A Short History of Operator Theory
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Mathematical and Physical Interpretations of Fractional Derivatives and Integrals
R. Hilfer Mathematical and Physical Interpretations of Fractional Derivatives and Integrals Abstract: Brief descriptions of various mathematical and physical interpretations of fractional derivatives and integrals have been collected into this chapter as points of reference and departure for deeper studies. “Mathematical interpre- tation” in the title means a brief description of the basic mathematical idea underlying a precise definition. “Physical interpretation” means a brief descrip- tion of the physical theory underlying an identification of the fractional order with a known physical quantity. Numerous interpretations had to be left out due to page limitations. Only a crude, rough and ready description is given for each interpretation. For precise theorems and proofs an extensive list of references can serve as a starting point. Keywords: fractional derivatives and integrals, Riemann-Liouville integrals, Weyl integrals, Riesz potentials, operational calculus, functional calculus, Mikusinski calculus, Hille-Phillips calculus, Riesz-Dunford calculus, classification, phase transitions, time evolution, anomalous diffusion, continuous time random walks Mathematics Subject Classification 2010: 26A33, 34A08, 35R11 published in: Handbook of Fractional Calculus: Basic Theory, Vol. 1, Ch. 3, pp. 47-86, de Gruyter, Berlin (2019) ISBN 978-3-11-057162-2 R. Hilfer, ICP, Fakultät für Mathematik und Physik, Universität Stuttgart, Allmandring 3, 70569 Stuttgart, Deutschland DOI https://doi.org/10.1515/9783110571622 Interpretations 1 1 Prolegomena -
Operator Theory in the C -Algebra Framework
Operator theory in the C∗-algebra framework S.L.Woronowicz and K.Napi´orkowski Department of Mathematical Methods in Physics Faculty of Physics Warsaw University Ho˙za74, 00-682 Warszawa Abstract Properties of operators affiliated with a C∗-algebra are studied. A functional calculus of normal elements is constructed. Representations of locally compact groups in a C∗-algebra are considered. Generaliza- tions of Stone and Nelson theorems are investigated. It is shown that the tensor product of affiliated elements is affiliated with the tensor product of corresponding algebras. 1 0 Introduction Let H be a Hilbert space and CB(H) be the algebra of all compact operators acting on H. It was pointed out in [17] that the classical theory of unbounded closed operators acting in H [8, 9, 3] is in a sense related to CB(H). It seems to be interesting to replace in this context CB(H) by any non-unital C∗- algebra. A step in this direction is done in the present paper. We shall deal with the following topics: the functional calculus of normal elements (Section 1), the representation theory of Lie groups including the Stone theorem (Sections 2,3 and 4) and the extensions of symmetric elements (Section 5). Section 6 contains elementary results related to tensor products. The perturbation theory (in the spirit of T.Kato) is not covered in this pa- per. The elementary results in this direction are contained the first author’s previous paper (cf. [17, Examples 1, 2 and 3 pp. 412–413]). To fix the notation we remind the basic definitions and results [17]. -
On Fractional Whittaker Equation and Operational Calculus
J. Math. Sci. Univ. Tokyo 20 (2013), 127–146. On Fractional Whittaker Equation and Operational Calculus By M. M. Rodrigues and N. Vieira Abstract. This paper is intended to investigate a fractional dif- ferential Whittaker’s equation of order 2α, with α ∈]0, 1], involving the Riemann-Liouville derivative. We seek a possible solution in terms of power series by using operational approach for the Laplace and Mellin transform. A recurrence relation for coefficients is obtained. The exis- tence and uniqueness of solutions is discussed via Banach fixed point theorem. 1. Introduction The Whittaker functions arise as solutions of the Whittaker differential equation (see [4], Vol.1)). These functions have acquired a significant in- creasing due to its frequent use in applications of mathematics to physical and technical problems [1]. Moreover, they are closely related to the con- fluent hypergeometric functions, which play an important role in various branches of applied mathematics and theoretical physics, for instance, fluid mechanics, electromagnetic diffraction theory, and atomic structure theory. This justifies the continuous effort in studying properties of these functions and in gathering information about them. As far as the authors aware, there were no attempts to study the corresponding fractional Whittaker equation (see below). Fractional differential equations are widely used for modeling anomalous relaxation and diffusion phenomena (see [3], Ch. 5, [6], Ch. 2). A systematic development of the analytic theory of fractional differential equations with variable coefficients can be found, for instance, in the books of Samko, Kilbas and Marichev (see [13], Ch. 3). 2010 Mathematics Subject Classification. Primary 35R11; Secondary 34B30, 42A38, 47H10. -
