Multivariate Operator Theory
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Corona Problem
数理解析研究所講究録 1487 巻 2006 年 13-26 13 Geometry of Banach Spaces Predual to $H^{\infty}$ and Corona Problem Alexander Brudnyi* Department of Mathematics and Statistics University of Calgary, Calgary Canada Abstract In this paper we present a new general approach to the multi-dimensional corona problem for the ball and polydisk. We also describe some properties of the Banach spaces predual to the spaces $H^{\infty}$ defined on these domains. 1. Corona Problem 1.1. In the present paper we discuss one of the fundamental problems in the theory of uniform algebras known as the corona problem. Such a problem was originally raised by S. Kakutani in 1941 and asked whether the open unit disk $\mathrm{D}\subset \mathbb{C}$ is dense in the maximal ideal space of the algebra $H^{\infty}(\mathrm{D})$ of bounded holomorphic functions on $\mathrm{D}$ with the supremum norm. This problem was answered affirmatively by L. Carleson in 1962 [C1]. Let us recall that for a uniform algebra $A$ of continuous functions defined on a Hausdorff topological space $X$ , the maximal ideal space $\mathcal{M}(A)$ is the set of all nonzero homomorphisms $Aarrow \mathbb{C}$ equipped with the weak*-topology of the dual space $A^{*}$ , known as the Gelfand topology. Then $\mathcal{M}(A)$ is a compact Hausdorff space. Any function $f\in A$ can be considered as a function on $\mathcal{M}(A)$ by means of the Celfand transform: $\hat{f}(m):=m(f)$ , $m\in \mathcal{M}(A)$ . If $A$ separates points on $X$ , then $X$ is naturally embedded into $\mathcal{M}(A),$ $xrightarrow\delta_{x}$ , $\delta_{x}$ $X$ $x\in X$ , where is the evaluation functional at $x$ . -
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]. -
RAPORT DE AUTOEVALUARE Perioada: 1.01.2003 – 30.11.2007
RAPORT DE AUTOEVALUARE perioada: 1.01.2003 – 30.11.2007 1. Datele de autentificare al unitatii de cercetare – dezvoltare: 1.1 Institutul de Matematica “Simion Stoilow” al Academiei Romane 1.2 Institutie de stat cu personalitate juridica in subordinea Academiei Romane. 1.3 Infiintata prin HG. 255 / 8.03.1990 (publicata in MO p.I nr. 220/1992) 1.4 Numarul de inregistrare in Registrul Potentialilor Contractori: 2685 1.5 Director: CSI. Dr. Vasile Brinzanescu 1.6 Adresa: Calea Grivitei, nr. 21, Sector 1, Bucuresti 1.7 Tel. 319.65.06 Fax. 319.65.05 Web. http:www.imar.ro e-mail [email protected] 2. Domeniul de specialitate: 2.1 Conform clasificarii UNESCO: 1100 2.2 Conform clasificarii CAEN: 7310 3. Starea Unitatii de cercetare – dezvoltare: ● Misiunea unitatii de cercetare – dezvoltare: cercetare stiintifica pe probleme de matematica pura si aplicata, precum si pe probleme de matematica in cadrul unor cercetari interdisciplinare ● Modul de valorificare al rezultatelor: articole stiintifice in reviste de specialitate sau in volume comunicari in cadrul unor manifestari stiintifice nationale si internationale monografii si tratate stiintifice ● Situatia financiara: nu are datorii la bugetul de stat. 1 4. Indeplinirea criteriilor primare de performanta: (Detalii in Anexa la raport.) Criteriul Factor de Numar TOTAL importanta A1 CRITERII PRIMARE DE PERFORMANTA 27555.55 1 Lucrări ştiinţifice/tehnice in reviste de specialitate cotate ISI . 30 494 14820 2 Factor de impact cumulat al lucrărilor cotate ISI. 5 341.11 1705.55 3 Citări in reviste de specialitate cotate ISI. 5 2200 11000 4 Brevete de invenţie. 30 1 30 5 Citari in sitemul ISI ale cercetarilor brevetate. -
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. -
Banach Algebras
Banach Algebras Yurii Khomskii Bachelor Thesis Department of Mathematics, Leiden University Supervisor: Dr. Marcel de Jeu April 18, 2005 i Contents Foreword iv 1. Algebraic Concepts 1 1.1. Preliminaries . 1 1.2. Regular Ideals . 3 1.3. Adjoining an Identity . 4 1.4. Quasi-inverses . 8 2. Banach Algebras 10 2.1. Preliminaries of Normed and Banach Algebras . 10 2.2. Inversion and Quasi-inversion in Banach Algebras . 14 3. Spectra 18 3.1. Preliminaries . 18 3.2. Polynomial Spectral Mapping Theorem and the Spectral Radius Formula . 22 4. Gelfand Representation Theory 25 4.1. Multiplicative Linear Functionals and the Maximal Ideal Space . 25 4.2. The Gelfand Topology . 30 4.3. The Gelfand Representation . 31 4.4. The Radical and Semi-simplicity . 33 4.5. Generators of Banach algebras . 34 5. Examples of Gelfand Representations 36 5.1. C (X ) for X compact and Hausdorff . 36 5.2. C 0(X ) for X locally compact and Hausdorff. 41 5.3. Stone-Cecˇ h compactification . 42 5.4. A(D) . 44 5.5. AC (Γ) . 46 5.6. H 1 . 47 ii iii Foreword The study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra multiplication operation. A standard exam- ple was the space of bounded linear operators on a Banach space, but another important one was function spaces (of continuous, bounded, vanishing at infin- ity etc. functions as well as functions with absolutely convergent Fourier series). Nowadays Banach algebras is a wide discipline with a variety of specializations and applications. -
