calculus on a von Neumann algebra ??ℚ$$\mathfrak{M}_{\mathbb{Q}}$$ over the Adele ring ??ℚ$$\mathbb{A}_{\mathbb{Q}}$$, and free probability on a certain subalgebra Φ$$\Phi $$ of the algebra ??,$$\mathcal{A},$$ consisting of all arithmetic functions equipped with the functional addition and convolution are studied. By showing that the Adelic calculus over ??ℚ$$\mathbb{A}_{\mathbb{Q}}$$ is understood as a free probability on a certain von Neumann algebra ??ℚ$$\mathfrak{M}_{\mathbb{Q}}$$, the connections with a system of natural free-probabilistic models on the subalgebra Φ$$\Phi $$ in ??$$\mathcal{A}$$ are considered. In particular, the subalgebra Φ$$\Phi $$ is generated by the Euler totient function ϕ.

In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy–Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered as both a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated with the vectorial part of the Cauchy–Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter’s theorem on one hand and the Cauchy–Kovalevskaya extension theorem on the other hand. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert, and Radon transforms, which are important both within the theoretical framework and in view of possible applications.

Generalizing the complex one-dimensional function theory the class of quaternion-valued functions, defined in domains of ℝ4$$\mathbb{R}^{4}$$, will be considered. The null solutions of a generalized Cauchy–Riemann operator are defined as the ℍ$$\mathbb{H}$$-holomorphic functions. They show a lot of analogies to the properties of classical holomorphic functions. An operator calculus is studied that leads to integral theorems and integral representations such as the Cauchy integral representation, the Borel–Pompeiu representation, and formulas of Plemelj–Sokhotski type. Also a Bergman–Hodge decomposition in the space of square integrable functions can be obtained. Finally, it is demonstrated how these tools can be applied to the solution of non-linear boundary value problems.

The purpose of this chapter is to describe the settings in which one can set up coherent generalizations to several variables of the classical single variable complex analysis and to point out some basic properties of the function spaces that naturally occur in such settings. Particular emphasis is placed on defining proper counterparts in higher dimension of the concept of holomorphic functions and of the Cauchy–Riemann operator. The resulting function theory deals with monogenic functions and Euclidean Dirac or Cauchy–Riemann operators. Its richness is illustrated by the existence of a general Cauchy–Pompeiu Integral Representation Formula. Actually, even more general formulas will be initially derived for first order homogeneous differential operators with constant coefficients in a unital associative Banach algebra. The point that will be made is that the operators and algebras that yield the simplest forms of such Integral Representation Formulas with Remainders are essentially the Cauchy–Riemann and Dirac operators associated with Euclidean Clifford algebras. As yet another generalization of a classical result in single variable complex analysis one gets a Quantitative Hartogs–Rosenthal Theorem concerned with uniform approximation on compact sets by monogenic functions. An equally important goal of the chapter is to outline some of the contributions that led to the discovery and development of Quaternion and Clifford analysis. The last section includes several concluding remarks and a few additional references that outline the full scope of some of the early and current developments, and help in further explorations of other lines of research in quaternion and Clifford analysis.

Firstly, it is recalled briefly what are the analogues of the usual Cauchy–Riemann operators of complex analysis for quaternion-valued functions. Two of them, the Fueter and the Moisil–Théodoresco operators, are related to the Laplace equation, and one more is related to the Helmholtz operator. Secondly, it is shown that each of the following four theories can be embedded into one of the above quaternionic ones: holomorphic functions in ℂ2$$\mathbb{C}^{2}$$, vector analysis, time-harmonic electromagnetic, and time-harmonic spinor fields. It is illustrated with specific examples that such embeddings prove to be rather fruitful for the conventional theories.

Clifford analysis, which covers a great part of the quaternionic analysis, is known as the theory of monogenic functions and represents a refinement of classical harmonic analysis and a generalization of complex analysis to higher dimensions. In this chapter a condensed account is given of a selection of studies connected to the Cauchy transform on the non-smooth boundary of a bounded domain in Euclidean space, in a Clifford and quaternionic analysis context. Particular emphasis will be placed on the study of the so-called jump problem (reconstruction problem) for monogenic functions, that is, the problem of reconstructing a monogenic function in both the interior and exterior of the domain vanishes at infinity and has a prescribed jump across the boundary.

