DOCSLIB.ORG

Vector-Valued Holomorphic Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. VECTOR-VALUED HOLOMORPHIC FUNCTIONS ∞ n f(z)= an(z − z0) n=0 converges absolutely for |z − z0| <r,where 1 f(z) an := dz. − n+1 2πi |z−z0|=r (z z0) We also mention a special form of the identity theorem. Proposition A.2 (Identity theorem for holomorphic functions). Let Y be a closed subspace of a Banach space X.LetΩ be a connected open set in C and f :Ω→ X be holomorphic. Assume that there exists a convergent sequence (zn)n∈N ⊂ Ω such that limn→∞ zn ∈ Ω and f(zn) ∈ Y for all n ∈ N.Thenf(z) ∈ Y for all z ∈ Ω. Note that for Y = {0}, we obtain the usual form of the identity theorem. ∗ 0 ∗ ∗ ∗ ∗ Proof. Let x ∈ Y := {y ∈ X : y, y =0(y ∈ Y )}.Thenx ◦ f(zn) = 0 for all n ∈ N. It follows from the scalar identity theorem that x∗ ◦ f(z) = 0 for all z ∈ Ω. Hence, f(z) ∈ Y 00 = Y for all z ∈ Ω. In the following we show that every weakly holomorphic function is holomor- phic. Actually, we will prove a slightly more general assertion which turns out to be useful. A subset N of X∗ is called norming if ∗ x 1 := sup | x, x | x∗∈N defines an equivalent norm on X. A function f :Ω→ X is called locally bounded if supK f(z) < ∞ for all compact subsets K of Ω. Proposition A.3. Let Ω ⊂ C be open and let f :Ω→ X be locally bounded such that x∗ ◦ f is holomorphic for all x∗ ∈ N,whereN is a norming subset of X∗. Then f is holomorphic. In particular, if X = L(Y,Z),whereY,Z are Banach spaces, and if f :Ω→ X is locally bounded, then the following are equivalent: (i) f is holomorphic. (ii) f(·)y is holomorphic for all y ∈ Y . (iii) f(·)y, z∗ is holomorphic for all y ∈ Y, z∗ ∈ Z∗. Proof. We can assume that x 1 = x for all x ∈ X. In order to show holomorphy at z0 ∈ Ω we can assume that z0 = 0, replacing Ω by Ω − z0 otherwise. For small h, k ∈ C\{0},let f(h) − f(0) f(k) − f(0) u(h, k):= − . h k A. VECTOR-VALUED HOLOMORPHIC FUNCTIONS 463 We have to show that for ε>0 there exists δ>0 such that u(h, k) ≤ε whenever |h|≤δ and |k|≤δ.Letr>0 such that B(0, 2r) ⊂ Ωand M := sup f(z) < ∞. z∈B(0,2r) Then by Cauchy’s integral formula, for |z| <r,|h|≤r, |k|≤r, h, k =0 ,x∗ ∈ N, 1 1 1 1 1 1 1 u(h, k),x∗ = f(z),x∗ − − − dz 2πi h z − h z k z − k z |z|=2r h − k f(z),x∗ = dz. 2πi |z|=2r z(z − h)(z − k) Hence, | u(h, k),x∗ |≤|h − k|M/r2.SinceN is norming, we deduce that M u(h, k) ≤|h − k| . r2 This proves the claim. Corollary A.4. Let Ω ⊂ C be a connected open set and Ω0 ⊂ Ω be open. Let h :Ω0 → X be holomorphic. Assume that there exists a norming subset N of ∗ ∗ X such that for all x ∈ N there exists a holomorphic extension Hx∗ :Ω→ C ∗ of x ◦ h.Ifsupx∗∈N |Hx∗ (z)| < ∞,thenh has a unique holomorphic extension z∈Ω H :Ω→ X. Proof. Again we assume that · 1 = · .Let Y := y =(yx∗ )x∗∈N ⊂ C : y ∞ := sup |yx∗ | < ∞ , x∗∈N and let H :Ω→ Y be given by H(z):=(Hx∗ (z))x∗∈N . It follows from Proposition ∗ A.3 that H is holomorphic. By x ∈ X → ( x, x )x∗∈N , one defines an isometric injection from X into Y .SinceH(z) ∈ X for z ∈ Ω0, it follows from the identity theorem (Proposition A.2) that H(z) ∈ X for all z ∈ Ω. We will extend Proposition A.3 considerably in Theorem A.7. Before that we prove Vitali’s theorem. Theorem A.5 (Vitali). Let Ω ⊂ C be open and connected. Let fn :Ω→ X be holomorphic (n ∈ N) such that sup fn(z) < ∞ n∈N z∈B(z0,r) whenever B(z0,r) ⊂ Ω. Assume that the set ' ( Ω := z ∈ Ω: lim fn(z) exists 0 n→∞ 464 A. VECTOR-VALUED HOLOMORPHIC FUNCTIONS has a limit point in Ω. Then there exists a holomorphic function f :Ω→ X such that (k) (k) f (z) = lim fn (z) n→∞ uniformly on all compact subsets of Ω for all k ∈ N0. ∞ ∞ Proof. Let l (X):={x =(xn)n∈N ⊂ X : x ∞ := sup xn < ∞}.Thenl (X) is a Banach space for the norm · ∞ and the space c(X) of all convergent sequences is a closed subspace of l∞(X). Consider the function F :Ω→ l∞(X)givenby F (z)=(fn(z))n∈N. It follows from Proposition A.3 that F is holomorphic. (One ∞ may take N to be the space of all functionals on l (X) of the form (xn)n∈N → ∗ ∗ ∗ ∗ xk,x where k ∈ N,x ∈ X , x ≤1). Since F (z) ∈ c(X) for all z ∈ Ω0, it follows from the identity theorem (Proposition A.2) that F (z) ∈ c(X) for all z ∈ Ω. Consider the mapping φ ∈L(c(X),X)givenbyφ((xn)n∈N) = limn→∞ xn. Then f = limn→∞ fn = φ ◦ F :Ω→ X is holomorphic. Finally, we prove uniform convergence on compact sets. Let B(z0,r) ⊂ Ωand k ∈ N0. It follows from (A.2) that 1 1 fn(w) f (k)(z)= dw. n − k+1 k! 2πi |w−z0|=r (w z) (k) Now the dominated convergence theorem implies that fn (z) converges uniformly (k) on B(z0,r/2) to f (z). Since every compact subset of Ω can be covered by a finite number of discs, the claim follows. If in Vitali’s theorem (fn) is a net instead of a sequence, the proof shows that f(z) = lim fn(z) exists for all z ∈ Ω and defines a holomorphic function f :Ω→ X. Next we recall a well known theorem from functional analysis. Theorem A.6 (Krein-Smulyan). Let X be a Banach space and W be a subspace of the dual space X∗. Denote by B∗ the closed unit ball of X∗.ThenW is weak* closed if and only if W ∩ B∗ is weak* closed. For a proof, see [Meg98, Theorem 2.7.11]. Now we obtain the following convenient criterion for holomorphy. Theorem A.7. Let Ω ⊂ C be open and connected, and let f :Ω→ X be a locally bounded function. Assume that W ⊂ X∗ is a separating subspace such that x∗ ◦ f is holomorphic for all x∗ ∈ W .Thenf is holomorphic. Here, W is called separating if x, x∗ =0forallx∗ ∈ W implies x =0 (x ∈ X). Proof. Let Y := {x∗ ∈ X∗ : x∗ ◦ f is holomorphic}.SinceW ⊂ Y , the subspace Y is weak* dense. It follows from Vitali’s theorem (applied to nets if X is not A. VECTOR-VALUED HOLOMORPHIC FUNCTIONS 465 separable) that Y ∩ B∗ is weak* closed. Now it follows from the Krein-Smulyan theorem that Y = X∗. Hence, f is holomorphic by Proposition A.3. Notes: Usually, Vitali’s theorem is proved with the help of Montel’s theorem which is only valid in finite dimensions. A vector-valued version is proved in the book of Hille and Phillips [HP57] by a quite complicated power-series argument going back to Liouville. The very simple proof given here is due to Arendt and Nikolski [AN00] who also proved Theorem A.7 (see also [AN06]). Appendix B Closed Operators Let X be a complex Banach space. An operator on X is a linear map A : D(A) → X,whereD(A) is a linear subspace of X, known as the domain of A.Therange Ran A,andthekernel Ker A,ofA are defined by: Ran A := {Ax : x ∈ D(A)}, Ker A := {x ∈ D(A):Ax =0}.