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. -
The Corona Problem for Kernel Multiplier Algebras 3
Washington University in St. Louis Washington University Open Scholarship Mathematics Faculty Publications Mathematics and Statistics 12-2016 The orC ona Problem for Kernel Multiplier Algebras Eric T. Sawyer Brett .D Wick Washington University in St. Louis, [email protected] Follow this and additional works at: https://openscholarship.wustl.edu/math_facpubs Part of the Algebra Commons, and the Analysis Commons Recommended Citation Sawyer, Eric T. and Wick, Brett .,D "The orC ona Problem for Kernel Multiplier Algebras" (2016). Mathematics Faculty Publications. 39. https://openscholarship.wustl.edu/math_facpubs/39 This Article is brought to you for free and open access by the Mathematics and Statistics at Washington University Open Scholarship. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of Washington University Open Scholarship. For more information, please contact [email protected]. THE CORONA PROBLEM FOR KERNEL MULTIPLIER ALGEBRAS ERIC T. SAWYER† AND BRETT D. WICK‡ Abstract. We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in Cn, and for the algebra of bounded analytic functions on certain strictly pseudoconvex domains and polydiscs in higher dimensions as well. This alternate Toeplitz corona theorem extends to more general Hilbert function spaces where it does not require the complete Pick property. Instead, the kernel functions kx (y) of certain Hilbert function spaces H are assumed to be invertible multipliers on H, and then we continue a research thread begun by Agler and McCarthy in 1999, and continued by Amar in 2003, and most recently by Trent and Wick in 2009. -
Paschke Dilations
Paschke Dilations Abraham Westerbaan Bas Westerbaan Radboud Universiteit Nijmegen Radboud Universiteit Nijmegen [email protected] [email protected] In 1973 Paschke defined a factorization for completely positive maps between C∗-algebras. In this paper we show that for normal maps between von Neumann algebras, this factorization has a univer- sal property, and coincides with Stinespring’s dilation for normal maps into B(H ). The Stinespring Dilation Theorem[17] entails that every normal completely positive linear map (NCP- map) j : A ! B(H ) is of the form A p / B(K ) V ∗(·)V / B(H ) where V : H ! K is a bounded operator and p a normal unital ∗-homomorphism (NMIU-map). Stinespring’s theorem is fun- damental in the study of quantum information and quantum computing: it is used to prove entropy inequalities (e.g. [10]), bounds on optimal cloners (e.g. [20]), full completeness of quantum program- ming languages (e.g. [16]), security of quantum key distribution (e.g. [8]), analyze quantum alternation (e.g. [1]), to categorify quantum processes (e.g. [14]) and as an axiom to single out quantum theory among information processing theories.[2] A fair overview of all uses of Stinespring’s theorem and its consequences would warrant a separate article of its own. One wonders: is the Stinespring dilation categorical in some way? Can the Stinespring dilation theorem be generalized to arbitrary NCP-maps j : A ! B? In this paper we answer both questions in the affirmative. We use the dilation introduced by Paschke[11] for arbitrary NCP-maps, and we show that it coincides with Stinespring’s dilation (a fact not shown before) by introducing a universal property for Paschke’s dilation, which Stinespring’s dilation also satisfies. -
Operator Theory Operator Algebras and Applications
http://dx.doi.org/10.1090/pspum/051.1 Operato r Theor y Operato r Algebra s an d Application s PROCEEDING S O F SYMPOSI A IN PUR E MATHEMATIC S Volum e 51 , Par t 1 Operato r Theor y Operato r Algebra s an d Application s Willia m B . Arveso n an d Ronal d G . Douglas , Editor s AMERICA N MATHEMATICA L SOCIET Y PROVIDENCE , RHOD E ISLAN D PROCEEDINGS OF THE SUMMER RESEARCH INSTITUTE ON OPERATOR THEORY/OPERATOR ALGEBRAS AND APPLICATIONS HELD AT UNIVERSITY OF NEW HAMPSHIRE DURHAM, NEW HAMPSHIRE JULY 3-23, 1988 with support from the National Science Foundation, Grant DMS-8714162 1980 Mathematics Subject Classification (1985 Revision). Primary 46L, 47A, 47B, 58G. Library of Congress Cataloging-in-Publication Data Operator theory: operator algebras and applications/William B. Arveson and Ronald G. Dou• glas, editors. p. cm.—(Proceedings of symposia in pure mathematics; v. 51) ISBN 0-8218-1486-9 I. Operator algebras — Congresses. 2. Operator theory — Congresses. I. Arveson, William. II. Douglas, Ronald G. III. Series. QA326.067 1990 90-33771 512'.55-dc20 CIP COPYING AND REPRINTING. Individual readers of this publication, and nonprofit li• braries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publi• cation (including abstracts) is permitted only under license from the American Mathematical Society. -
The Quantum Mechanics of Supersymmetry
