Operator theory on quaternionic hilbert spaces volume 7 issue 2 neil charles powers. The symmetry properties of constituents of holomorphic functions and their derivatives of all orders are proved. Quaternionic structure encyclopedia of mathematics. So, is an antiisomorphism of the quaternionic banach spaces b lv. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called as with complex and real analysis, it is possible to study the concepts of analyticity, holomorphy, harmonicity and conformality in the context of. Quaternionic hilbert space, functional calculus, slice functions, spectral map. A quaternionic structure on a real vector space is a module structure over the skewfield of quaternions, that is, a subalgebra of the algebra of endomorphisms of induced by two anticommutative complex structures on cf. Frames for banach spaces, in the functional and harmonic analysis. Of course the ftir analysis is the actual method of identification of functional group. The theory of slice regular functions over quaternions is the central subject of the present volume. A new approach to slice regularity on real algebras, hypercomplex analysis and.
Recall that we deal with quaternionic holomorphic functions constructed from the corresponding holomorphic functions by a direct replacement of a complex variable by a quaternionic in expressions for holomorphic functions without change of a functional dependence form. That functional calculus is based on the theory of slice regular functions and on a cauchy formula which holds for particular domains where the admissible functions have power series expansions. Quaternionic and clifford calculus for physicists and. Kurepa, 9783540178330, available at book depository with free delivery worldwide. A quaternionic structure on a manifold is induced by a special quaternionic structure if and only if the bundle is trivial. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Mathematics operator algebras, mathematics functional analysis, 16d20, 46b04, 46c05, 46l05, 81s99. Pdf on quaternionic functional analysis researchgate. Inner product spaces and krein spaces in the quaternionic setting. A quaternionic structure on a manifold can be regarded as a structure, and a special quaternionic structure as a structure in the sense of the theory of structures cf. We show that the elementary quaternion functions introduced by hamilton as well as the quaternion logarithm function possess such a derivative. Michael parfenov, on properties of holomorphic functions in quaternionic analysis.
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain andor range. In this article, we will show that the category of quaternion vector spaces. X is a normal linear operator, then as is well known, a functional calculus can be defined with the standard eigenvalue definition, eigenvalues being the values. Quaternionicanalysis,representationtheoryand physics igorfrenkelandmatveilibine december1,2018 abstract we develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. Its geometric meaning is also more obvious as the rotation axis and angle can be trivially recovered. It is assumed that the reader is familiar with the basics of complex function theory in one variable, functional analysis and algebra. In some recent works we have developed a new functional calculus for bounded and unbounded quaternionic operators acting on a quaternionic banach space. To develop schur analysis in the slice hyperholomophic setting, a number of facts and results from quaternionic functional analysis for which no references were available are needed. As for the adjective quaternionic, note that we have quaternion group though references often use ionic for the designation of that group. Adler, quaternionic quantum mechanics and quantum fields, international series of monographs on physics, vol. We use cookies to distinguish you from other users and to provide you with a better experience on our websites. A survey of quaternionic analysis christopher stover january 5, 2014 abstract the group h of hamilton quaternions can be described, algebraically, as a fourdimensional associative normed division algebra over the ring r of real numbers.
Adequate quaternionic generalization of complex differentiability. Fractional powers of quaternionic operators and katos formula using slice hyperholomorphicity. The sarscov2 spike s glycoprotein promotes entry into host cells and is the main target of neutralizing. The minkowski space reformulation of quaternionic analysis brings us into a thorough study of minkowski space realization of unitary representations of the conformal group su2, 2 by h. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. Sarscov2 is a newly emerged coronavirus responsible for the current covid19 pandemic that has resulted in more than one million infections and 73,000 deaths11,22. Quaternionic signal analysis has also been used in image processing to construct. On some properties of the quaternionic functional calculus. We draw the conclusions from the earlier presented quaternionic generalization of cauchyriemanns equations. After the course functional analysis 2 i was fascinated by the idea of a functional calculus.
This book defines and examines the counterpart of schur functions and schur analysis in the slice hyperholomorphic setting. This recent theory has expanded rapidly, producing a variety of new results that have caught the attention of the international research community. Deformation quantization in quantum mechanics and quantum field theory hirshfeld, allen, 2003. On properties of holomorphic functions in quaternionic analysis. Segal on the foundational role of representation theory of su2, 2 in physics. The functional analytic and representation theoretic foundations of wavelet theory. There functional analysis is put on a par with geometric analysis. Functions of a complex variable target groups research. This is a precopyediting, authorproduced pdf of an article accepted for publication in. Speranca, explicit constructions over the exotic 8sphere pdf, pdf the proof of prop. The functional analytic and representation theoretic. Larson, frames for banach spaces, in the functional and harmonic analysis of wavelets and frames san antonio, tx, 1999, contemp. Seminar on functional analysis and global analysis. Fabrizio colombo, phd professore associato am politecnico di milano durch jonathan.
