Hilbert Space Operators and Functional Integrals
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MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster
MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster Cotter, Simon and Roberts, Gareth and Stuart, Andrew and White, David 2013 MIMS EPrint: 2014.71 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 Statistical Science 2013, Vol. 28, No. 3, 424–446 DOI: 10.1214/13-STS421 c Institute of Mathematical Statistics, 2013 MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster S. L. Cotter, G. O. Roberts, A. M. Stuart and D. White Abstract. Many problems arising in applications result in the need to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become arbitrarily slow under the mesh refinement dictated by nonparametric description of the un- known function. We describe an approach to modifying a whole range of MCMC methods, applicable whenever the target measure has density with respect to a Gaussian process or Gaussian random field reference measure, which ensures that their speed of convergence is robust under mesh refinement. Gaussian processes or random fields are fields whose marginal distri- butions, when evaluated at any finite set of N points, are RN -valued Gaussians. The algorithmic approach that we describe is applicable not only when the desired probability measure has density with respect to a Gaussian process or Gaussian random field reference measure, but also to some useful non-Gaussian reference measures constructed through random truncation. -
Distinguished Property in Tensor Products and Weak* Dual Spaces
axioms Article Distinguished Property in Tensor Products and Weak* Dual Spaces Salvador López-Alfonso 1 , Manuel López-Pellicer 2,* and Santiago Moll-López 3 1 Department of Architectural Constructions, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 2 Emeritus and IUMPA, Universitat Politècnica de València, 46022 Valencia, Spain 3 Department of Applied Mathematics, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] * Correspondence: [email protected] 0 Abstract: A local convex space E is said to be distinguished if its strong dual Eb has the topology 0 0 0 0 b(E , (Eb) ), i.e., if Eb is barrelled. The distinguished property of the local convex space Cp(X) of real- valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space Cp(X) is distinguished if and only if any function f 2 RX belongs to the pointwise closure of a pointwise bounded set in C(X). The extensively studied distinguished properties in the injective tensor products Cp(X) ⊗# E and in Cp(X, E) contrasts with the few distinguished properties of injective tensor products related to the dual space Lp(X) of Cp(X) endowed with the weak* topology, as well as to the weak* dual of Cp(X, E). To partially fill this gap, some distinguished properties in the injective tensor product space Lp(X) ⊗# E are presented and a characterization of the distinguished property of the weak* dual of Cp(X, E) for wide classes of spaces X and E is provided. -
Inverse and Implicit Function Theorems for Noncommutative
Inverse and Implicit Function Theorems for Noncommutative Functions on Operator Domains Mark E. Mancuso Abstract Classically, a noncommutative function is defined on a graded domain of tuples of square matrices. In this note, we introduce a notion of a noncommutative function defined on a domain Ω ⊂ B(H)d, where H is an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to injectivity. Keywords: Noncommutive functions, operator noncommutative functions, free anal- ysis, inverse and implicit function theorems, strong operator topology, dilation theory. MSC (2010): Primary 46L52; Secondary 47A56, 47J07. INTRODUCTION Polynomials in d noncommuting indeterminates can naturally be evaluated on d-tuples of square matrices of any size. The resulting function is graded (tuples of n × n ma- trices are mapped to n × n matrices) and preserves direct sums and similarities. Along with polynomials, noncommutative rational functions and power series, the convergence of which has been studied for example in [9], [14], [15], serve as prototypical examples of a more general class of functions called noncommutative functions. The theory of non- commutative functions finds its origin in the 1973 work of J. L. Taylor [17], who studied arXiv:1804.01040v2 [math.FA] 7 Aug 2019 the functional calculus of noncommuting operators. Roughly speaking, noncommutative functions are to polynomials in noncommuting variables as holomorphic functions from complex analysis are to polynomials in commuting variables. -
Perturbations of Operators with Application to Testing Equality of Covariance Operators
