Notes on Infinite Determinants of Hilbert Space Operators
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L2-Theory of Semiclassical Psdos: Hilbert-Schmidt And
2 LECTURE 13: L -THEORY OF SEMICLASSICAL PSDOS: HILBERT-SCHMIDT AND TRACE CLASS OPERATORS In today's lecture we start with some abstract properties of Hilbert-Schmidt operators and trace class operators. Then we will investigate the following natural t questions: for which class of symbols, the quantized operator Op~(a) is a Hilbert- Schmidt or trace class operator? How to compute its trace? 1. Hilbert-Schmidt and Trace class operators: Abstract theory { Definitions via singular values. As we have mentioned last time, we are aiming at developing the spectral theory for semiclassical pseudodifferential operators. For a 2 S(m), where m is a decaying t 2 n symbol, we showed that Op~(a) is a compact operator acting on L (R ). As a t consequence, either Op~(a) has only finitely many nonzero eigenvalues, or it has infinitely nonzero eigenvalues which can be arranged to a sequence whose norms converges to 0. However, in general the eigenvalues of a compact operator A are non-real. A very simple way to get real eigenvalues is to consider the operator A∗A, which is 1 a compact self-adjoint linear operator acting on L2(Rn). Thus the eigenvalues of A∗A can be list2 in decreasing order as 2 2 2 s1 ≥ s2 ≥ s3 ≥ · · · : The numbers sj (which will always be taken to be the positive one) are called the singular values of A. Definition 1.1. We say a compact operator3 A is a Hilbert-Schmidt operator if !1=2 X 2 (1) kAkHS := sj < +1: j The number kAkHS is called the Hilbert-Schmidt norm (or Schatten 2-norm) of A. -
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. -
Clustering and the Three-Point Function
Clustering and the Three-Point Function Yunfeng Jianga, Shota Komatsub, Ivan Kostovc1, Didina Serbanc a Institut f¨urTheoretische Physik, ETH Z¨urich, Wolfgang Pauli Strasse 27, CH-8093 Z¨urich,Switzerland b Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada c Institut de Physique Th´eorique,DSM, CEA, URA2306 CNRS Saclay, F-91191 Gif-sur-Yvette, France [email protected], [email protected], ivan.kostov & [email protected] Abstract We develop analytical methods for computing the structure constant for three heavy operators, starting from the recently proposed hexagon approach. Such a structure constant is a semiclassical object, with the scale set by the inverse length of the operators playing the role of the Planck constant. We reformulate the hexagon expansion in terms of multiple contour integrals and recast it as a sum over clusters generated by the residues of the measure of integration. We test the method on two examples. First, we compute the asymptotic three-point function of heavy fields at any coupling and show the result in the semiclassical limit matches both the string theory computation at strong coupling and the arXiv:1604.03575v2 [hep-th] 22 May 2016 tree-level results obtained before. Second, in the case of one non-BPS and two BPS operators at strong coupling we sum up all wrapping corrections associated with the opposite bridge to the non-trivial operator, or the \bottom" mirror channel. We also give an alternative interpretation of the results in terms of a gas of fermions and show that they can be expressed compactly as an operator- valued super-determinant. -
Operators and Special Functions in Random Matrix Theory
Operators and Special Functions in Random Matrix Theory Andrew McCafferty, MSci Submitted for the degree of Doctor of Philosophy at Lancaster University March 2008 Operators and Special Functions in Random Matrix Theory Andrew McCafferty, MSci Submitted for the degree of Doctor of Philosophy at Lancaster University, March 2008 Abstract The Fredholm determinants of integral operators with kernel of the form A(x)B(y) A(y)B(x) − x y − arise in probabilistic calculations in Random Matrix Theory. These were ex- tensively studied by Tracy and Widom, so we refer to them as Tracy–Widom operators. We prove that the integral operator with Jacobi kernel converges in trace norm to the integral operator with Bessel kernel under a hard edge scaling, using limits derived from convergence of differential equation coef- ficients. The eigenvectors of an operator with kernel of Tracy–Widom type can sometimes be deduced via a commuting differential operator. We show that no such operator exists for TW integral operators acting on L2(R). There are analogous operators for discrete random matrix ensembles, and we give sufficient conditions for these to be expressed as the square of a Han- kel operator: writing an operator in this way aids calculation of Fredholm determinants. We also give a new example of discrete TW operator which can be expressed as the sum of a Hankel square and a Toeplitz operator. Previously unsolvable equations are dealt with by threats of reprisals . Woody Allen 2 Acknowledgements I would like to thank many people for helping me through what has sometimes been a difficult three years. -
