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Fabrizio Colombo

Publications and source records attributed to Fabrizio Colombo.

At least 73 records · Page 4Linked to original sources
arXiv2016

On the Bargmann-Radon transform in the monogenic setting

In this paper we introduce and study a Bargmann-Radon transform on the real monogenic Bargmann module. This transform is defined as the projection of the real Bargmann module on the closed submodule of monogenic functions spanned by the monogenic plane waves. We prove that this projection can be written in integral form in terms the so-called Bargmann-Radon kernel. Moreover, we have a characterization formula for the Bargmann-Radon transform of a function in the real Bargmann module in terms of its complex extension and then its restriction to the nullcone in $\mathbb C^m$. We also show that the formula holds for the Szegő-Radon transform that we introduced in [4] (see list of references). Finally, we define the dual transform and we provide an inversion formula.

math.CV↗
arXiv2016

Entire slice regular functions

Entire functions in one complex variable are extremely relevant in several areas ranging from the study of convolution equations to special functions. An analog of entire functions in the quaternionic setting can be defined in the slice regular setting, a framework which includes polynomials and power series of the quaternionic variable. In the first chapters of this work we introduce and discuss the algebra and the analysis of slice regular functions. In addition to offering a self-contained introduction to the theory of slice-regular functions, these chapters also contain a few new results (for example we complete the discussion on lower bounds for slice regular functions initiated with the Ehrenpreis-Malgrange, by adding a brand new Cartan-type theorem). The core of the work is Chapter 5, where we study the growth of entire slice regular functions, and we show how such growth is related to the coefficients of the power series expansions that these functions have. It should be noted that the proofs we offer are not simple reconstructions of the holomorphic case. Indeed, the non-commutative setting creates a series of non-trivial problems. Also the counting of the zeros is not trivial because of the presence of spherical zeros which have infinite cardinality. We prove the analog of Jensen and Carathéodory theorems in this setting.

math.CV↗
arXiv2016

Fractional powers of quaternionic operators and Kato's formula using slice hyperholomorphicity

In this paper we introduce fractional powers of quaternionic operators. Their definition is based on the theory of slice-hyperholomorphic functions and on the $S$-resolvent operators of the quaternionic functional calculus. The integral representation formulas of the fractional powers and the quaternionic version of Kato's formula are based on the notion of $S$-spectrum of a quaternionic operator. The proofs of several properties of the fractional powers of quaternionic operators rely on the $S$-resolvent equation. This equation, which is very important and of independent interest, has already been introduced in the case of bounded quaternionic operators, but for the case of unbounded operators some additional considerations have to be taken into account. Moreover, we introduce a new series expansion for the pseudo-resolvent, which is of independent interest and allows to investigate the behavior of the $S$-resolvents close to the $S$-spectrum. The paper is addressed to researchers working in operator theory and in complex analysis.

math.FA↗
arXiv2016

Adaptative decomposition: the case of the Drury-Arveson space

The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space $\mathbf H_2$ of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and products have counterparts in the unit ball of $\mathbb C^N$, and this fact allows us to extend in the present paper the maximum selection principle to the case of functions in the Drury-Arveson space of functions analytic in the unit ball of $\mathbb C^N$. This will give rise to an algorithm which is a variation in this higher dimensional case of the greedy algorithm. We also introduce infinite Blaschke products in this setting and study their convergence.

math.CV↗
arXiv2016

A Cauchy kernel for the Hermitian submonogenic system

Hermitian monogenic functions are the null solutions of two complex Dirac type operators. The system of these complex Dirac operators is overdetermined and may be reduced to constraints for the Cauchy datum together with what we called the Hermitian submonogenic system (see [8],[9]). This last system is no longer overdetermined and it has properties that are similar to those of the standard Dirac operator in Euclidean space, such as a Cauchy-Kowalevski extension theorem and Vekua type solutions. In this paper, we investigate plane wave solutions of the Hermitian submonogenic system, leading to the construction of a Cauchy kernel. We also establish a Stokes type formula that, when applied to the Cauchy kernel provides an integral representation formula for Hermitian submonogenic functions.

math.CV↗
arXiv2015

Schatten class and Berezin transform of quaternionic linear operators

In this paper we introduce the Schatten class of operators and the Berezin transform of operators in the quaternionic setting. The first topic is of great importance in operator theory but it is also necessary to study the second one because we need the notion of trace class operators, which is 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 for 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.

math.FA↗
arXiv2015

Functions of the infinitesimal generator of a strongly continuous quaternionic group

The analogue of the Riesz-Dunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. In this paper we suppose that $T$ is the infinitesimal generator of a strongly continuous group of operators $(\mathcal{Z}_T(t))_{t \in \mathbb{R}}$ and we show how we can define bounded operators $f(T)$, where $f$ belongs to a class of functions which is larger than the class of slice regular functions, using the quaternionic Laplace-Stieltjes transform. This class will include functions that are slice regular on the $S$-spectrum of $T$ but not necessarily at infinity. Moreover, we establish the relation of $f(T)$ with the quaternionic functional calculus and we study the problem of finding the inverse of $f(T)$.

