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In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity. [ABSTRACT FROM AUTHOR]

In this paper we use techniques in Fock spaces theory and compute how the Segal-Bargmann transform acts on special wave functions obtained by multiplying superoscillating sequences with normalized Hermite functions. It turns out that these special wave functions can be constructed also by computing the approximating sequence of the normalized Hermite functions. First, we start by treating the case when a superoscillating sequence is multiplied by the Gaussian function. Then, we extend these calculations to the case of normalized Hermite functions leading to interesting relations with Weyl operators. In particular, we show that the Segal-Bargmann transform maps superoscillating sequences onto a superposition of coherent states. Following this approach, the computations lead to a specific linear combination of the normalized reproducing kernels (coherent states) of the Fock space. As a consequence, we obtain two new integral Bargmann-type representations of superoscillating sequences. We also investigate some results relating superoscillation functions with Weyl operators and Fourier transform. [ABSTRACT FROM AUTHOR]

In the last decade there has been a growing interest in superoscillations in various fields of mathematics, physics and engineering. However, while in applications as optics the local oscillatory behaviour is the important property, some convergence to a plane wave is the standard characterizing feature of a superoscillating function in mathematics and quantum mechanics. Also there exists a certain discrepancy between the representation of superoscillations either as generalized Fourier series, as certain integrals or via special functions. The aim of this work is to close these gaps and give a general definition of superoscillations, covering the well-known examples in the existing literature. Superoscillations will be defined as sequences of holomorphic functions, which admit integral representations with respect to complex Borel measures and converge to a plane wave in the space \mathcal {A}_1(\mathbb {C}) of entire functions of exponential type. [ABSTRACT FROM AUTHOR]

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. These functions appear in various fields of science and technology, in particular they were discovered in quantum mechanics in the context of weak values introduced by Y. Aharonov and collaborators. The evolution problem of superoscillatory functions as initial conditions for the Schrödinger equation is intensively studied nowadays and the supershift property of the solution of Schrödinger equation encodes the persistence of superoscillatory phenomenon during the evolution. In this paper, we prove that the evolution of a superoscillatory initial datum for spinning particles in a magnetic field has the supershift property. Our techniques are based on the exact propagator of spinning particles, the associated infinite order differential operators and their continuity on suitable spaces of entire functions with growth conditions. [ABSTRACT FROM AUTHOR]

Triangular operators are an essential tool in the study of nonselfadjoint operators that appear in different fields with a wide range of applications. Although the development of a quaternionic counterpart for this theory started at the beginning of this century, the lack of a proper spectral theory combined with problems caused by the underlying noncommutative structure prevented its real development for a long time. In this paper, we give criteria for a quaternionic linear operator to have a triangular representation, namely, under which conditions such operators can be represented as a sum of a diagonal operator with a Volterra operator. To this effect, we investigate quaternionic Volterra operators based on the quaternionic spectral theory arising from the S‐spectrum. This allow us to obtain conditions when a non‐selfadjoint operator admits a triangular representation. [ABSTRACT FROM AUTHOR]

We use methods from the Fock space and Segal–Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods and in support vector machine classification algorithms. Complex analysis techniques allow us to consider several notions linked to the radial basis function (RBF) kernels, such as the feature space and the feature map, using the so-called Segal–Bargmann transform. We also show how the RBF kernels can be related to some of the most used operators in quantum mechanics and time frequency analysis; specifically, we prove the connections of such kernels with creation, annihilation, Fourier, translation, modulation, and Weyl operators. For the Weyl operators, we also study a semigroup property in this case. [ABSTRACT FROM AUTHOR]

We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given. [ABSTRACT FROM AUTHOR]

In this paper we define a new function theory of slice monogenic functions of a Clifford variable using the S-functional calculus for Clifford numbers. Previous attempts of such a function theory were obstructed by the fact that Clifford algebras, of sufficiently high order, have zero divisors. The fact that Clifford algebras have zero divisors does not pose any difficulty whatsoever with respect to our approach. The new class of functions introduced in this paper will be called the class of slice monogenic Clifford functions to stress the fact that they are defined on open sets of the Clifford algebra Rn. The methodology can be generalized, for example, to handle the case of noncommuting matrix variables. [ABSTRACT FROM AUTHOR]