This chapter focuses on the use of Clifford analysis techniques as an encompassing and unifying tool to study higher spin generalizations of the classical Dirac operator. These operators belong to a complete family of conformally invariant first-order differential operators, acting on functions taking their values in an irreducible representation for the spin group (the double cover for the orthogonal group). Their existence follows from a standard classification result due to Fegan (Q. J. Math. 27:513–538, 1976), and a canonical way to construct them is to use the technique of Stein–Weiss gradients. This then gives rise to two kinds of differential operators defined on irreducible tensor fields, the standard language used in, e.g., theoretical physics, where higher spin operators appear in the equations of motion for elementary particles having arbitrary half-integer spin: on the one hand, there are the (elliptic) generalizations of the Dirac operator, acting as endomorphisms on the space of smooth functions with values in a fixed module (i.e., preserving the values), and on the other hand there are the invariant operators acting between functions taking values in different modules for the spin group (the so-called twistor operators and their duals). In this chapter, both types of higher spin operators will be defined on spinor-valued functions of a matrix variable (i.e., in several vector variables): this has the advantage that the resulting equations become more transparent, and it allows using techniques for Clifford analysis in several variables. In particular, it provides an elegant framework to develop a function theory for the aforementioned operators, such as a full description of the (polynomial) null solutions and analogues of the classical Cauchy integral formula.

Fueter’s Theorem offers a method that conveys certain holomorphic functions in one complex variable to quaternionic regular (monogenic) functions. Ever since the theorem was proved in 1935, it underwent several main generalizations. Those are not only based on its own interest, but also motivated by applications found in other areas of mathematics, such as functional calculus of operators. This article serves as a survey on Fueter’s Theorem, its generalizations and applications.

This chapter introduces contemporary Clifford analysis as a local function theory of first-order systems of PDEs invariant under various Lie groups. A concept of a symmetry of a system of partial differential equations is the key point of view, it makes it possible to use many efficient tools from the theory of representations of simple Lie groups. A systematic approach is based on a choice of a Klein geometry (a homogeneous space M ≃ G∕P with P being a Lie subgroup of a Lie group G) and on a notion of a homogeneous (invariant) differential operator acting among sections of associated homogeneous vector bundles. The main example is the conformal group G=Spin(m+1,1)$$G = Spin(m + 1,1)$$ acting on the sphere Sm and the Dirac operator. The chapter contains a description of basic properties of solutions of such systems and lists many various examples of the aforementioned scheme. The introductory sections describe the Clifford algebra, its spinor representations, the conformal group of the Euclidean space, the Fegan classification of the conformally invariant first order differential operators, and a series of examples of such operators appearing in the Clifford analysis. They include the Dirac equation for spinor-valued functions, the Hodge and Moisil–Théodoresco systems for differential forms, the Hermitian Clifford analysis, the quaternionic Clifford analysis, the (generalized) Rarita–Schwinger equations, and the massless fields of higher spin. A different point of view to these first-order systems presents them as special solutions of the (twisted) Dirac equation. The second part of this chapter contains a description of basic properties of solutions of the Dirac equation, including the Fischer decomposition of spinor-valued polynomials, the Howe duality, the Taylor and the Laurent series for monogenic functions. The last two sections contains a description of the Gelfand–Tsetlin bases for the spaces of (solid) spherical monogenics and a discussion of possible future direction of research in Clifford analysis.

This article discusses how the theory of Fueter regular functions on quaternions can be extended to the case of several variables. This can be done in two different (complementary) ways. One can follow the traditional approach to several complex variables developed in the first part of the twentieth century, and construct suitable generalizations of the Cauchy–Fueter formula to the setting of several variables. In this way one obtains an analog of the Bochner–Martinelli formula for regular functions of several quaternionic variables, and from that starting point one can develop most of the fundamental results of the theory. On the other hand, one can take a more algebraic point of view, in line with the general ideas of Ehrenpreis on solutions to systems of linear constant coefficients partial differential equations, and exploit the fact that regular functions in several variables are infinitely differentiable functions that satisfy a reasonably simple overdetermined system of differential equations. By using this characterization, and the fundamental ideas pioneered by Ehrenpreis and Palamodov, one can construct a sheaf theoretical approach to regular functions of several quaternionic variables that rather immediately allows one to discover important global properties of such functions, and indeed to develop a rigorous theory of hyperfunctions in the quaternionic domain. This article further shows how this process can be adapted to variations of Fueter regularity such as biregularity and Moisil–Theodorescu regularity, as well as to the case of monogenic functions of several vector variables.Quaternion(s) Finally the article considers the notion of slice monogeneity and slice regularity, and shows how they can also be extended to several variables. The theories in these cases are very recent, and rapidly developing.

This paper gives an overview of some basic results on Hermitian Clifford analysis, a refinement of classical Clifford analysis dealing with functions in the kernel of two mutually adjoint Dirac operators invariant under the action of the unitary group. The set of these functions, called Hermitian monogenic, contains the set of holomorphic functions in several complex variables. The paper discusses, among other results, the Fischer decomposition, the Cauchy–Kovalevskaya extension problem, the axiomatic radial algebra, and also some algebraic analysis of the system associated with Hermitian monogenic functions. While the Cauchy–Kovalevskaya extension problem can be carried out for the Hermitian monogenic system, this system imposes severe constraints on the initial Cauchy data. There exists a subsystem of the Hermitian monogenic system in which these constraints can be avoided. This subsystem, called submonogenic system, will also be discussed in the paper.