American Mathematical Society TRANSLATIONS Series 2 • Volume 130
American Mathematical Society TRANSLATIONS Series 2 • Volume 130 One-Dimensional Inverse Problems of Mathematical Physics aM „ °m American Mathematical Society One-Dimensional Inverse Problems of Mathematical Physics http://dx.doi.org/10.1090/trans2/130 American Mathematical Society TRANSLATIONS Series 2 • Volume 130 One-Dimensional Inverse Problems of Mathematical Physics By M. M. Lavrent'ev K. G. Reznitskaya V. G. Yakhno do, n American Mathematical Society r, --1 Providence, Rhode Island Translated by J. R. SCHULENBERGER Translation edited by LEV J. LEIFMAN 1980 Mathematics Subject Classification (1985 Revision): Primary 35K05, 35L05, 35R30. Abstract. Problems of determining a variable coefficient and right side for hyperbolic and parabolic equations on the basis of known solutions at fixed points of space for all times are considered in this monograph. Here the desired coefficient of the equation is a function of only one coordinate, while the desired right side is a function only of time. On the basis of solution of direct problems the inverse problems are reduced to nonlinear operator equations for which uniqueness and in some cases also existence questions are investigated. The problems studied have applied importance, since they are models for interpreting data of geophysical prospecting by seismic and electric means. The monograph is of interest to mathematicians concerned with mathematical physics. Bibliography: 75 titles. Library of Congress Cataloging-in-Publication Data Lavrent'ev, M. M. (Mlkhail Milchanovich) One-dimensional inverse problems of mathematical physics. (American Mathematical Society translations; ser. 2, v. 130) Translation of: Odnomernye obratnye zadachi matematicheskoi Bibliography: p. 67. 1. Inverse problems (Differential equations) 2. Mathematical physics. -
On the Origin and Early History of Functional Analysis
U.U.D.M. Project Report 2008:1 On the origin and early history of functional analysis Jens Lindström Examensarbete i matematik, 30 hp Handledare och examinator: Sten Kaijser Januari 2008 Department of Mathematics Uppsala University Abstract In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18th and 19th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of infinite systems of equations and determinants by Helge von Koch were used in Ivar Fredholm’s 1900 paper on the integral equation b Z ϕ(s) = f(s) + λ K(s, t)f(t)dt (1) a which resulted in a vast study of integral equations. One of the most enthusiastic followers of Fredholm and integral equation theory was David Hilbert, and we will see how he further developed the theory of integral equations and spectral theory. The concept introduced by Fredholm to study sets of transformations, or operators, made Maurice Fr´echet realize that the focus should be shifted from particular objects to sets of objects and the algebraic properties of these sets. This led him to introduce abstract spaces and we will see how he introduced the axioms that defines them. Finally, we will investigate how the Lebesgue theory of integration were used by Frigyes Riesz who was able to connect all theory of Fredholm, Fr´echet and Lebesgue to form a general theory, and a new discipline of mathematics, now known as functional analysis. -
Floer Homology on Symplectic Manifolds
Floer Homology on Symplectic Manifolds KWONG, Kwok Kun A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Philosophy in Mathematics c The Chinese University of Hong Kong August 2008 The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate School. Thesis/Assessment Committee Professor Wan Yau Heng Tom (Chair) Professor Au Kwok Keung Thomas (Thesis Supervisor) Professor Tam Luen Fai (Committee Member) Professor Dusa McDuff (External Examiner) Floer Homology on Symplectic Manifolds i Abstract The Floer homology was invented by A. Floer to solve the famous Arnold conjecture, which gives the lower bound of the fixed points of a Hamiltonian symplectomorphism. Floer’s theory can be regarded as an infinite dimensional version of Morse theory. The aim of this dissertation is to give an exposition on Floer homology on symplectic manifolds. We will investigate the similarities and differences between the classical Morse theory and Floer’s theory. We will also explain the relation between the Floer homology and the topology of the underlying manifold. Floer Homology on Symplectic Manifolds ii ``` ½(ÝArnold øî×bÛñøÝFÝóê ×Íì§A. FloerxñÝFloer!§¡XÝ9 Floer!§¡Ú Morse§¡Ý×ÍP§îÌÍÍ¡Zº EøîÝFloer!®×Í+&ƺD¡BÎMorse§¡ Floer§¡Ý8«õ! ¬ÙÕFloer!ÍXòøc RÝn; Floer Homology on Symplectic Manifolds iii Acknowledgements I would like to thank my advisor Prof. Thomas Au Kwok Keung for his encouragement in writing this thesis. I am grateful to all my teachers. -