Superoscillations are band-limited functions that can oscillate faster than their fastest Fourier component. These functions (or sequences) appear in weak values in quantum mechanics and in many fields of science and technology such as optics, signal processing and antenna theory. In this paper, we introduce a new method to generate superoscillatory functions that allows us to construct explicitly a very large class of superoscillatory functions. [ABSTRACT FROM AUTHOR]

We consider the evolution, for a time-dependent Schrödinger equation, of the so-called Dirac comb. We show how this evolution allows us to recover explicitly (indeed optically) the values of the quadratic generalized Gauss sums. Moreover we use the phenomenon of superoscillatory sequences to prove that such Gauss sums can be asymptotically recovered from the values of the spectrum of any sufficiently regular function compactly supported on R. The fundamental tool we use is the so called Galilean transform that was introduced and studied in the context on non-linear time dependent Schrödinger equations. Furthermore, we utilize this tool to understand in detail the evolution of an exponential e i ω x in the case of a Schrödinger equation with time-independent periodic potential. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the research of our previous work [5] and considers quaternionic generalized Carathéodory functions and the related family of generalized positive functions. It is addressed to a wide audience which includes researchers in complex and hypercomplex analysis, in the theory of linear systems, but also electric engineers. For this reason it includes some results on generalized Carathéodory functions and their factorization in the classic complex case which might be of independent interest. An important new result is a pseudo-spectral factorization and we also discuss some interpolation problems in the class of quaternionic generalized positive functions. [ABSTRACT FROM AUTHOR]

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The study of the evolution of superoscillations as initial datum of field equations requires the notion of supershift, which generalizes the concept of superoscillations. The present paper has a dual purpose. The first one is to give an updated and self-contained explanation of the strategy to study the evolution of superoscillations by referring to the quantum-mechanical Schrödinger equation and its variations. The second purpose is to treat the Dirac equation in relativistic quantum theory. The treatment of the evolution of superoscillations for the Dirac equation can be deduced by recent results on the Klein–Gordon equation, but further additional considerations are in order, which are fully described in this paper. [ABSTRACT FROM AUTHOR]

In the recent years, there has been a lot of interest in fractional diffusion and fractional evolution problems. The spectral theory on the S‐spectrum turned out to be an important tool to define new fractional diffusion operators stating from the Fourier law for nonhomogeneous materials. Precisely, let eℓ, eℓ=1,2,3 be orthogonal unit vectors in R3 and let Ω⊂R3 be a bounded open set with smooth boundary ∂Ω. Denoting by x_ a point in Ω, the heat equation is obtained replacing the Fourier law given by T=ax_∂xe1+bx_∂ye2+cx_∂ze3into the conservation of energy law. In this paper, we investigate the structure of the fractional powers of the vector operator T, with homogeneous Dirichlet boundary conditions. Recently, we have found sufficient conditions on the coefficients a, b, c:Ω→R such that the fractional powers of T exist in the sense of the S‐spectrum approach. In this paper, we show that under a different set of conditions on the coefficients a, b, c, the fractional powers of T have a different structure. [ABSTRACT FROM AUTHOR]