Load more

Recommended publications
  • Lecture Notes in Mathematics
    Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Mathematisches Institut der Universit~it und Max-Planck-lnstitut for Mathematik, Bonn - voL 5 Adviser: E Hirzebruch 1111 Arbeitstagung Bonn 1984 Proceedings of the meeting held by the Max-Planck-lnstitut fur Mathematik, Bonn June 15-22, 1984 Edited by E Hirzebruch, J. Schwermer and S. Suter I IIII Springer-Verlag Berlin Heidelberg New York Tokyo Herausgeber Friedrich Hirzebruch Joachim Schwermer Silke Suter Max-Planck-lnstitut fLir Mathematik Gottfried-Claren-Str. 26 5300 Bonn 3, Federal Republic of Germany AMS-Subject Classification (1980): 10D15, 10D21, 10F99, 12D30, 14H10, 14H40, 14K22, 17B65, 20G35, 22E47, 22E65, 32G15, 53C20, 57 N13, 58F19 ISBN 3-54045195-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15195-8 Springer-Verlag New York Heidelberg Berlin Tokyo CIP-Kurztitelaufnahme der Deutschen Bibliothek. Mathematische Arbeitstagung <25. 1984. Bonn>: Arbeitstagung Bonn: 1984; proceedings of the meeting, held in Bonn, June 15-22, 1984 / [25. Math. Arbeitstagung]. Ed. by E Hirzebruch ... - Berlin; Heidelberg; NewYork; Tokyo: Springer, 1985. (Lecture notes in mathematics; Vol. 1t11: Subseries: Mathematisches I nstitut der U niversit~it und Max-Planck-lnstitut for Mathematik Bonn; VoL 5) ISBN 3-540-t5195-8 (Berlin...) ISBN 0-387q5195-8 (NewYork ...) NE: Hirzebruch, Friedrich [Hrsg.]; Lecture notes in mathematics / Subseries: Mathematischee Institut der UniversitAt und Max-Planck-lnstitut fur Mathematik Bonn; HST This work ts subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re~use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.
    [Show full text]
  • 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.
    [Show full text]
  • Sample Questions for Preliminary Complex Analysis Exam
    SAMPLE QUESTIONS FOR PRELIMINARY COMPLEX ANALYSIS EXAM VERSION 2.0 Contents 1. Complex numbers and functions 1 2. Definition of holomorphic function 1 3. Complex Integrals and the Cauchy Integral Formula 2 4. Sequences and series, Taylor series, and series of analytic functions 2 5. Identity Theorem 3 6. Schwarz Lemma and Cauchy Inequalities 4 7. Liouville's Theorem 4 8. Laurent series and singularities 4 9. Residue Theorem 5 10. Contour Integrals 5 11. Argument Principle 6 12. Rouch´e'sTheorem 6 13. Conformal maps 7 14. Analytic Continuation 7 15. Suggested Practice Exams 8 1. Complex numbers and functions (1.1) Write all values of ii in the form a + bi. 2 (1.2) Prove thatp sin z = z has infinitely many complex solutions. (1.3) Find log 3 + i, using the principal branch. 2. Definition of holomorphic function (2.1) Find all v : R2 ! R2 such that for z = x + iy, f(z) = (x3 − 3xy2) + iv(x; y) is analytic. (2.2) Prove that if g : C ! C is a C1 function, the following two definitions of \holomor- phic" are the same: @g (a) @z = 0 0 2 2 (b) the derivative transformation g (z0): R ! R is C-linear, for all z0 2 C. That 0 0 is, g (z0)mw = mwg (z0) for all w 2 C, where mw is the linear transformation given by complex multiplication by w. @h (2.3) True or false: If h is an entire function such that @z 6= 0 everywhere, then h is injective. 1 2 VERSION 2.0 (2.4) Find all possible a; b 2 R such that f (x; y) = x2 +iaxy +by2, x; y 2 R is holomorphic as a function of z = x + iy.