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. -
Preduals for Spaces of Operators Involving Hilbert Spaces and Trace
PREDUALS FOR SPACES OF OPERATORS INVOLVING HILBERT SPACES AND TRACE-CLASS OPERATORS HANNES THIEL Abstract. Continuing the study of preduals of spaces L(H,Y ) of bounded, linear maps, we consider the situation that H is a Hilbert space. We establish a natural correspondence between isometric preduals of L(H,Y ) and isometric preduals of Y . The main ingredient is a Tomiyama-type result which shows that every contractive projection that complements L(H,Y ) in its bidual is automatically a right L(H)-module map. As an application, we show that isometric preduals of L(S1), the algebra of operators on the space of trace-class operators, correspond to isometric preduals of S1 itself (and there is an abundance of them). On the other hand, the compact operators are the unique predual of S1 making its multiplication separately weak∗ continuous. 1. Introduction An isometric predual of a Banach space X is a Banach space F together with an isometric isomorphism X =∼ F ∗. Every predual induces a weak∗ topology. Due to the importance of weak∗ topologies, it is interesting to study the existence and uniqueness of preduals; see the survey [God89] and the references therein. Given Banach spaces X and Y , let L(X, Y ) denote the space of operators from X to Y . Every isometric predual of Y induces an isometric predual of L(X, Y ): If Y =∼ F ∗, then L(X, Y ) =∼ (X⊗ˆ F )∗. Problem 1.1. Find conditions on X and Y guaranteeing that every isometric predual of L(X, Y ) is induced from an isometric predual of Y . -
Arxiv:Math/0312267V1
(MODIFIED) FREDHOLM DETERMINANTS FOR OPERATORS WITH MATRIX-VALUED SEMI-SEPARABLE INTEGRAL KERNELS REVISITED FRITZ GESZTESY AND KONSTANTIN A. MAKAROV Dedicated with great pleasure to Eduard R. Tsekanovskii on the occasion of his 65th birthday. Abstract. We revisit the computation of (2-modified) Fredholm determi- nants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Green’s functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants. We illustrate our approach by identifying classical objects such as the Jost function for half-line Schr¨odinger operators and the inverse transmission coeffi- cient for Schr¨odinger operators on the real line as Fredholm determinants, and rederiving the well-known expressions for them in due course. We also apply our formalism to Floquet theory of Schr¨odinger operators, and upon identify- ing the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant. Finally, we rederive the explicit formula for the 2-modified Fredholm de- terminant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener–Hopf -
Determinants of Elliptic Pseudo-Differential Operators 3
DETERMINANTS OF ELLIPTIC PSEUDO-DIFFERENTIAL OPERATORS MAXIM KONTSEVICH AND SIMEON VISHIK March 1994 Abstract. Determinants of invertible pseudo-differential operators (PDOs) close to positive self-adjoint ones are defined through the zeta-function regularization. We define a multiplicative anomaly as the ratio det(AB)/(det(A) det(B)) con- sidered as a function on pairs of elliptic PDOs. We obtained an explicit formula for the multiplicative anomaly in terms of symbols of operators. For a certain natural class of PDOs on odd-dimensional manifolds generalizing the class of elliptic dif- ferential operators, the multiplicative anomaly is identically 1. For elliptic PDOs from this class a holomorphic determinant and a determinant for zero orders PDOs are introduced. Using various algebraic, analytic, and topological tools we study local and global properties of the multiplicative anomaly and of the determinant Lie group closely related with it. The Lie algebra for the determinant Lie group has a description in terms of symbols only. Our main discovery is that there is a quadratic non-linearity hidden in the defi- nition of determinants of PDOs through zeta-functions. The natural explanation of this non-linearity follows from complex-analytic prop- erties of a new trace functional TR on PDOs of non-integer orders. Using TR we easily reproduce known facts about noncommutative residues of PDOs and obtain arXiv:hep-th/9404046v1 7 Apr 1994 several new results. In particular, we describe a structure of derivatives of zeta- functions at zero as of functions on logarithms of elliptic PDOs. We propose several definitions extending zeta-regularized determinants to gen- eral elliptic PDOs. -