• Review of studies on ceftaroline fosamil, a fifth-generation cephalosporin with anti-MRSA activity. • Prescribed off-label in bacteremia, endocarditis, osteoarticular infections, hospital-acquired pneumonia, meningitis. • 77% clinical success with off-label ceftaroline and few (9%) adverse events (AEs) reported but not all studies reported AEs. • Studies used different doses/dosing intervals because of renal adjustment and off-label use of higher doses. • Quality of included studies is low because of the retrospective nature of studies, case series and small sample sizes. • More evidence of ceftaroline efficacy and safety required from randomized controlled trials and PK/PD studies. Ceftaroline fosamil is a fifth-generation cephalosporin with anti-methicillin-resistant Staphylococcus aureus (MRSA) activity. It has been approved by the EMA and FDA for the treatment of adults and children with community-acquired bacterial pneumonia (CABP) and acute bacterial skin and skin structure infections (ABSSSI). However, ceftaroline fosamil has a broad spectrum of activity, and a good safety and tolerability profile, so is frequently used off-label. The aim of this systematic review was to summarize the safety and efficacy of off-label use of ceftaroline. The review was conducted according to PRISMA guidelines. MEDLINE, EMBASE and CENTRAL databases (2010-2018) were searched using as the main term ceftaroline fosamil and its synonyms in combination with names of infectious diseases of interest. A total of 21 studies with 1901 patients were included: the most common off-label indications for ceftaroline use were bacteremia (n =595), endocarditis (n =171), osteoarticular infections (n =368), hospital-acquired pneumonia (n =115) and meningitis (n =23). The most common reasons for off-label use were persistent or recurrent infection after standard treatment or non-susceptibility to vancomycin and daptomycin. Clinical success was evaluated in 933 patients, and 724 (77%) of these reached this positive outcome. Incidence of adverse events (AEs) was reported in 11 studies. In 83 (9%) cases there were AEs related to the use of ceftaroline; the most common reported AEs were nausea, vomiting, diarrhea, rash and neutropenia. The review results show that ceftaroline may be used in clinical settings other than those currently approved; however, the use of ceftaroline in these contexts deserves further investigation. [ABSTRACT FROM AUTHOR]

In this paper we discuss the approximation of hyperfunctions in one variable using sequences of low frequency hyperfunctions. In particular, we show that this superoscillation property holds for compactly supported hyperfunctions. Finally, we show that the ap- proximation is also possible for tempered hyperfunctions. [ABSTRACT FROM AUTHOR]

The theory of quaternionic operators has applications in several different fields, such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of a spectrum. In fact, in quaternionic operator theory the classical notion of a resolvent operator and the one of a spectrum need to be replaced by the two S-resolvent operators and the S-spectrum. This is a consequence of the noncommutativity of the quaternionic setting. Indeed, the S-spectrum of a quaternionic linear operator T is given by the noninvertibility of a second order operator. This presents new challenges which make our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of S-spectrum and of slice hyperholomorphicity of the S-resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator. [ABSTRACT FROM AUTHOR]

Let eℓ, for ℓ = 1,2,3, be orthogonal unit vectors in R3 and let Ω⊂R3 be a bounded open set with smooth boundary ∂Ω. Denoting by x_ a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: T=a(x_)∂xe1+b(x_)∂ye2+c(x_)∂ze3,into the conservation of energy law, here a, b, c:Ω→R are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, c:Ω→R the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the additive splitting associated to the quaternionic Cauchy transform defined by the Cauchy formula of slice hyperholomorphic functions. Moreover, we introduce and study the analogue of the fundamental solution of the global operator of slice hyperholomorphic functions. We state our results in the quaternionic setting but several results hold for Clifford algebra-valued function with minor changes in the proofs. [ABSTRACT FROM AUTHOR]

Abstract In this paper we provide realizations of Herglotz functions in the quaternionic case where hyperholomorphic functions are meant in the sense of Fueter. We consider the counterparts of the Arveson space and of the Herglotz–Agler functions by introducing suitable positive kernels. Crucial tools are the so-called Fueter variables and the CK-product. Our realization results also show that positivity implies analyticity. [ABSTRACT FROM AUTHOR]

