Pdf in this article, we will show that the category of quaternion vector spaces, the category of both onesided and two sided quaternion hilbert. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Montreal quaternionic wavelets on quaternionic hilbert spaces the standard wavelet group can be identi. Functional analysis immediately available upon purchase as print book shipments may be delayed due to the covid19 crisis. Close this message to accept cookies or find out how to manage your cookie settings.
A variation on uncertainty principle and logarithmic uncertainty principle for continuous quaternion wavelet transforms bahri, mawardi and ashino, ryuichi, abstract and applied analysis, 2017. As for the adjective quaternionic, note that we have quaternion group though references often use ionic for the designation of that group. Eriksson and others developed in a number of papers a modi. But textbooks today frequently use complex variable in preference to complex analysis. In the same hilbert space, then we study the fredholm operators and the fredholm index. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. There functional analysis is put on a par with geometric analysis. But additionally, what i did with my quantum dots were also interesting. Operator theory on quaternionic hilbert spaces volume 7 issue 2 neil charles powers. These are developed in the present chapter, largely taken from 35. Kurepa, 9783540178330, available at book depository with free delivery worldwide. 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. About the author in addition to functional analysis, second edition, walter rudin is the author of two other books. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.
Slice functional calculus in quaternionic hilbert spaces unitn. The functional analytic and representation theoretic. Mr 33599 daniel alpay, the schur algorithm, reproducing kernel spaces and system theory, smfams texts and monographs, vol. On properties of holomorphic functions in quaternionic. 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. The sarscov2 spike s glycoprotein promotes entry into host cells and is the main target of neutralizing. After the course functional analysis 2 i was fascinated by the idea of a functional calculus. We draw the conclusions from the earlier presented quaternionic generalization of cauchyriemanns equations.
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. Speranca, explicit constructions over the exotic 8sphere pdf, pdf the proof of prop. On properties of holomorphic functions in quaternionic analysis. Vaccine and therapeutic discovery efforts are paramount to curb the pandemic spread of this zoonotic virus. Quaternionic analysis mathematical proceedings of the. 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. Operator theory on quaternionic hilbert spaces bulletin.
Functions of the infinitesimal generator of a strongly. The general expressions for constituents of holomorphic functions as well as the relations between them are deduced. 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. Regular functions of a quaternionic variable graziano. Volume 164, issue 1 pages 1180 10 may 1999 download full issue. Functions of a complex variable target groups research. We use cookies to distinguish you from other users and to provide you with a better experience on our websites. This is a precopyediting, authorproduced pdf of an article accepted for publication in. 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. The extension of the wavelet theory to quaternion variable function spaces would be very beneficial in the solution of many problems in physics. Slice hyperholomorphic schur analysis springerlink. Fractional powers of quaternionic operators and katos formula using slice hyperholomorphicity. Seminar on functional analysis and global analysis organized by kenro furutani and takao kobayashi 21.
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. 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. A quaternion is a 4tuple, which is a more concise representation than a rotation matrix. Mathematics free fulltext quaternionic blaschke group. 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. Michael parfenov, on properties of holomorphic functions in quaternionic analysis. 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. Of course the ftir analysis is the actual method of identification of functional group. 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.
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 demonstrations, we will use this observ ation to. The third part represents the core of the book and explores quaternionic schur analysis and its various applications. A quaternionic structure on a manifold is induced by a special quaternionic structure if and only if the bundle is trivial. Let x be a finitedimensional c algebra over the field of complex numbers c. Deformation quantization in quantum mechanics and quantum field theory hirshfeld, allen, 2003. Noteworthy fiber products with the quaternionic hopf fibration, notably exotic 7spheres, are discussed in llohann d. Quaternionic signal analysis has also been used in image processing to construct. Adequate quaternionic generalization of complex differentiability. The endomorphisms are called standard generators of the quaternionic structure, and the basis of defined by them is called the standard basis. Journal of functional analysis vol 164, issue 1, pages 1.
Fabrizio colombo, phd professore associato am politecnico di milano durch jonathan. A survey of quaternionic analysis florida state university. 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. On some properties of the quaternionic functional calculus. He wrote principles of mathematical analysis while he was a c. Quaternionicanalysis,representationtheoryand physics. The sfunctional calculus has two versions one for quaternionicvalued functions and one for clifford algebravalued functions and can be considered the. Adler, quaternionic quantum mechanics and quantum fields, international series of monographs on physics, vol. The theory of slice regular functions over quaternions is the central subject of the present volume. Analytic functional calculus in quaternionic framework. 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. Segal on the foundational role of representation theory of su2, 2 in physics.
Quaternionic structure encyclopedia of mathematics. This paper is meant to be a survey on these numbers. 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. They imply, explicitly or implicitly, the leftand right.
Pdf on quaternionic functional analysis researchgate. For full derivatives it is a consequence of uniting. Many of the algebraic and geometric properties of complex analytic functions are not present in quaternionic analysis. Quaternionic and clifford calculus for physicists and.
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. A a quaternion is a 4tuple, which is a more concise representation than a rotation matrix. Slice hyperholomorphic functions and the quaternionic. 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. However, the mentioned continuous functional calculi are defined only for bounded normal operators.
Seminar on functional analysis and global analysis. Larson, frames for banach spaces, in the functional and harmonic analysis of wavelets and frames san antonio, tx, 1999, contemp. Ams transactions of the american mathematical society. 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. It is assumed that the reader is familiar with the basics of complex function theory in one variable, functional analysis and algebra. 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. 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. In some recent works we have developed a new functional calculus for bounded and unbounded quaternionic operators acting on a quaternionic banach space. In mathematics, quaternionic analysis is the study of functions with quaternions as the domain andor range. In this section, we introduce functional calculus, and then in the next section we introduce quaternionic calgebras. The symmetry properties of constituents of holomorphic functions and their derivatives of all orders are proved. Pdf basics of functional analysis with bicomplex scalars. 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. We show that the elementary quaternion functions introduced by hamilton as well as the quaternion logarithm function possess such a derivative.
Frames for banach spaces, in the functional and harmonic analysis. 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. There exist various versions of quaternionic analysis. The richness of the theory of functions over the complex. Quaternionic hilbert space, functional calculus, slice functions, spectral map. Its geometric meaning is also more obvious as the rotation axis and angle can be trivially recovered. This recent theory has expanded rapidly, producing a variety of new results that have caught the attention of the international research community. 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. Mathematics operator algebras, mathematics functional analysis, 16d20, 46b04, 46c05, 46l05, 81s99. Not all differential operators encode the same information about shape. For bounded right linear operators, in a right quaternionic hilbert space with a left multiplication defined on it, we study the approximate spoint spectrum.
On the formulations of the quaternionic functional calculus. We discuss regular functions, that are counterpart of complex holomorphic functions for quaternionic setting. So, is an antiisomorphism of the quaternionic banach spaces b lv. At the same time, the theory has already developed. The functional analytic and representation theoretic foundations of wavelet theory. 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. In this article, we will show that the category of quaternion vector spaces. A new approach to slice regularity on real algebras, hypercomplex analysis and. This book defines and examines the counterpart of schur functions and schur analysis in the slice hyperholomorphic setting. Slice hyperholomorphic functions and the quaternionic functional.
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