Basic Theory of Fredholm Operators Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze 3E Série, Tome 21, No 2 (1967), P
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze MARTIN SCHECHTER Basic theory of Fredholm operators Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 21, no 2 (1967), p. 261-280 <http://www.numdam.org/item?id=ASNSP_1967_3_21_2_261_0> © Scuola Normale Superiore, Pisa, 1967, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ BASIC THEORY OF FREDHOLM OPERATORS (*) MARTIN SOHECHTER 1. Introduction. " A linear operator A from a Banach space X to a Banach space Y is called a Fredholm operator if 1. A is closed 2. the domain D (A) of A is dense in X 3. a (A), the dimension of the null space N (A) of A, is finite 4. .R (A), the range of A, is closed in Y 5. ~ (A), the codimension of R (A) in Y, is finite. The terminology stems from the classical Fredholm theory of integral equations. Special types of Fredholm operators were considered by many authors since that time, but systematic treatments were not given until the work of Atkinson [1]~ Gohberg [2, 3, 4] and Yood [5]. These papers conside- red bounded operators. -
Computing the Singular Value Decomposition to High Relative Accuracy
Computing the Singular Value Decomposition to High Relative Accuracy James Demmel Department of Mathematics Department of Electrical Engineering and Computer Science University of California - Berkeley [email protected] Plamen Koev Department of Mathematics University of California - Berkeley [email protected] Structured Matrices In Operator Theory, Numerical Analysis, Control, Signal and Image Processing Boulder, Colorado June 27-July 1, 1999 Supported by NSF and DOE INTRODUCTION • High Relative Accuracy means computing the correct SIGN and LEADING DIGITS • Singular Value Decomposition (SVD): A = UΣV T where U, V are orthogonal, σ1 σ2 Σ = and σ1 ≥ σ2 ≥ . σn ≥ 0 .. . σn • GOAL: Compute all σi with high relative accuracy, even when σi σ1 • It all comes down to being able to compute determi- nants to high relative accuracy. Example: 100 by 100 Hilbert Matrix H(i, j) = 1/(i + j − 1) • Singular values range from 1 down to 10−150 • Old algorithm, New Algorithm, both in 16 digits Singular values of Hilb(100), blue = accurate, red = usual 0 10 −20 10 −40 10 −60 10 −80 10 −100 10 −120 10 −140 10 0 10 20 30 40 50 60 70 80 90 100 • D = log(cond(A)) = log(σ1/σn) (here D = 150) • Cost of Old algorithm = O(n3D2) • Cost of New algorithm = O(n3), independent of D – Run in double, not bignums as in Mathematica – New hundreds of times faster than Old • When does it work? Not for all matrices ... • Why bother? Why do we want tiny singular values accurately? 1. When they are determined accurately by the data • Hilbert: H(i, j) = 1/(i + j − 1) • Cauchy: C(i, j) = 1/(xi + yj) 2. -
PRINCIPLES of APPLIED MATHEMATICS Transformation and Approximation Revised Edition
PRINCIPLES OF To Kristine, Sammy, and Justin, APPLIED Still my best friends after all these years. MATHEMATICS Transformation and Approximation Revised Edition JAMES P. KEENER University of Utah Salt Lake City, Utah ADVANCED BOOK PROGRAM I ewI I I ~ Member of the Perseus Books Group Contents Preface to First Edition xi Preface to Second Edition xvii 1 Finite Dimensional Vector Spaces 1 1.1 Linear Vector Spaces ....... 1 1.2 Spectral Theory for Matrices . 9 1.3 Geometrical Significance of Eigenvalues 17 1.4 Fredholm Alternative Theorem . 24 1.5 Least Squares Solutions-Pseudo Inverses . 25 1.5.1 The Problem of Procrustes .... 41 Many of the designations used by manufacturers and sellers to distinguish their 42 products are claimed as trademarks. Where those designations appear in this 1.6 Applications of Eigenvalues and Eigenfunctions book and Westview Press was aware of a trademark claim, the designations have 1.6.1 Exponentiation of Matrices . 42 been printed in initial capital letters. 1.6.2 The Power Method and Positive Matrices 43 45 Library of Congress Catalog Card Number: 99-067545 1.6.3 Iteration Methods Further Reading . 47 ISBN: 0-7382-0129-4 Problems for Chapter 1 49 Copyright© 2000 by James P. Keener 2 Function Spaces 59 Westview Press books are available at special discounts for bulk purchases in the 2.1 Complete Vector Spaces .... 59 U.S. by corporations, institutions, and other organizations. For more informa 2.1.1 Sobolev Spaces ..... 65 tion, please contact the Special Markets Department at the Perseus Books Group, 2.2 Approximation in Hilbert Spaces 67 11 Cambridge Center, Cambridge, MA 02142, or call (617) 252-5298. -
An Analytic Exact Form of the Unit Step Function