Abstract: Care transitions represent a common source of drug errors and confusions. The purpose of our prospective observational study was to assess the prevalence of medication discrepancies at care transitions, along with potentially inappropriate medications and potential drug‐drug interactions, in an internal medicine unit of an Italian hospital. Adverse drug reactions that occurred in the 30‐day period after the discharge from the hospital were included. A related‐samples McNemar test was performed for evaluating the effects of hospitalization on the above‐mentioned measures of drug prescribing. Medication discrepancies were frequent both on admission (93.4% [95%CI 0.8749, 0.9713]) and at discharge (78.7% [95%CI 0.7035, 0.8558]), with a significant difference between transition times (−14.7% [95%CI −21.82%, −7.69%]; P < .001)]. A high potentially inappropriate medication use prevalence was revealed without differences between care transitions. Potential drug‐drug interactions were more frequent at admission to the hospital, with a significant difference of 8.2% in the distribution of patients with potential drug‐drug interactions between care transitions. None of the adverse drug reactions recorded on follow‐up was related to unintentional discrepancies, and the prevalence rate of patients with potentially inappropriate medication‐related adverse drug reactions ranged between 4.9% and 6.9%, and the prevalence rate of patients with drug‐drug interaction–related adverse drug reactions was 4.1% of patients. This study is important to raise awareness of the potential dangers medication discrepancies, potentially inappropriate medications, and potential drug‐drug interactions could have on older adults. Clinicians and clinical pharmacologists must collaborate to improve patient care and minimize drug‐related clinical outcomes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study the fractional Hardy spaces of the half space and of the unit ball in the quaternionic setting. In particular, we discuss their properties of invariance and of factorization in terms of functions in the Hardy space of the half space in the first case, and in terms of a suitable reproducing kernel Hilbert space in the case of the unit ball. [ABSTRACT FROM AUTHOR]

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 C m . We also show that the formula holds for the Szegő–Radon transform that we introduced in Colombo et al. (2016) and we define the dual transform and we provide an inversion formula. Finally, in Theorem 5.6, we prove an integral formula for the monogenic part of an entire function. [ABSTRACT FROM AUTHOR]

This study compares the effect of the modified early warning score (MEWS) versus a new early warning system (Niguarda MEWS) for detecting instability and criticality in hospital medical departments. A retrospective observational study was conducted in the Internal Medicine ward of Niguarda Ca' Granda Hospital in Milan between November 2013 and October 2014. MEWS and Niguarda-MEWS were gathered using: systolic blood pressure, respiratory frequency, heart rate, temperature, level of consciousness, oxygen saturation, creatinine level, hematocrit level and age. In order to determine if the patient was critical or not the MEWS criticality cut-off value chosen was 3, while in the Niguarda MEWS it was 6. The primary outcome was the correlation between the critical level of the two scores and in-hospital mortality. The secondary endpoint was the correlation between a specific disease and the two scores. In the study, 471 patients were included, using both the MEWS and the Niguarda MEWS score at admittance: 33.4% of patients turned out to be critically ill using the former, 40.98% when using the latter. Therefore, the specificity of scores was 70% for MEWS and 73% for Niguarda MEWS, the sensitivity 58% for MEWS and 63% for Niguarda MEWS, Niguarda MEWS area under the curve (AUC): 0.736, MEWS AUC: 0.670. For the secondary outcome, the new score is higher for genitourinary and respiratory diseases. Niguarda-MEWS could be an optimal tool to detect criticality and instability in order to address the patient to the right level of care. [ABSTRACT FROM AUTHOR]

In this paper we consider functions in the Hardy space H2p×q defined in the unit disc of matrix-valued functions. We show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably modified matrix-valued Blaschke products, in an adaptive way. The procedure is based on a generalization to the matrix-valued case of the maximum selection principle which involves not only selections of suitable points in the unit disc but also suitable orthogonal projections. We show that the maximum selection principle gives rise to a convergent algorithm. Finally, we discuss the case of real-valued signals. [ABSTRACT FROM AUTHOR]