This survey is intended as an overview of discrete Clifford analysis and its current developments. Since in the discrete case one has to replace the partial derivative with two difference operators, backward and forward partial difference, one needs to modify the main tools for a development of a discrete function theory, such as the replacement of a real Clifford algebra by a complexified Clifford algebra or of the classic Weyl relations by so-called S-Weyl relations. The main results, like Cauchy integral formula, Fischer decomposition, CK-extension, and Taylor series, will be derived. To give a better idea of the differences between the discrete and continuous case, this chapter contains the problem of discrete Hardy spaces as well as some discrete objects which do not have an equivalent object in continuous Clifford analysis, such as the CK-extension of a discrete Delta function.

The main purpose of this chapter is to offer an overview to show how the theory of holomorphic functions of one complex variable can be successfully extended to the setting of real alternative algebras. Thus, the purpose of this chapter is to show how notions of holomorphicity (which will be called hyperholomorphicity in this case) can be properly defined when the field of complex numbers is replaced by what are generically referred to as its hypercomplex generalizations. However, not all possible hypercomplex generalizations will be considered, and, more specifically, this chapter presents various definitions of slice hyperholomorphic functions for functions whose values lie in real alternative algebras with a unit. After recalling the basic definitions from the theory of real alternative algebras, it will be shown how some of the most important hypercomplex algebras fall within this context; in particular the category of real alternative algebras includes quaternions, octonions, and of course Clifford Algebras. On the basis of this theory, this chapter then introduces the basic results in the theory of slice regular functions for quaternions and octonions, as developed in Gentili et al. (Regular Functions of a Quaternionic Variable. Springer, New York, 2013), as well as the corresponding results for slice monogenic functions, as developed in Colombo et al. (Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions. Birkhauser, Basel, 2011). The most important result that will be discussed is the Cauchy formula for slice hyperholomorphic functions, which rests on the so-called noncommutative Cauchy kernel series introduced originally in Colombo and Sabadini (Hypercomplex Analysis. Birkhauser, Basel, 2009). The results that will be described are the foundation for any advanced study of the subject. This chapter then focuses on a different, comprehensive approach to slice hyperholomorphicity, developed in the last few years in Ghiloni and Perotti (Adv Math 226:1662–1691, 2011), and the connections between the two theories are explored and discussed in detail.

This chapter gives an overview of the theory of hypercomplex Fourier transforms, which are generalized Fourier transforms in the context of Clifford analysis. The emphasis lies on three different approaches that are currently receiving a lot of attention: the eigenfunction approach, the generalized roots of −1 approach, and the characters of the spin group approach. The eigenfunction approach prescribes complex eigenvalues to the L₂ basis consisting of the Clifford–Hermite functions and is therefore strongly connected to the representation theory of the Lie superalgebra ??????(1|2)$$\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\vert 2)$$. The roots of −1 approach consists of replacing all occurrences of the imaginary unit in the classical Fourier transform by roots of −1 belonging to a suitable Clifford algebra. The resulting transforms are often used in engineering. The third approach uses characters to generalize the classical Fourier transform to the setting of the group Spin(4), resp. Spin(6) for application in image processing. For each approach, precise definitions of the transforms under consideration are given, important special cases are highlighted, and a summary of the most important results is given. Also directions for further research are indicated.

Clifford analysis is a higher dimensional functions theory for the Dirac operator and builds a bridge between complex function theory and harmonic analysis. The construction of wavelets is done in three different ways. Firstly, a monogenic mother wavelet is obtained from monogenic extensions (Cauchy–Kovalevskaya extensions) of special functions like Hermite and Laguerre polynomials. Based on the kernel function, Cauchy wavelets are also monogenic but not square integrable in the usual sense. On the other hand, these wavelets and their kernels are connected to the Cauchy–Riemann equations in the upper half space as well as to Bergman and Hardy spaces. Secondly, a group theoretical approach is used to construct wavelets. This approach considers pure dilations and rotations as group actions on the unit sphere. It can be generalized by using the action of the Spin group because the Spin group is a double cover of the rotation group, whereas dilations arise from Möbius transformations. Here, Clifford analysis gives the tools to construct wavelets. Finally, an application to image processing based on monogenic wavelets is considered. Here, the starting point are scalar-valued functions and the resulting Clifford wavelets are boundary values of monogenic functions in the upper half space. One proceeds in two steps. First choose a real- or complex-valued primary wavelet and then construct from that using the Riesz transform = Hilbert transform Clifford wavelets and Clifford wavelet frames.