Mathematics and Statistics 2(7): 235-237, 2014 http://www.hrpub.org DOI: 10.13189/ms.2014.020702 An Analytic Exact Form of the Unit Step Function J. Venetis Section of Mechanics, Faculty of Applied Mathematics and Physical Sciences, National Technical University of Athens *Corresponding Author: [email protected] Copyright © 2014 Horizon Research Publishing All rights reserved. Abstract In this paper, the author obtains an analytic Meanwhile, there are many smooth analytic exact form of the unit step function, which is also known as approximations to the unit step function as it can be seen in Heaviside function and constitutes a fundamental concept of the literature [4,5,6]. Besides, Sullivan et al [7] obtained a the Operational Calculus. Particularly, this function is linear algebraic approximation to this function by means of a equivalently expressed in a closed form as the summation of linear combination of exponential functions. two inverse trigonometric functions. The novelty of this However, the majority of all these approaches lead to work is that the exact representation which is proposed here closed – form representations consisting of non - elementary is not performed in terms of non – elementary special special functions, e.g. Logistic function, Hyperfunction, or functions, e.g. Dirac delta function or Error function and Error function and also most of its algebraic exact forms are also is neither the limit of a function, nor the limit of a expressed in terms generalized integrals or infinitesimal sequence of functions with point wise or uniform terms, something that complicates the related computational convergence. Therefore it may be much more appropriate in procedures. -
Variational Calculus of Supervariables and Related Algebraic Structures1
Variational Calculus of Supervariables and Related Algebraic Structures1 Xiaoping Xu Department of Mathematics, The Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong2 Abstract We establish a formal variational calculus of supervariables, which is a combination of the bosonic theory of Gel’fand-Dikii and the fermionic theory in our earlier work. Certain interesting new algebraic structures are found in connection with Hamiltonian superoperators in terms of our theory. In particular, we find connections between Hamiltonian superoperators and Novikov- Poisson algebras that we introduced in our earlier work in order to establish a tensor theory of Novikov algebras. Furthermore, we prove that an odd linear Hamiltonian superoperator in our variational calculus induces a Lie superalgebra, which is a natural generalization of the Super-Virasoro algebra under certain conditions. 1 Introduction Formal variational calculus was introduced by Gel’fand and Dikii [GDi1-2] in studying Hamiltonian systems related to certain nonlinear partial differential equation, such as the arXiv:math/9911191v1 [math.QA] 24 Nov 1999 KdV equations. Invoking the variational derivatives, they found certain interesting Pois- son structures. Moreover, Gel’fand and Dorfman [GDo] found more connections between Hamiltonian operators and algebraic structures. Balinskii and Novikov [BN] studied sim- ilar Poisson structures from another point of view. The nature of Gel’fand and Dikii’s formal variational calculus is bosonic. In [X3], we presented a general frame of Hamiltonian superoperators and a purely fermionic formal variational calculus. Our work [X3] was based on pure algebraic analogy. In this paper, we shall present a formal variational calculus of supervariables, which is a combination 11991 Mathematical Subject Classification. -
Vector-Valued Holomorphic Functions
Appendix A Vector-valued Holomorphic Functions Let X be a Banach space and let Ω ⊂ C be an open set. A function f :Ω→ X is holomorphic if f(z0 + h) − f(z0) f (z0) := lim (A.1) h→0 h h∈C\{0} exists for all z0 ∈ Ω. If f is holomorphic, then f is continuous and weakly holomorphic (i.e. x∗ ◦ f ∗ ∗ is holomorphic for all x ∈ X ). If Γ := {γ(t):t ∈ [a,b]} is a finite, piecewise smooth contour in Ω, we can form the contour integral f(z) dz. This coincides Γ b with the Bochner integral a f(γ(t))γ (t) dt (see Section 1.1). Similarly we can define integrals over infinite contours when the corresponding Bochner integral is absolutely convergent. Since 1 2 f(z) dz, x∗ = f(z),x∗ dz, Γ Γ many properties of holomorphic functions and contour integrals may be extended from the scalar to the vector-valued case, by applying the Hahn-Banach theorem. For example, Cauchy’s theorem is valid, and also Cauchy’s integral formula: 1 f(z) f(w)= dz (A.2) − 2πi |z−z0|=r z w whenever f is holomorphic in Ω, the closed ball B(z0,r) is contained in Ω and w ∈ B(z0,r). As in the scalar case one deduces Taylor’s theorem from this. Proposition A.1. Let f :Ω→ X be holomorphic, where Ω ⊂ C is open. Let z0 ∈ Ω,r>0 such that B(z0,r) ⊂ Ω.Then W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, 461 Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7, © Springer Basel AG 2011 462 A.