Slice regular functions on real alternative algebras

In this paper we develop a theory of slice regular functions on a real alternative algebra A. Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in A and we prove a Cauchy integral formula for slice functions of class C¹.     

The Fueter mapping theorem in integral form and the ℱ‐functional calculus

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n‐tuples of commuting operators (called ?‐functional calculus) based on a new notion of spectrum, called ?‐spectrum, for the n‐tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ?‐functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.     

Schatten class and Berezin transform of quaternionic linear operators

In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S‐spectrum of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S‐functional calculus. Copyright © 2016 John Wiley & Sons, Ltd.     

The quaternionic evolution operator

In the recent years, the notion of slice regular functions has allowed the introduction of a quaternionic functional calculus. In this paper, motivated also by the applications in quaternionic quantum mechanics, see Adler (1995) [1], we study the quaternionic semigroups and groups generated by a quaternionic (bounded or unbounded) linear operator T=T₀+iT₁+jT₂+kT₃. It is crucial to note that we consider operators with components T?(?=0,1,2,3) that do not necessarily commute. Among other results, we prove the quaternionic version of the classical Hille–Phillips–Yosida theorem. This result is based on the fact that the Laplace transform of the quaternionic semigroup etT is the S-resolvent operator , the quaternionic analogue of the classical resolvent operator. The noncommutative setting entails that the results we obtain are somewhat different from their analogues in the complex setting. In particular, we have four possible formulations according to the use of left or right slice regular functions for left or right linear operators.     

Krein–Langer Factorization and Related Topics in the Slice Hyperholomorphic Setting

We study various aspects of Schur analysis in the slice hyperholomorphic setting. We present two sets of results: first, we give new results on the functional calculus for slice hyperholomorphic functions. In particular, we introduce and study some properties of the Riesz projectors. Then we prove a Beurling–Lax type theorem, the so-called structure theorem. A crucial fact which allows to prove our results is the fact that the right spectrum of a quaternionic linear operator and the point S-spectrum coincide. Finally, we study the Krein–Langer factorization for slice hyperholomorphic generalized Schur functions. Both the Beurling–Lax type theorem and the Krein–Langer factorization are far-reaching results which have not been proved in the quaternionic setting using notions of hyperholomorphy other than slice hyperholomorphy.     

On Geometric Aspects of Quaternionic and Octonionic Slice Regular Functions

The aim of this paper is twofold. On the one hand, we enrich from a geometrical point of view the theory of octonionic slice regular functions. We first prove a boundary Schwarz lemma for slice regular self-mappings of the open unit ball of the octonionic space. As applications, we obtain two Landau–Toeplitz type theorems for slice regular functions with respect to regular diameter and slice diameter, respectively, together with a Cauchy type estimate. Along with these results, we introduce some new and useful ideas, which also allow us to prove the minimum principle and one version of the open mapping theorem. On the other hand, we adopt a completely new approach to strengthen a version of boundary Schwarz lemma first proved in Ren and Wang (Trans Am Math Soc 369:861–885, 2017) for quaternionic slice regular functions. Our quaternionic boundary Schwarz lemma with optimal estimate improves considerably a well-known Osserman type estimate and provides additionally all the extremal functions.     

The Cauchy formula with s-monogenic kernel and a functional calculus for noncommuting operators

The new notion of slice monogenic functions introduced in the paper [F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math. 171 (2009) 385-403] led us to define a new functional calculus for an n-tuple of not necessarily commuting operators, see [F. Colombo, I. Sabadini, D.C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal. 254 (2008) 2255-2274]. In this paper we prove a Cauchy formula with slice monogenic kernel for the slice monogenic functions. This new Cauchy formula is the fundamental tool to prove that our functional calculus apply to a more general setting. Moreover, we deduce some fundamental properties of the functional calculus, for example: some algebraic properties, the Spectral Mapping Theorem and the Spectral Radius Theorem.     

A Cauchy kernel for slice regular functions

In this article, we show how to construct a regular, non-commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results.     

Monogenic plane waves and the W‐functional calculus

In this paper, we introduce some integral transforms that map slice monogenic functions to monogenic functions. We then show that one of these integral transforms, which is based on the Cauchy formula of slice monogenic functions, is useful to define a functional calculus depending on a parameter for n‐tuples of bounded operators. Copyright © 2015 John Wiley & Sons, Ltd.     

On some notions of spectra for quaternionic operators and for n-tuples of operators

The S-spectrum has been introduced for the definition of the S-functional calculus that includes both the quaternionic functional calculus and a calculus for n-tuples of nonnecessarily commuting operators. The notion of right spectrum for right linear quaternionic operators has been widely used in the literature, especially in the context of quaternionic quantum mechanics. Moreover, several results in linear algebra, like the spectral theorem for quaternionic matrices, involve the right spectrum. In this Note we prove that the two notions of S-spectrum and of right spectrum coincide.     

On subnormal operators

LetS be a pure subnormal operator such thatC*(S), theC*-algebra generated byS, is generated by a unilateral shiftU of multiplicity 1. We obtain conditions under which 5 is unitarily equivalent toα + βU, α andβ being scalars orS hasC*-spectral inclusion property. It is also proved that if in addition,S hasC*-spectral inclusion property, then so does its dualT andC*(T) is generated by a unilateral shift of multiplicity 1. Finally, a characterization of quasinormal operators among pure subnormal operators is obtained.     