The quaternionic analogue of the Riesz-Dunford functional calculus and the theory of semigroups and groups of linear quaternionic operators have recently been introduced and studied. In this paper, we suppose that is the quaternionic infinitesimal generator of a strongly continuous group of operators and we show how we can define bounded operators , where belongs to a class of functions that is larger than the one to which the quaternionic functional calculus applies, using the quaternionic Laplace-Stieltjes transform. This class includes functions that are slice regular on the -spectrum of but not necessarily at infinity. Moreover, we establish the relation between and the quaternionic functional calculus and we study the problem of finding the inverse of . [ABSTRACT FROM AUTHOR]

In this paper, we start the study of stochastic processes over the skew field of quaternions. We discuss the relation between positive definite functions and the covariance of centered Gaussian processes and the construction of stochastic processes and their derivatives. The use of perfect spaces and strong algebras and the notion of Fock space are crucial in this framework. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 addressed to researchers working in operator theory and in hypercomplex analysis. [ABSTRACT FROM AUTHOR]

In this paper we extend the H ∞ functional calculus to quaternionic operators and to n -tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called S -functional calculus. The S -functional calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford functional calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S -functional calculus is based on the notion of S -spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H ∞ functional calculus based on the notion of S -spectrum for both quaternionic operators and for n -tuples of noncommuting operators. We remark that the H ∞ functional calculus for ( n + 1 ) -tuples of operators applies, in particular, to the Dirac operator. [ABSTRACT FROM AUTHOR]

In this paper we study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space R N = { ( x 1 , x 2 , … ) : x d ∈ R } endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner–Minlos theorem. A tool for proving this result is a classical matricial analogue of the Bochner–Minlos theorem, which we believe is new. We will see that in all these various settings the integral representation is with respect to a quaternion-valued measure which has certain symmetry properties. [ABSTRACT FROM AUTHOR]

Spectral theory on the S -spectrum was born out of the need to give quaternionic quantum mechanics a precise mathematical foundation (Birkhoff and von Neumann [8] showed that a general set theoretic formulation of quantum mechanics can be realized on real, complex or quaternionic Hilbert spaces). Then it turned out that spectral theory on S -spectrum has important applications in several fields such as fractional diffusion problems and, moreover, it allows one to define several functional calculi for n -tuples of noncommuting operators. With this paper we show that the spectral theory on the S -spectrum is much more general and it contains, just as particular cases, the complex, the quaternionic and the Clifford settings. In fact, the S -spectrum is well defined for objects in an algebra that has a complex structure and for operators in general Banach modules. We show that the abstract formulation of the S -functional calculus goes beyond quaternionic and Clifford analysis, indeed the S -functional calculus has a certain universality property. This fact makes the spectral theory on the S -spectrum applicable to several fields of operator theory and allows one to define functions of noncommuting matrix variables, and operator variables, as a particular case. [ABSTRACT FROM AUTHOR]

Using Zeilberger generating function formula for the values of a discrete analytic function in a quadrant we make connections with the theory of structured reproducing kernel spaces, structured matrices and a generalized moment problem. An important role is played by a Krein space realization result of Dijksma, Langer and de Snoo for functions analytic in a neighborhood of the origin. A key observation is that, using a simple Moebius transform, one can reduce the study of discrete analytic functions in the upper right quadrant to problems of function theory in the open unit disk. As an example, we associate to each finite positive measure on [ 0 , 2 π ] a discrete analytic function on the right-upper quarter plane with values on the lattice defining a positive definite function. Emphasis is put on the rational case, both when an underlying Carathéodory function is rational and when, in the positive case, the spectral function is rational. The rational case and the general case are linked via the existence of a unitary dilation, possibly in a Krein space. [ABSTRACT FROM AUTHOR]

We define a version of the Radon transform for monogenic functions which is based on Szegő kernels. The corresponding Szegő–Radon projection is abstractly defined as the orthogonal projection of a Hilbert module of left monogenic functions onto a suitable closed submodule of functions depending only on two variables. We also establish the inversion formula based on the dual transform. [ABSTRACT FROM AUTHOR]