S0002-9947-1980-0570783-0.Pdf
transactions of the american mathematical society Volume 260, Number 1, July 1980 NONSTANDARD EXTENSIONS OF TRANSFORMATIONS BETWEEN BANACH SPACES BY D. G. TACON Abstract. Let X and Y be (infinite-dimensional) Banach spaces and denote their nonstandard hulls with respect to an Nj -saturated enlargement by X and Y respectively. If 9> (X, Y) denotes the space of bounded linear transformations then a subset S of elements of ® (X, Y) extends naturally to a subset S of *$>(X, Y). This paper studies the behaviour of various kinds of transformations under this extension and introduces, in this context, the concepts of super weakly compact, super strictly singular and socially compact operators. It shows that (® (X, Y))~ c 'S (X, Y) provided X and Y are infinite dimensional and contrasts this with the inclusion %(H) c (%(H))~ where %(H) denotes the space of compact operators on a Hubert space. The nonstandard hull of a uniform space was introduced by Luxemburg [14] and the construction, in the case of normed spaces, has since been investigated in a number of papers (see, for example [4], [6], [7], [8], [9], [10], [15]). Our intention here is to examine the behaviour of single bounded linear transformations and classes of bounded linear transformations under the nonstandard hull extension. We recount the principal ideas. Consider a normed space (X, || • ||) embedded in a set theoretical structure 911 and consider an N, -sa tura ted enlargement *91t of 9H. An element p G *X is said to be finite if *||p|| is a finite element of *R, the set of such elements being denoted by fin(*Ar). -
Spectral Theory for Polynomially Demicompact Operators
Filomat 33:7 (2019), 2017–2030 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1907017B University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Spectral Theory for Polynomially Demicompact Operators Fatma Ben Brahima, Aref Jeribia, Bilel Krichena aDepartment of Mathematics. Faculty of Sciences of Sfax. University of Sfax Abstract.In this article, we introduce the notion of polynomial demicompactness and we use it to give some results on Fredholm operators and to establish a fine description of some essential spectra of a closed densely defined linear operator. Our work is a generalization of many known ones in the literature. 1. Introduction Throughout this paper, X and Y denote two infinite dimensional complex Banach spaces. We denote by (X; Y) (resp. (X; Y)) the set of all closed densely defined (resp. bounded) linear operators acting fromC X into Y. TheL subspace of all compact (resp. weakly compact) operators of (X; Y) is denoted by (X; Y) ( resp. (X; Y)). For T (X; Y), we denote by α(T) the dimension ofL the kernel (T) andK by β(T) the codimensionW of the range2 C (T) in Y. The next sets of upper semi-Fredholm, lowerN semi-Fredholm, Fredholm and semi-FredholmR operators from X into Y are, respectively, defined by Φ (X; Y) = T (X; Y) such that α(T) < and (T) closed in Y , + f 2 C 1 R g Φ (X; Y) = T (X; Y) such that β(T) < and (T) closed in Y , − f 2 C 1 R g Φ(X; Y):= Φ (X; Y) Φ+(X; Y), − \ and Φ (X; Y):= Φ (X; Y) Φ+(X; Y). -
On the Characterization of Trace Class Representations and Schwartz Operators
On the characterization of trace class representations and Schwartz operators Gerrit van Dijk, ∗ Karl-Hermann Neeb, y Hadi Salmasian,z Christoph Zellner x January 3, 2016 Abstract In this note we collect several characterizations of unitary representa- tions (π; H) of a finite dimensional Lie group G which are trace class, i.e., for each compactly supported smooth function f on G, the operator π(f) is trace class. In particular we derive the new result that, for some m 2 N, m all operators π(f), f 2 Cc (G), are trace class. As a consequence the cor- responding distribution character θπ is of finite order. We further show π is trace class if and only if every operator A, which is smoothing in the sense that AH ⊆ H1, is trace class and that this in turn is equivalent to the Fr´echet space H1 being nuclear, which in turn is equivalent to the re- alizability of the Gaussian measure of H on the space H−∞ of distribution vectors. Finally we show that, even for infinite dimensional Fr´echet{Lie groups, A and A∗ are smoothing if and only if A is a Schwartz operator, i.e., all products of A with operators from the derived representation are bounded. Introduction Let (π; H) be a (strongly continuous) unitary representation of the (possibly infinite dimensional) Lie group G (with an exponential function). Let H1 be its subspace of smooth vectors. On this space we obtain by d dπ(x)v = π(exp tx)v dt t=0 the derived representation of g which we extend naturally to a representation of the enveloping algebra U(g), also denoted dπ.