Operator theory on quaternionic hilbert spaces bulletin. In this article, we will show that the category of quaternion vector spaces, the category of both onesided and two sided quaternion hilbert spaces and the category of quatern, the category of real hilbert spaces and the category. As demonstrations, we will use this observ ation to. In this section, we introduce functional calculus, and then in the next section we introduce quaternionic calgebras.
As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. For example, the prevailing direction of quaternionic analysis 3 constructs the reg ular functions in an indirect way by means of expressions combining harmonic functions of four real variables and analytic functions of a complex variable. For full derivatives it is a consequence of uniting. Montreal quaternionic wavelets on quaternionic hilbert spaces the standard wavelet group can be identi.
But additionally, what i did with my quantum dots were also interesting. The sfunctional calculus has two versions one for quaternionicvalued functions and one for clifford algebravalued functions and can be considered the. This paper is meant to be a survey on these numbers. Eriksson and others developed in a number of papers a modi. A quaternion is a 4tuple, which is a more concise representation than a rotation matrix. Journal of functional analysis vol 164, issue 1, pages 1. On the formulations of the quaternionic functional calculus. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. Mathematics free fulltext quaternionic blaschke group. With an ever increasing literature on quaternionic and clifford analysis the need for an accessible and applicable book on the subject has never been greater. The richness of the theory of functions over the complex. We discuss regular functions, that are counterpart of complex holomorphic functions for quaternionic setting. At the same time, the theory has already developed. Pdf basics of functional analysis with bicomplex scalars.
The general expressions for constituents of holomorphic functions as well as the relations between them are deduced. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic hilbert space. The extension of the wavelet theory to quaternion variable function spaces would be very beneficial in the solution of many problems in physics. Functional analysis immediately available upon purchase as print book shipments may be delayed due to the covid19 crisis. The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other nontrivial real associative division algebra, namely the quaternions. Seminar on functional analysis and global analysis organized by kenro furutani and takao kobayashi 21. Principles of mathematical analysis and real and complex analysis, whose widespread use is illustrated by the fact that they have been translated into a total of languages. In the complex case, the blaschke group was introduced and studied.
I intend to show a way of generating frames by means of operators based on joint work with dave larson. Volume 164, issue 1 pages 1180 10 may 1999 download full issue. These are developed in the present chapter, largely taken from 35. On properties of holomorphic functions in quaternionic.
In this paper we suppose that t is the infinitesimal generator of a strongly continuous group of operators zt tt2r and we show how we can define bounded operators ft, where f belongs to a class of functions which. Noteworthy fiber products with the quaternionic hopf fibration, notably exotic 7spheres, are discussed in llohann d. Functions of the infinitesimal generator of a strongly. But textbooks today frequently use complex variable in preference to complex analysis. Ams transactions of the american mathematical society. For bounded right linear operators, in a right quaternionic hilbert space with a left multiplication defined on it, we study the approximate spoint spectrum.
It turned out that in the complex case this group plays important role in the construction of analytic wavelets and multiresolution analysis in different analytic function spaces. We will also give a riesz representation theorem for quaternion hilbert spaces and will extend two results of kulkarni namely, we will give the full. Slice hyperholomorphic functions and the quaternionic. Vaccine and therapeutic discovery efforts are paramount to curb the pandemic spread of this zoonotic virus. The aim of this work is to define a continuous functional calculus in quaternionic hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. They imply, explicitly or implicitly, the leftand right. Quaternionicanalysis,representationtheoryand physics. Many of the algebraic and geometric properties of complex analytic functions are not present in quaternionic analysis. A a quaternion is a 4tuple, which is a more concise representation than a rotation matrix. As a consequence, the class of analytic functions are generalized to include all polynomial functions of a single quaternionic variable with right acting quaternionic coefficients.
A survey of quaternionic analysis florida state university. In the same hilbert space, then we study the fredholm operators and the fredholm index. A variation on uncertainty principle and logarithmic uncertainty principle for continuous quaternion wavelet transforms bahri, mawardi and ashino, ryuichi, abstract and applied analysis, 2017. Slice hyperholomorphic functions and the quaternionic functional.
The third part represents the core of the book and explores quaternionic schur analysis and its various applications. Close this message to accept cookies or find out how to manage your cookie settings. Slice hyperholomorphic schur analysis springerlink. A dirac operator for extrinsic shape analysis hsuehti derek liu1 alec jacobson2 keenan crane1 1carnegie mellon university 2university of toronto min max laplacebeltrami relative dirac operator purely intrinsic purely extrinsic figure 1.
The analogue of the rieszdunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. About the author in addition to functional analysis, second edition, walter rudin is the author of two other books. Pdf in this article, we will show that the category of quaternion vector spaces, the category of both onesided and two sided quaternion hilbert. Regular functions of a quaternionic variable graziano. Analytic functional calculus in quaternionic framework.
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