On some notions of convergence for n‐tuples of operators

The aim of this paper is to show that we can extend the notion of convergence in the norm‐resolvent sense to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of S‐resolvent operator. With this notion, we can define bounded functions of unbounded operators using the S‐functional calculus for n‐tuples of noncommuting operators. The same notion can be extended to the case of the F‐resolvent operator, which is the basis of the F‐functional calculus, a monogenic functional calculus for n‐tuples of commuting operators. We also prove some properties of the F‐functional calculus, which are of independent interest. Copyright © 2013 John Wiley & Sons, Ltd.     

The (<Emphasis Type="Italic">S</Emphasis>)<Subscript>+</Subscript> condition on generalized variational inequalities

In this paper, we derive some existence results for generalized variational inequalities associated with mappings satisfying the (S)₊ condition. The relation between the (S)₊ and (S)₊¹ conditions is discussed. As an application, we also consider multivalued complementarity problems associated with mappings satisfying the (S)₊ condition, and prove a theorem to characterize the solvability of such problems in terms of exceptional families of elements.     

Approximation by polynomials on quaternionic compact sets

In this paper, we obtain several extensions to the quaternionic setting of some results concerning the approximation by polynomials of functions continuous on a compact set and holomorphic in its interior. To this end, we prove an analog of the Riemann mapping theorem for a subclass of open sets, whose validity involves precisely the slice regular functions for which the composition remains slice regular. The results include approximation on compact starlike sets and compact axially symmetric sets. The cases of some concrete particular sets are described in details, including also quantitative estimates. Copyright © 2014 John Wiley & Sons, Ltd.     

The homogeneous q-difference operator

We introduce a q-differential operator D_xy on functions in two variables which turns out to be suitable for dealing with the homogeneous form of the q-binomial theorem as studied by Andrews, Goldman, and Rota, Roman, Ihrig, and Ismail, et al. The homogeneous versions of the q-binomial theorem and the Cauchy identity are often useful for their specializations of the two parameters. Using this operator, we derive an equivalent form of the Goldman–Rota binomial identity and show that it is a homogeneous generalization of the q-Vandermonde identity. Moreover, the inverse identity of Goldman and Rota also follows from our unified identity. We also obtain the q-Leibniz formula for this operator. In the last section, we introduce the homogeneous Rogers–Szegö polynomials and derive their generating function by using the homogeneous q-shift operator.     

Flatness properties of diagonal acts over monoids

If S is a monoid, the right S-act S×S, equipped with componentwise S-action, is called the diagonal act of S. The question of when this act is cyclic or finitely generated has been a subject of interest for many years, but so far there has been no explicit work devoted to flatness properties of diagonal acts. Considered as a right S-act, the monoid S is free, and thus is also projective, flat, weakly flat, and so on. In 1991, Bulman-Fleming gave conditions on S under which all right acts S ^I (for I a non-empty set) are projective (or, equivalently, when all products of projective right S-acts are projective). At approximately the same time, Victoria Gould solved the corresponding problem for strong flatness. Implicitly, Gould’s result also answers the question for condition (P) and condition (E). For products of flats, weakly flats, etc. to again have the same property, there are some published results as well. The specific questions of when S×S has certain flatness properties have so far not been considered. In this paper, we will address these problems. S. Bulman-Fleming research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494. Some of the results in this article are contained in the M.Math. thesis of A. Gilmour, University of Waterloo (2007).     

A Functional Calculus for n-Tuples of Noncommuting Operators

We employ the notion of slice monogenic functions to define a new functional calculus for an n-tuple of not necessarily commuting operators. This calculus is consistent with the Riesz-Dunford calculus for a single operator. Received: October, 2007. Accepted: February, 2008.     

The Weierstrass factorization theorem for slice regular functions over the quaternions

The class of slice regular functions of a quaternionic variable has been recently introduced and is intensively studied, as a quaternionic analogue of the class of holomorphic functions. Unlike other classes of quaternionic functions, this one contains natural quaternionic polynomials and power series. Its study has already produced a rather rich theory having steady foundations and interesting applications. The main purpose of this article is to prove a Weierstrass factorization theorem for slice regular functions. This result holds in a formulation that reflects the peculiarities of the quaternionic setting and the structure of the zero set of such functions. Some preliminary material that we need to prove has its own independent interest, like the study of a quaternionic logarithm and the convergence of infinite products of quaternionic functions.     

Quaternionic Monge-Ampère equations

The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in ℍ ⁿ . We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].     

Groupoid C *-Algebras and Index Theory on Manifolds with Singularities

The simplest case of a manifold with singularities is a manifold M with boundary, together with an identification M M × P, where P is a fixed manifold. The associated singular space is obtained by collapsing P to a point. When P = Z/k or S ¹, we show how to attach to such a space a noncommutative C ^*-algebra that captures the extra structure. We then use this C ^*-algebra to give a new proof of the Freed–Melrose Z/k-index theorem and a proof of an index theorem for manifolds with S ¹ singularities. Our proofs apply to the real as well as to the complex case. Applications are given to the study of metrics of positive scalar curvature.