    [Show full text]
  • Chapter 1 Holomorphic and Real Analytic Calculus
    This version: 28/02/2014 Chapter 1 Holomorphic and real analytic calculus In this chapter we develop basic analysis in the holomorphic and real analytic settings. We do this, for the most part, simultaneously, and as a result some of the ways we do things are a little unconventional, especially when compared to the standard holomorphic treatments in, for example, texts like [Fritzsche and Grauert 2002, Gunning and Rossi 1965,H ormander¨ 1973, Krantz 1992, Laurent-Thiebaut´ 2011, Range 1986, Taylor 2002]. We assume, of course, a thorough acquaintance with real analysis such as one might find in [Abraham, Marsden, and Ratiu 1988, Chapter 2] and single variable complex analysis such as one might find in [Conway 1978]. 1.1 Holomorphic and real analytic functions Holomorphic and real analytic functions are defined as being locally prescribed by a convergent power series. We, therefore, begin by describing formal (i.e., not depending on any sort of convergence) power series. We then indicate how the usual notion of a Taylor series gives rise to a formal power series, and we prove Borel’s Theorem which says that, in the real case, all formal power series arise as Taylor series. This leads us to consider convergence of power series, and then finally to consider holomorphic and real analytic functions. Much of what we say here is a fleshing out of some material from Chapter 2 of [Krantz and Parks 2002], adapted to cover both the holomorphic and real analytic cases simultaneously. 1.1.1 Fn Because much of what we say in this chapter applies simultaneously to the real and complex case, we shall adopt the convention of using the symbol F when we wish to refer to one of R or C.
    [Show full text]
  • Polynomial Stability of Operator Semigroups
    POLYNOMIAL STABILITY OF OPERATOR SEMIGROUPS ANDRAS´ BATKAI,´ KLAUS-JOCHEN ENGEL, JAN PRUSS,¨ AND ROLAND SCHNAUBELT Abstract. We investigate polynomial decay of classical solutions of linear evolution equations. For bounded C0{semigroups on a Banach space this property is closely re- lated to polynomial growth estimates of the resolvent of the generator. For systems of commuting normal operators polynomial decay is characterized in terms of the location of the generator spectrum. The results are applied to systems of coupled wave-type equations. 1. Introduction The asymptotic theory of operator semigroups provides powerful tools for the investi- gation of the (exponential) convergence to 0 of mild and classical solutions of the linear Cauchy problem (1.1) u0(t) + Au(t) = 0; t ≥ 0; u(0) = x; where −A generates the strongly continuous operator semigroup (T (t))t≥0 on a Banach space X. In Section 2 we briefly review these results in order to provide the background for our paper. However, weakly damped systems of linear wave equations can exhibit a type of be- haviour not satisfactorily covered by semigroup theory so far: Classical solutions of (1.1) may converge to 0 polynomially, but not exponentially. Formulated in the framework of the evolution equation (1.1), certain systems lead to decay estimates of the form (1.2) kT (t)xk ≤ Ct−βkAαxk; x 2 D(Aα); t > 0; for some constants α; β > 0. Such results were obtained in the recent papers [1], [2], [10], [15]; see also the references therein. These authors used energy type estimates which are more or less closely related to the specific problem posed on a Hilbert space.
    [Show full text]
  • Math 656 • Main Theorems in Complex Analysis • Victor Matveev
    Math 656 • Main Theorems in Complex Analysis • Victor Matveev ANALYTICITY: CAUCHY‐RIEMANN EQUATIONS (Theorem 2.1.1); review CRE in polar coordinates. Proof: you should know at least how to prove that CRE are necessary for complex differentiability (very easy to show). The proof that they are sufficient is also easy if you recall the concept of differentiability of real functions in Rn . INTEGRAL THEOREMS: There are 6 theorems corresponding to 6 arrows in my hand‐out. Only 4 of them are independent theorems, while the other two are redundant corollaries, including the important (yet redundant) Morera's Theorem (2.6.5). Cauchy‐Goursat Theorem is the main integral theorem, and can be formulated in several completely equivalent ways: 1. Integral of a function analytic in a simply‐connected domain D is zero for any Jordan contour in D 2. If a function is analytic inside and on a Jordan contour C, its integral over C is zero. 3. Integral over a Jordan contour C is invariant with respect to smooth deformation of C that does not cross singularities of the integrand. 4. Integrals over open contours CAB and C’AB connecting A to B are equal if CAB can be smoothly deformed into C’AB without crossing singularities of the integrand. 5. Integral over a disjoint but piece‐wise smooth contour (consisting of properly oriented external part plus internal pieces) enclosing no singularities of the integrand, is zero (proved by cross‐cutting or the deformation theorem above). Cauchy's Integral Formula (2.6.1) is the next important theorem; you should know its proof, its extended version and corollaries: Liouville Theorem 2.6.4: to prove it, apply the extended version of C.I.F.