Perturbations of Operators with Application to Testing Equality of Covariance Operators. by Krishna Kaphle, M.S. A Dissertation In Mathematics and Statistics Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philoshopy Approved Frits H. Ruymgaart Linda J. Allen Petros Hadjicostas Peggy Gordon Miller Interim Dean of the Graduate School August, 2011 c 2011, Krishna Kaphle Texas Tech University, Krishna Kaphle, August-2011 ACKNOWLEDGEMENTS This dissertation would not have been possible without continuous support of my advisor Horn Professor Dr. Firts H. Ruymgaart. I am heartily thankful to Dr. Ruym- gaart for his encouragement, and supervision from the research problem identification to the present level of my dissertation research. His continuous supervision enabled me not only to bring my dissertation in this stage but also to develop an understand- ing of the subject. I am lucky to have a great person like him as my advisor. I am grateful to Horn. Professor Dr. Linda J. Allen for her help throughout my stay at Texas Tech. I would also like to thank Dr. Petros Hadjicostas for encouragement and several valuable remarks during my research. I am also heartily thankful to Dr. David Gilliam for his support and help on writing Matlab code. I am pleased to thank those who made this dissertation possible. Among those, very important people are Dr. Kent Pearce, chair, Dr. Ram Iyer, graduate advisor, and all the staffs of Department of Mathematics and Statistics, who gave me the moral, financial, and technological supports required during my study. -
Algebra Homomorphisms and the Functional Calculus
Pacific Journal of Mathematics ALGEBRA HOMOMORPHISMS AND THE FUNCTIONAL CALCULUS MARK PHILLIP THOMAS Vol. 79, No. 1 May 1978 PACIFIC JOURNAL OF MATHEMATICS Vol. 79, No. 1, 1978 ALGEBRA HOMOMORPHISMS AND THE FUNCTIONAL CALCULUS MARC THOMAS Let b be a fixed element of a commutative Banach algebra with unit. Suppose σ(b) has at most countably many connected components. We give necessary and sufficient conditions for b to possess a discontinuous functional calculus. Throughout, let B be a commutative Banach algebra with unit 1 and let rad (B) denote the radical of B. Let b be a fixed element of B. Let έ? denote the LF space of germs of functions analytic in a neighborhood of σ(6). By a functional calculus for b we mean an algebra homomorphism θr from έ? to B such that θ\z) = b and θ\l) = 1. We do not require θr to be continuous. It is well-known that if θ' is continuous, then it is equal to θ, the usual functional calculus obtained by integration around contours i.e., θ{f) = -±τ \ f(t)(f - ]dt for f eέ?, Γ a contour about σ(b) [1, 1.4.8, Theorem 3]. In this paper we investigate the conditions under which a functional calculus & is necessarily continuous, i.e., when θ is the unique functional calculus. In the first section we work with sufficient conditions. If S is any closed subspace of B such that bS Q S, we let D(b, S) denote the largest algebraic subspace of S satisfying (6 — X)D(b, S) = D(b, S)f all λeC. -
The Weyl Functional Calculus F(4)
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 4, 240-267 (1969) The Weyl Functional Calculus ROBERT F. V. ANDERSON Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Communicated by Edward Nelson Received July, 1968 I. INTRODUCTION In this paper a functional calculus for an n-tuple of noncommuting self-adjoint operators on a Banach space will be proposed and examined. The von Neumann spectral theorem for self-adjoint operators on a Hilbert space leads to a functional calculus which assigns to a self- adjoint operator A and a real Borel-measurable function f of a real variable, another self-adjoint operatorf(A). This calculus generalizes in a natural way to an n-tuple of com- muting self-adjoint operators A = (A, ,..., A,), because their spectral families commute. In particular, if v is an eigenvector of A, ,..., A, with eigenvalues A, ,..., A, , respectively, and f a continuous function of n real variables, f(A 1 ,..., A,) u = f(b , . U 0. This principle fails when the operators don’t commute; instead we have the Uncertainty Principle, and so on. However, the Fourier inversion formula can also be used to define the von Neumann functional cakuius for a single operator: f(4) = (37F2 1 (Sf)(&) exp( -2 f A,) 4 E’ where This definition generalizes naturally to an n-tuple of operators, commutative or not, as follows: in the n-dimensional Fourier inversion 240 THE WEYL FUNCTIONAL CALCULUS 241 formula, the free variables x 1 ,..,, x, are replaced by self-adjoint operators A, ,..., A, . -
Embedding Nuclear Spaces in Products of an Arbitrary Banach Space