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 lack of a suitable notion of spectrum was a major obstruction to fully understand the spectral theorem for quaternionic normal operators. A prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded antiself adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The pigeonhole principle: "If you put three pigeons in two pigeonholes, at least two of the pigeons end up in the same hole," is an obvious yet fundamental principle of nature as it captures the very essence of counting. Here however we show that in quantum mechanics this is not true! We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box. Furthermore, we show that the above "quantum pigeonhole principle" is only one of a host of related quantum effects, and points to a very interesting structure of quantum mechanics that was hitherto unnoticed. Our results shed new light on the very notions of separability and correlations in quantum mechanics and on the nature of interactions. It also presents a new role for entanglement, complementary to the usual one. Finally, interferometric experiments that illustrate our effects are proposed. [ABSTRACT FROM AUTHOR]

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 paperwe 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 suitableweights are taken into account. Finally,we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane. [ABSTRACT FROM AUTHOR]

The theory of slice hyperholomorphic functions, introduced in recent years, has important applications in operator theory. The quaternionic version of this function theory and its Cauchy formula yield to a definition of the quaternionic version of the Riesz-Dunford functional calculus which is based on the notion of S-spectrum. This quaternionic functional calculus allows to define the quaternionic evolution operator which appears in the quaternionic version of quantum mechanics proposed by J. von Neumann and later developed by S. L. Adler. Generation results such as the Hille-Phillips-Yosida theorem have been recently proved. In this paper, we study the perturbation of the generator. The motivation of this study is that, as it happens in the classical case of closed complex linear operators, to verify the generation conditions of the Hille-Phillips-Yosida theorem, in the concrete cases, is often difficult. Thus in this paper we study the generation problem from the perturbation point of view. Precisely, given a quaternionic closed operator T that generates the evolution operator we study under which condition a closed operator P is such that T + P generates the evolution operator . This paper is addressed to people working in different research areas such as hypercomplex analysis and operator theory. [ABSTRACT FROM AUTHOR]

In this paper we study second order ordinary and partial differential equations with generalized Wentzell boundary condition. We prove, by a perturbation method, that certain second order ordinary differential operators generate an analytic semigroup in $$W^{1,p}([0,1])$$ and the same result has been extended for the degenerate operator $$Au(x):=x(1-x)u''(x)$$ . Finally, we prove that certain linear partial differential operators of the second order generate analytic semigroups in the space of continuous functions. [ABSTRACT FROM AUTHOR]

In the recent years the theory of slice hyperholomorphic functions has become an important tool to study two functional calculi for n -tuples of operators and also for its applications to Schur analysis. In particular, using the Cauchy formula for slice hyperholomorphic functions, it is possible to give the Fueter–Sce mapping theorem an integral representation. With this integral representation it has been defined a monogenic functional calculus for n -tuples of bounded commuting operators, the so called F -functional calculus. In this paper we show that it is possible to define this calculus also for n -tuples containing unbounded operators and we obtain an integral representation formula analogous to the one of the Riesz–Dunford functional calculus for unbounded operators acting on a complex Banach space. As we will see, it is not an easy task to provide the correct definition of the F -functional calculus in the unbounded case. This paper is addressed to a double audience, precisely to people with interests in hypercomplex analysis and also to people working in operator theory. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we provide an alternative method to construct the Fueter primitive of an axial monogenic function of degree k, which is complementary to the one used in [4]. As a byproduct, we obtain an explicit description of the kernel of the Fueter mapping. We also apply our method to obtain the Fueter primitives of the Cauchy kernels with singularities on the unit sphere. [Copyright &y& Elsevier]

In this paper we introduce and study a functional calculus for bicomplex linear bounded operators. The study is based on the decomposition of bicomplex numbers and of linear operators using the two nonreal idempotents. We show that, due to the presence of zero divisors in the bicomplex numbers, the spectrum of a bounded operator is unbounded. We therefore introduce a different spectrum (called reduced spectrum) which is bounded and turns out to be the right tool to construct the bicomplex holomorphic functional calculus. Finally we provide some properties of the calculus. [ABSTRACT FROM AUTHOR]