math.SP↗
arXiv2015

On power series expansions of the S-resolvent operator and the Taylor formula

The $S$-functional calculus is based on the theory of slice hyperholomorphic functions and it defines functions of $n$-tuples of not necessarily commuting operators or of quaternionic operators. This calculus relays on the notion of $S$-spectrum and of $S$-resolvent operator. Since most of the properties that hold for the Riesz-Dunford functional calculus extend to the S-functional calculus it can be considered its non commutative version. In this paper we show that the Taylor formula of the Riesz-Dunford functional calculus can be generalized to the S-functional calculus, the proof is not a trivial extension of the classical case because there are several obstructions due to the non commutativity of the setting in which we work that have to be overcome. To prove the Taylor formula we need to introduce a new series expansion of the $S$-resolvent operators associated to the sum of two $n$-tuples of operators. This result is a crucial step in the proof of our main results,but it is also of independent interest because it gives a new series expansion for the $S$-resolvent operators. This paper is devoted to researchers working in operators theory and hypercomplex analysis.

math.FA↗
arXiv2015

Wiener algebra for the quaternions

We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-Lévy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener-Hopf operators.

math.CV↗
arXiv2014

The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum

In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of $S$-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The $S$-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators. The fact that the correct notion of spectrum for quaternionic operators was not previously known has been one of the main obstructions to fully understanding the spectral theorem in this setting. A prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.

math.SP↗
arXiv2014

Generalized quaternionic Schur functions in the ball and half-space and Krein-Langer factorization

In this paper we prove a new version of Krein-Langer factorization theorem in the slice hyperholomorphic setting which is more general than the one proved in [D. Alpay, F. Colombo, I. Sabadini, Krein-Langer factorization and related topics in the slice hyperholomorphic setting, J. Geom. Anal., 24 (2014), 843--872]. We treat both the case of functions with $κ$ negative squares defined on subsets of the quaternionic unit ball or on subsets of the half space of quaternions with positive real part. A crucial tool in the proof of our results is the Schauder-Tychonoff theorem and an invariant subspace theorem for contractions in a Pontryagin space.

math.CV↗
arXiv2014

The Fock space in the slice hyperholomorphic setting

In this paper we introduce and study some basic properties of the Fock space (also known as Segal-Bargmann space) in the slice hyperholomorphic setting. We discuss both the case of slice regular functions over quaternions and also the case of slice monogenic functions with values in a Clifford algebra. In the specific setting of quaternions, we also introduce the full Fock space. This paper can be seen as the beginning of the study of infinite dimensional analysis in the quaternionic setting.

math.CV↗
arXiv2014

The C-property for slice regular functions and applications to the Bergman space

This paper has a twofold purpose: on one hand we deepen the study of slice regular functions by studying their behavior with respect to the so-called C-property and anti-C-property. We show that, for any fixed basis of the algebra of quaternions $\mathbb{H}$ any slice regular function decomposes into the sum of four slice regular components each of them satisfying the C-property. Then, we will use these results to show a reproducing property of the Bergman kernels of the second kind.

math.CV↗
arXiv2014

The Radon transform between monogenic and generalized slice monogenic functions

In [J. Bures, R. Lavicka, V. Soucek, Elements of quaternionic analysis and Radon transform, Textos de Matematica 42, Departamento de Matematica, Universidade de Coimbra, 2009], the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on R^(n+1) using the Radon and dual Radon transforms. The main aim of this paper is to further develop this approach. In fact, the Radon transform for functions with values in the Clifford algebra R_n is mapping solutions of the generalized Cauchy-Riemann equation, i.e., monogenic functions, to a parametric family of holomorphic functions with values in R_n and, analogously, the dual Radon transform is mapping parametric families of holomorphic functions as above to monogenic functions. The parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so-called slice monogenic functions. An important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms.

math.CV↗
arXiv2014

Further properties of the Bergman spaces of slice regular functions

In this paper we continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paper we mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitable weights are taken into account. Finally, we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane.

math.CV↗
arXiv2014

Formulations of the F-functional calculus and some consequences

In this paper we introduce the two possible formulations of the F-functional calculus which are based on the Fueter-Sce mapping theorem in integral form and we introduce the pseudo F-resolvent equation. In the case of dimension 3 we prove the F-resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the F-functional calculus.

math.FA↗
arXiv2013

A new resolvent equation for the S-functional calculus

The S-functional calculus is a functional calculus for $(n+1)$-tuples of non necessarily commuting operators that can be considered a higher dimensional version of the classical Riesz-Dunford functional calculus for a single operator. In this last calculus, the resolvent equation plays an important role in the proof of several results. Associated with the S-functional calculus there are two resolvent operators: the left $S_L^{-1}(s,T)$ and the right one $S_R^{-1}(s,T)$, where $s=(s_0,s_1,\ldots,s_n)\in \mathbb{R}^{n+1}$ and $T=(T_0,T_1,\ldots,T_n)$ is an $(n+1)$-tuple of non commuting operators. These two S-resolvent operators satisfy the S-resolvent equations $S_L^{-1}(s,T)s-TS_L^{-1}(s,T)=\mathcal{I}$, and $sS_R^{-1}(s,T)-S_R^{-1}(s,T)T=\mathcal{I}$, respectively, where $\mathcal{I}$ denotes the identity operator. These equations allows to prove some properties of the S-functional calculus. In this paper we prove a new resolvent equation for the S-functional calculus which is the analogue of the classical resolvent equation. It is interesting to note that the equation involves both the left and the right S-resolvent operators simultaneously.

math.FA↗