This survey is intended as a mathematical overview of the monogenic signalMonogenic signals theory, its current developments as well as of its applications to various problems. Several approaches to the concept of monogenic signal are given, followed by a short discussion on its advantages and drawbacks. Among those, one highlights the definitions arising from different concepts of Hilbert transforms, as well as the concept of the monogenic signal as a solution to a boundary value problem. In the last part, some recent developments in the field are presented.

The aim of this paper is to provide a brief panorama of the subjects that have been considered in the development of quaternionic functional analysis. Since there are many similarities with classic functional analysis over real and complex numbers, it is focused on showing peculiarities which arise when one works with the basic objects in quaternionic situation: the notion of linearity, norm, inner product, linear operators and functionals, eigenvalues and eigenvectors.

This chapter is a survey on recent developments in quaternionic Schur analysis. The first part is based on functions which are slice hyperholomorphic in the unit ball of the quaternions, and have modulus bounded by 1. These functions, which by analogy to the complex case are called Schur multipliers, are shown to be (as in the complex case) the source of a wide range of problems of general interest. They also suggest new problems in quaternionic operator theory, especially in the setting of indefinite inner product spaces. This chapter gives an overview on rational functions and their realizations, on the Hardy space of the unit ball, on the half-space of quaternions with positive real part, and on Schur multipliers, also discussing related results. For the purpose of comparison this chapter presents also another approach to Schur analysis in the quaternionic setting, in the framework of Fueter series. To ease the presentation most of the chapter is written for the scalar case, but the reader should be aware that the appropriate setting is often that of vector-valued functions.

This paper is a survey on some functional calculi constructed with slice hyperholomorphic functions, mainly on the S-functional calculus. This is a functional calculus, defined for n-tuples of not necessarily commuting operators, and it is based on the recent theory of slice hyperholomorphic functions. Its version for commuting operators, called the SC-functional calculus, and its quaternionic version, called the quaternionic functional calculus, are presented, as well as the so-called F-functional calculus, based on the Fueter–Sce mapping theorem in integral form.Since the theory of hyperholomorphic functions is quite recent, the paper contains the main results of this function theory that are necessary to introduce these calculi, as well as the Fueter–Sce mapping theorem in integral form.

The monogenic functional calculusMonogenic functioncalculus is a means of constructing functions of a finite system of bounded or unbounded operators. For bounded noncommuting operators, a polynomial function produces a polynomial of the operators in which all possible operator orderings are equally weighted. For example, for two bounded selfadjoint operators A₁, A₂, the operator 12(A1A2+A2A1)$$\frac{1} {2}(A_{1}A_{2} + A_{2}A_{1})$$ is associated with polynomial z↦z1z2$$z\mapsto z_{1}z_{2}$$ in two variables by the monogenic functional calculus. The same formula applies just when the spectrum σ(ξ1A1+ξ2A2)$$\sigma (\xi _{1}A_{1} +\xi _{2}A_{2})$$ of a finite linear combination ξ1A1+ξ2A2$$\xi _{1}A_{1} +\xi _{2}A_{2}$$ of A₁ and A₂ is a subset of the real numbers for any ξ1,ξ2∈ℝ$$\xi _{1},\xi _{2} \in \mathbb{R}$$.The article begins with a discussion of Clifford algebras and Clifford analysis and points out the connection with Weyl’s functional calculus for a finite system of selfadjoint operators in Hilbert space. The construction of the Cauchy kernel in the monogenic functional calculus is achieved with the plane wave decomposition of the Cauchy kernel in Clifford analysis. The formulation applies to any finite system (A1,…,An)$$(A_{1},\ldots,A_{n})$$ of closed unbounded operators such that the spectrum σξ1A1+…+ξnAn$$\sigma \left (\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\right )$$ of the closed operator ξ1A1+…+ξnAn$$\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}$$ is contained in a two-sided sector in ℂ$$\mathbb{C}$$ for almost all ξ∈ℝn$$\xi \in \mathbb{R}^{n}$$. The connection with harmonic analysis and irregular boundary value problems is emphasized. The monogenic functional calculus may be applied to the finite commuting system DΣ=i(D1,…,Dn)$$D_{\Sigma } = i(D_{1},\ldots,D_{n})$$ of differentiating operators on a Lipschitz surface Σ$$\Sigma $$ so that an H∞-functional calculus f↦f(DΣ)$$f\mapsto f(D_{\Sigma })$$ is obtained for functions f uniformly bounded and left monogenic in a sector containing almost all tangent planes to Σ$$\Sigma $$.