proceedings of the american mathematical society Volume 34, Number 1, July 1972 EMBEDDING NUCLEAR SPACES IN PRODUCTS OF AN ARBITRARY BANACH SPACE STEPHEN A. SAXON Abstract. It is proved that if E is an arbitrary nuclear space and F is an arbitrary infinite-dimensional Banach space, then there exists a fundamental (basic) system V of balanced, convex neigh- borhoods of zero for E such that, for each Kin i*~, the normed space Ev is isomorphic to a subspace of F. The result for F=lv (1 ^/>^oo) was proved by A. Grothendieck. This paper is an outgrowth of an interest in varieties of topological vector spaces [2] stimulated by J. Diestel and S. Morris, and is in response to their most helpful discussions and questions. The main theorem, valid for arbitrary infinite-dimensional Banach spaces, was first proved by A. Grothendieck [3] (also, see [5, p. 101]) for the Banach spaces /„ (1 ^p^ co) and later by J. Diestel for the Banach space c0. Our demonstration relies on two profound results of T. Kömura and Y. Kömura [4] and C. Bessaga and A. Pelczyriski [1], respectively: (i) A locally convex space is nuclear if and only if it is isomorphic to a subspace of a product space (s)1, where / is an indexing set and (s) is the Fréchet space of all rapidly decreasing sequences. (ii) Every infinite-dimensional Banach space contains a closed infinite- dimensional subspace which has a Schauder basis.1 Recall that for a balanced, convex neighborhood V of zero in a locally convex space E, Ev is a normed space which is norm-isomorphic to (M,p\M), where/? is the gauge of V and M is a maximal linear subspace of £ on which/; is a norm; Ev is the completion of Ev. -
The Holomorphic Functional Calculus Approach to Operator Semigroups
THE HOLOMORPHIC FUNCTIONAL CALCULUS APPROACH TO OPERATOR SEMIGROUPS CHARLES BATTY, MARKUS HAASE, AND JUNAID MUBEEN In honour of B´elaSz.-Nagy (29 July 1913 { 21 December 1998) on the occasion of his centenary Abstract. In this article we construct a holomorphic functional calculus for operators of half-plane type and show how key facts of semigroup theory (Hille- Yosida and Gomilko-Shi-Feng generation theorems, Trotter-Kato approxima- tion theorem, Euler approximation formula, Gearhart-Pr¨usstheorem) can be elegantly obtained in this framework. Then we discuss the notions of bounded H1-calculus and m-bounded calculus on half-planes and their relation to weak bounded variation conditions over vertical lines for powers of the resolvent. Finally we discuss the Hilbert space case, where semigroup generation is char- acterised by the operator having a strong m-bounded calculus on a half-plane. 1. Introduction The theory of strongly continuous semigroups is more than 60 years old, with the fundamental result of Hille and Yosida dating back to 1948. Several monographs and textbooks cover material which is now canonical, each of them having its own particular point of view. One may focus entirely on the semigroup, and consider the generator as a derived concept (as in [12]) or one may start with the generator and view the semigroup as the inverse Laplace transform of the resolvent (as in [2]). Right from the beginning, functional calculus methods played an important role in semigroup theory. Namely, given a C0-semigroup T = (T (t))t≥0 which is uni- formly bounded, say, one can form the averages Z 1 Tµ := T (s) µ(ds) (strong integral) 0 for µ 2 M(R+) a complex Radon measure of bounded variation on R+ = [0; 1). -
JOINT MEASURES and CROSS-COVARIANCE OPERATORS(L)
TRANSACTIONSOF THE AMERICANMATHEMATICAL SOCIETY Volume 186, December 1973 JOINT MEASURESAND CROSS-COVARIANCE OPERATORS(l) BY CHARLESR. BAKER ABSTRACT. Let H. (resp., H ) be a real and separable Hubert space with Borel O'field T (resp., rj, and let (H. X //-, T, X T.) be the product measurable space generated by the measurable rectangles. This paper develops relations between probability measures on (H. x H , V x T.), i.e., joint measures, and the projections of such measures on (H., T.) and (H , Y ). In particular, the class of all joint Gaussian measures having two specified Gaussian measures as projections is characterized, and conditions are ob- tained for two joint Gaussian measures to be mutually absolutely continuous. The cross-covariance operator of a joint measure plays a major role in these results and these operators are characterized. Introduction. Let H. (resp., A7 ) be a real and separable Hubert space with inner product (•, *)j (resp.,(', -)2) and Borel o-field Tj (resp., TJ. Let Tj x T denote the o-field generated by the measurable rectangles AxB,A£rj,B£ I"^. Define H. x //, = {(u, v): u in H., v in H A. f/j x H is a real linear space, with addition and scalar multiplication defined by (u, v) + (z, y) = (u + z, v + y) and k(u, v) = (ka, k\). H. x H2 is a separable Hubert space under the inner product [•, •] defined by [(u, v), (t, z)] = (u, t)j + (v, z)2; moreover, the open sets under the norm obtained from this inner product generate Tj x T2 [10]. -