    [Show full text]
  • From Generalized Fourier Transforms to Coupled Supersymmetry
    FROM GENERALIZED FOURIER TRANSFORMS TO COUPLED SUPERSYMMETRY A Dissertation Presented to the Faculty of the Department of Mathematics University of Houston In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy By Cameron Louis Williams May 2017 FROM GENERALIZED FOURIER TRANSFORMS TO COUPLED SUPERSYMMETRY Cameron Louis Williams Approved: Dr. Bernhard G. Bodmann, Chairman Committee Members: Dr. Donald J. Kouri, Co-Chairman Dr. Mehrdad Kalantar Dr. Demetrio Labate Dr. John R. Klauder Department of Physics, University of Florida Dean, College of Natural Sciences and Mathematics ii Acknowledgements I would like to first thank my doctoral advisors Professors Donald Kouri and Bernhard Bodmann. I have had the opportunity to work with them for eight years. I started working under their tutelage as an undergraduate in mathematics at the University of Houston. During my undergraduate career, we pursued many different topics, each of which has been very rewarding and heavily impacted this thesis. The work in this thesis began the summer before my senior year and continued while I was pursuing my master's at the University of Waterloo. I returned to the University of Houston for my doctorate to finish the work we began during my undergraduate studies. It has since flourished in ways none of us had ever expected or even dreamed, due in no small part to Professors Kouri and Bodmann. I would not have been able to achieve this doctorate without their mentoring. I would like to thank my committee members Professors Mehrdad Kalantar and Demetrio Labate from the University of Houston and Professor John Klauder from the University of Florida for their helpful comments and corrections to this thesis.
    [Show full text]
  • Recent Developments in Some Non-Self-Adjoint Problems of Mathematical Physics
    RECENT DEVELOPMENTS IN SOME NON-SELF-ADJOINT PROBLEMS OF MATHEMATICAL PHYSICS C. L. DOLPH 1. Introduction. In view of the fact that a relatively small per­ centage of the membership of the American Mathematical Society have an opportunity to become acquainted with or follow the ever increasing activity in the fields of applied and immediately applicable mathematics, it is perhaps useful to attempt an assessment of recent trends in the areas with which I have some familiarity, namely in the field of diffraction and scattering theory and that part of transport theory where similar techniques are employed. While one would need the prescience of a Hubert to adequately carry out even this limited program, it is possible to pick out a few concepts and techniques which underlie many of the recent developments. The areas under consideration received a tremendous impetus as a result of the radar development of the last war and more recently as a result of reactor theory and the attempts to achieve controlled fusion. In many instances, it was the physicists and not the mathematicians who led the way to new methods which in some cases are only for­ mally understood mathematically to this day even though they may have been widely applied with frequent success. This area is also char­ acterized by the fact that the tools necessary to establish recent re­ sults have frequently been available for many years. This does not detract from the achievement of the authors who found this kind of result but once again illustrates the healthy influence physical prob­ lems can have on the development of mathematics and the dangers inherent when the two get too widely divorced.
    [Show full text]
  • D. Van Nostrand Company, Inc. Toronto New York London New York
    Introduction to Abstract Harmonic Analysis by LYNN H. LOOMIS Associate Professor of Mathematics Harvard University 1953 D. VAN NOSTRAND COMPANY, INC. TORONTO NEW YORK LONDON NEW YORK D. Van Nostrand Company, Inc., 250 Fourth Avenue, New York 3 TORONTO D. Van Nostrand Company (Canada), Ltd., 25 Hollinger Rd., Toronto LONDON Macmillan & Company, Ltd., St. Martin's Street, London, W.C. 2 COPYRIGHT, 1953, BV D. VAN NOSTRAND COMPANY, INC. Published simultaneously in Canada by D. VAN NOSTRAND COMPANY (Canada), LTD. All Rights Reserved This book, or any parts thereof, may not be reproduced in any form without written per- mission from the author and the publisher. Library of Congress Catalogue Card No. 53-5459 PRINTED IN THE UNITED STATES OF AMERICA PREFACE This book is an outcome of a course given at Harvard first by G. W. Mackey and later by the author. The original course was modeled on Weil's book [48] and covered essentially the material of that book with modifications. As Gelfand's theory of Banach algebras and its applicability to harmonic analysis on groups became better k lown, the methods and content of the course in- evitably shifted in this direction, and the present volume concerns itself almost exclusively with this point of view. Thus our devel- opment of the subject centers around Chapters IV and V, in which the elementary theory of Banach algebras is worked out, and groups are relegated to the supporting role of being the prin- cipal application. A typical result of this shift in emphasis and method is our treatment of the Plancherel theorem.