The Quantum Mechanics of Supersymmetry A Senior Project presented to the Faculty of the Physics Department California Polytechnic State University, San Luis Obispo In Partial Fulfillment of the Requirements for the Degree Bachelor of Science by Joshua Gearhart May, 2012 ©2012 Joshua William Gearhart I. Introduction Most, if not all, physicists are familiar with the idea of a grand unified theory (GUT). The general idea behind a GUT is to have a single framework that succinctly describes all of the fundamental interactions of nature: gravity, electromagnetism (EM), strong nuclear, weak nuclear. The current dominating model of physics is the Standard Model (SM), which unifies the EM, strong, and weak interactions with a reasonable level of success, but has no mechanism to explain gravity. It is precisely this problem that modern GUTs seek to resolve. Unfortunately, there are about as many GUTs as there are physicists in the field. That being said, there appear to be some recurring elements in many such theories. One of the more intriguing aspects to many GUTs is the idea of supersymmetry (SUSY). The fundamental idea behind SUSY is that for every elementary particle, there exists a “superpartner” particle that differs in spin by a half unit. As a reminder, all elementary particles possess an intrinsic angular momentum characterized by their spin number. Particles with integer spin (s = 0, 1, 2, …) are called bosons and particles with half-integer spin (s = 1/2, 3/2, 5/2, …) are called fermions. Although they will not be discussed in detail here, bosons and fermions possess many unique and interesting properties due to their spins and are experimentally easy to differentiate. -
Surface Operators 4 1.2 Classification of Surface Operators 5 1.3 Surface Operators in 4D = 2 Gauge Theory 7 N 1.4 Their Role in AGT Correspondence 9
Preprint typeset in JHEP style - HYPER VERSION Surface Operators Sergei Gukov1,2 1 California Institute of Technology, Pasadena, CA 91125, USA 2 Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 Abstract: We give an introduction and a broad survey of surface operators in 4d gauge theories, with a particular emphasis on aspects relevant to AGT correspondence. One of the main goals is to highlight the boundary between what we know and what we don’t know about surface operators. To this end, the survey contains many open questions and suggests various directions for future research. Although this article is mostly a review, we did include a number of new results, previously unpublished. CALT-TH-2014-170 arXiv:1412.7127v1 [hep-th] 22 Dec 2014 Contents 1. What is a surface operator? 1 1.1 Construction of surface operators 4 1.2 Classification of surface operators 5 1.3 Surface operators in 4d = 2 gauge theory 7 N 1.4 Their role in AGT correspondence 9 2. Surface operators from higher dimensions 12 2.1 Brane constructions 13 2.2 Geometric engineering 13 2.3 Surface operators and BPS states 16 2.4 Relation to 3d-3d correspondence and integrable systems 18 3. Surface operators and line operators 19 3.1 Line operators and Hecke algebras 23 4. Superconformal index 29 5. Surface operators as order parameters 31 1. What is a surface operator? Surface operators (a.k.a. surface defects) in a four-dimensional gauge theory are operators supported on two-dimensional submanifolds in the space-time manifold M. -
Banach Space Theory
Banach space theory Razvan Anisca (Lakehead) Steve Dilworth (South Carolina) Edward Odell (UT Austin) Bunyamin¨ Sarı (North Texas) March 4–9, 2012 The workshop was largely motivated by the recent extraordinary work of Argyros and Haydon [AH] (dis- cussed below) which, following on the fundamental work of Gowers and Maurey in the 1990’s, continued the discovery of the incredible variety of possible Banach space structure. [AH] is connected strongly with operator theory and Banach algebras. The last couple of years were very fruitful for researchers and many stunning results were presented at the workshop and new consequences were observed as well. The testimo- nials of the participants, which included numeruous young researchers and graduate students, attest to the great success of the workshop. The following presents a brief overview in which it is possible to cite only a selection of highlights. Much more can be found in the files and the videos and we expect more developments as these are studied. Scalar plus compact and Invariant subspace problems A key theme of the workshop and one underlying a number of presentations, dates back to the remarkable construction in 1980 of J. Bourgain and F. Delbaen [BD]. They constructed a Banach space X with X∗ isomorphic to `1, yet c0 does not embed into X. The example struck researchers as quite special and too limited to be useful in solving other open problems. After almost 20 years some researchers [A], [H] began to think otherwise. Two years ago, S.A. Argyros and R. Haydon used the BD-construction to solve a famous problem in Banach spaces.