Hoo Functional Calculus of Second Order Elliptic Partial Differential Operators on LP Spaces
91 Hoo Functional Calculus of Second Order Elliptic Partial Differential Operators on LP Spaces Xuan Thinh Duong Abstract. Let L be a strongly elliptic partial differential operator of second order, with real coefficients on LP(n), 1 < p <"",with either Dirichlet, or Neumann, or "oblique" boundary conditions. Assume that n is an open, bounded domain with cz boWldary. By adding a oonstmt, if necessa_ry, we then establish an H., fnnctioMI. calculus which associates an operator m(L) to each bounded liolomorphic fooction m so that II m (L)II ~ M II m II.,., where M is a constmt independent of m. Under suitable asumpl.ions on L, we can also obtain a similar result in the case of Dirichlet boundary conditioos where n is a non-smooth domain. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 47A60, 42B15 , 35125 1. Introduction 1imd Notation We denote the sectors Se = { z e C I z = 0 or I arg z I $; 6 ) 0 S 9 = {z e ll::lz;!f;Oorlargzi<S} A linear operator L is of type ro in a Banach space X if L is closed, densely defined, G(L) is a subset of S00u{ "" } and for each S e (ro ,n:], there exists Ce < = such that ii (L-rl) "1 11 s; Ce I z 1-1 for all z ~ S ~· z:;<: 0. For 0 < !J. < x, denote H""(S ~) ={ f: S ~ ~ [I fanalytic and II fll~<-} 0 where II f II .. = sup { I f(z)l I z e S !!. } , • V(S ~) ={ f e H"" (S ~) i 3s>O, c 2:0 such that I f(z)i $; 1 ::~,:8 } • Let r be the contour defined by 92 -t exp(i8) for -- < t < 0 g(t) { t exp(i8) for Os:;t<+"" Assume that Lis of type ro, ro < 8 < :n:. -
Functional Properties of Hörmander's Space of Distributions Having A
Functional properties of Hörmander’s space of distributions having a specified wavefront set Yoann Dabrowski, Christian Brouder To cite this version: Yoann Dabrowski, Christian Brouder. Functional properties of Hörmander’s space of distributions having a specified wavefront set. 2014. hal-00850192v2 HAL Id: hal-00850192 https://hal.archives-ouvertes.fr/hal-00850192v2 Preprint submitted on 3 May 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Communications in Mathematical Physics manuscript No. (will be inserted by the editor) Functional properties of H¨ormander’s space of distributions having a specified wavefront set Yoann Dabrowski1, Christian Brouder2 1 Institut Camille Jordan UMR 5208, Universit´ede Lyon, Universit´eLyon 1, 43 bd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France 2 Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie, Sorbonne Univer- sit´es, UMR CNRS 7590, UPMC Univ. Paris 06, Mus´eum National d’Histoire Naturelle, IRD UMR 206, 4 place Jussieu, F-75005 Paris, France. Received: date / Accepted: date ′ Abstract: The space Γ of distributions having their wavefront sets in a closed cone Γ has become importantD in physics because of its role in the formulation of quantum field theory in curved spacetime. -
Data Representation from Finite to Infinite-Dimensional Settings
From Covariance Matrices to Covariance Operators Data Representation from Finite to Infinite-Dimensional Settings Ha` Quang Minh Pattern Analysis and Computer Vision (PAVIS) Istituto Italiano di Tecnologia, ITALY November 30, 2017 H.Q. Minh (IIT) Covariance matrices & covariance operators November 30, 2017 1 / 99 Acknowledgement Collaborators Marco San Biagio Vittorio Murino H.Q. Minh (IIT) Covariance matrices & covariance operators November 30, 2017 2 / 99 From finite to infinite dimensions H.Q. Minh (IIT) Covariance matrices & covariance operators November 30, 2017 3 / 99 Part I - Outline Finite-dimensional setting Covariance Matrices and Applications 1 Data Representation by Covariance Matrices 2 Geometry of SPD matrices 3 Machine Learning Methods on Covariance Matrices and Applications in Computer Vision H.Q. Minh (IIT) Covariance matrices & covariance operators November 30, 2017 4 / 99 Part II - Outline Infinite-dimensional setting Covariance Operators and Applications 1 Data Representation by Covariance Operators 2 Geometry of Covariance Operators 3 Machine Learning Methods on Covariance Operators and Applications in Computer Vision H.Q. Minh (IIT) Covariance matrices & covariance operators November 30, 2017 5 / 99 From finite to infinite-dimensional settings H.Q. Minh (IIT) Covariance matrices & covariance operators November 30, 2017 6 / 99 Part II - Outline Infinite-dimensional setting Covariance Operators and Applications 1 Data Representation by Covariance Operators 2 Geometry of Covariance Operators 3 Machine Learning Methods on Covariance