    [Show full text]
  • Complex Analysis
    Complex Analysis Joachim Wehler Ludwig-Maximilians-Universitat¨ Munchen¨ Department of Mathematics Summer Term 2019 - Lecture Notes DRAFT, Release 2.6 July 24, 2019 2 c Joachim Wehler, 2019 I prepared these notes for the students of my lecture. The present lecture took place during the summer term 2019 at the mathematical department of LMU (Ludwig-Maximilians-Universitat)¨ at Munich. The first input are notes taken from former lectures by Otto Forster. Compared to the oral lecture in class these written notes contain some additional material. In particular, at the end of several chapters I added an outlook to some more advanced topics from complex analysis of several variables and from the theory of complex spaces. Please report any errors or typos to [email protected] Release notes: • Release 2.6: Chapter 8: Minor revisions. • Release 2.5: Minor revisions. • Release 2.4: Chapter 8 completed. List of results added. • Release 2.3: Chapter 8, Section 8.2 added. • Release 2.2: Definition 7.6 corrected. • Release 2.1: Chapter 8, Section 8.1 added. References to problem sheet 12 removed • Release 2.0: Chapter 7 completed. Chapter 5, references to problem sheet 11 removed. • Release 1.9: Chapter 6 completed. References to problem sheet 12 to be removed later. • Release 1.8: Minor revisions. • Release 1.7: Chapter 6, Section 6.1 added. • Release 1.6: Chapter 5 completed. References to problem sheet 11 to be removed later. • Release 1.5: Chapter 5, Section 5.2 added. • Release 1.4: Proposition 5.7: typo corrected. • Release 1.3: Chapter 5, Section 5.1 added.
    [Show full text]
  • A Short History of Operator Theory
    7/17/12 History of Operator Theory A Short History of Operator Theory by Evans M. Harrell II © 2004. Unrestricted use is permitted, with proper attribution, for noncommercial purposes. In the first textbook on operator theory, Théorie des Opérations Linéaires, published in Warsaw 1932, Stefan Banach states that the subject of the book is the study of functions on spaces of infinite dimension, especially those he coyly refers to as spaces of type B, otherwise Banach spaces (definition). This was a good description for Banach, but tastes vary. I propose rather the "operational" definition that operators act like matrices. And what that means depends on who you are. If you are an engineering student, matrices are particular symbols you manipulate to solve linear systems. As a working engineer you may instead use Heaviside's operational calculus, in which you are permitted to do all sorts of dangerous manipulations of symbols for derivatives and what not, exactly as if they were matrices, in order to solve linear problems of applied analysis. About 90% of the time you will get the right answer, just like the student; somewhat more with experience. And that is good enough, if the bridges you build aren't where I drive. In mathematics the student of elementary analysis learns that matrices are linear functions relating finite- dimensional vector spaces, and conversely. As a working mathematician the analyst has lost all fear of minor matters like infinity, and will happy agree with Banach's definition. For the students of algebra, matrices are fun objects that can be added and multiplied, usually in flagrant disregard for the law (of commutativity).
    [Show full text]
  • Extensions of Positive Definite Functions
    Palle Jorgensen, Steen Pedersen, Feng Tian Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis July 10, 2015 arXiv:1507.02547v1 [math.FA] 9 Jul 2015 Springer 2 Palle E.T. Jorgensen Department of Mathematics The University of Iowa Iowa City, IA 52242-1419, U.S.A. Email address: [email protected] URL: http://www.math.uiowa.edu/~jorgen/ Steen Pedersen Department of Mathematics Wright State University Dayton, OH 45435, U.S.A. E-mail address: [email protected], [email protected] URL: http://www.wright.edu/~steen.pedersen/ Feng Tian Department of Mathematics Trine University One University Avenue, Angola, IN 46703, U.S.A. E-mail address: [email protected], [email protected] 3 Dedicated to the memory of William B. Arveson1 (22 November 1934 – 15 November 2011) Edward Nelson2 (May 4, 1932 – September 10, 2014) 1 William Arveson (1934 – 2011) worked in operator algebras and harmonic analysis, and his re- sults have been influential in our thinking, and in our approach to the particular extension questions we consider here. In fact, Arveson’s deep and pioneering work on completely positive maps may be thought of as a non-commutative variant of our present extension questions. We have chosen to give our results in the commutative setting, but readers with interests in non-commutative analysis will be able to make the connections. While the non-commutative theory was initially motivated by the more classical commutative theory, the tools involved are different, and there are not always direct links between theorems in one area and the other.
    [Show full text]