ON the DUAL of ORLICZ–LORENTZ SPACE 1. Introduction
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The Semi-M Property for Normed Riesz Spaces Compositio Mathematica, Tome 34, No 2 (1977), P
COMPOSITIO MATHEMATICA EP DE JONGE The semi-M property for normed Riesz spaces Compositio Mathematica, tome 34, no 2 (1977), p. 147-172 <http://www.numdam.org/item?id=CM_1977__34_2_147_0> © Foundation Compositio Mathematica, 1977, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion 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/ COMPOSITIO MATHEMATICA, Vol. 34, Fasc. 2, 1977, pag. 147-172 Noordhoff International Publishing Printed in the Netherlands THE SEMI-M PROPERTY FOR NORMED RIESZ SPACES Ep de Jonge 1. Introduction It is well-known that if (0394, F, IL) is a u-finite measure space and if 1 ~ p 00, then the Banach dual L *p of the Banach space Lp = Lp(0394, IL) can be identified with Lq = Lq(L1, 03BC), where p-1 + q-1 = 1. For p =00 the situation is different; the space Li is a linear subspace of L*, and only in a very trivial situation (the finite-dimensional case) we have Li = Lfi. Restricting ourselves to the real case, the Banach dual L *~ is a (real) Riesz space, i.e., a vector lattice, and Li is now a band in L*. The disjoint complement (i.e., the set of all elements in L* disjoint to all elements in LI) is also a band in L*, called the band of singular linear functionals on Loo. -
Burkholder's Inequalities in Noncommutative Lorentz
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 7, July 2010, Pages 2431–2441 S 0002-9939(10)10267-6 Article electronically published on March 24, 2010 BURKHOLDER’S INEQUALITIES IN NONCOMMUTATIVE LORENTZ SPACES YONG JIAO (Communicated by Marius Junge) Abstract. We prove Burkholder’s inequalities in noncommutative Lorentz spaces Lp,q(M), 1 <p<∞, 1 ≤ q<∞, associated with a von Neumann algebra M equipped with a faithful normal tracial state. These estimates generalize the classical inequalities in the commutative case. 1. Introduction Martingale inequalities and sums of independent random variables are important tools in classical harmonic analysis. A fundamental result due to Burkholder [1, 2] can be stated as follows. Given a probability space (Ω, F ,P), let {Fn}n≥1 be a nondecreasing sequence of σ-fields of F such that F = ∨Fn and let En be the conditional expectation operator relative to Fn. Given 2 ≤ p<∞ and an p L -bounded martingale f =(fn)n≥1, we have ∞ ∞ 1/2 1/p 2 p (1.1) fLp ≈ Ek−1(|df k| ) + |df k| . Lp Lp k=1 k=1 The first term on the right is called the conditioned square function of f, while the second is called the p-variation of f. Rosenthal’s inequalities [14] can be regarded as the particular case where the sequence df =(df 1,df2, ...) is a family of independent mean-zero random variables df k = ak. In this case it is easy to reduce Rosenthal’s inequalities to ∞ ∞ ∞ 1/2 1/p ≈ 2 p (1.2) ak Lp ak 2 + ak p . -
On Domain of Nörlund Matrix
mathematics Article On Domain of Nörlund Matrix Kuddusi Kayaduman 1,* and Fevzi Ya¸sar 2 1 Faculty of Arts and Sciences, Department of Mathematics, Gaziantep University, Gaziantep 27310, Turkey 2 ¸SehitlerMah. Cambazlar Sok. No:9, Kilis 79000, Turkey; [email protected] * Correspondence: [email protected] Received: 24 October 2018; Accepted: 16 November 2018; Published: 20 November 2018 Abstract: In 1978, the domain of the Nörlund matrix on the classical sequence spaces lp and l¥ was introduced by Wang, where 1 ≤ p < ¥. Tu˘gand Ba¸sarstudied the matrix domain of Nörlund mean on the sequence spaces f 0 and f in 2016. Additionally, Tu˘gdefined and investigated a new sequence space as the domain of the Nörlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space bs(Nt) and cs(Nt) and examined the domain of the Nörlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their a-, b-, g-duals, and characterized their matrix transformations on this space and into this space. Keywords: nörlund mean; nörlund transforms; difference matrix; a-, b-, g-duals; matrix transformations 1. Introduction 1.1. Background In the studies on the sequence space, creating a new sequence space and research on its properties have been important. Some researchers examined the algebraic properties of the sequence space while others investigated its place among other known spaces and its duals, and characterized the matrix transformations on this space. -
Arxiv:1911.05823V1 [Math.OA] 13 Nov 2019 Al Opc Oooia Pcsadcommutative and Spaces Topological Compact Cally Edt E Eut Ntplg,O Ipie Hi Ros Nteothe the in Study Proofs
TOEPLITZ EXTENSIONS IN NONCOMMUTATIVE TOPOLOGY AND MATHEMATICAL PHYSICS FRANCESCA ARICI AND BRAM MESLAND Abstract. We review the theory of Toeplitz extensions and their role in op- erator K-theory, including Kasparov’s bivariant K-theory. We then discuss the recent applications of Toeplitz algebras in the study of solid state sys- tems, focusing in particular on the bulk-edge correspondence for topological insulators. 1. Introduction Noncommutative topology is rooted in the equivalence of categories between lo- cally compact topological spaces and commutative C∗-algebras. This duality allows for a transfer of ideas, constructions, and results between topology and operator al- gebras. This interplay has been fruitful for the advancement of both fields. Notable examples are the Connes–Skandalis foliation index theorem [17], the K-theory proof of the Atiyah–Singer index theorem [3, 4], and Cuntz’s proof of Bott periodicity in K-theory [18]. Each of these demonstrates how techniques from operator algebras lead to new results in topology, or simplifies their proofs. In the other direction, Connes’ development of noncommutative geometry [14] by using techniques from Riemannian geometry to study C∗-algebras, led to the discovery of cyclic homol- ogy [13], a homology theory for noncommutative algebras that generalises de Rham cohomology. Noncommutative geometry and topology techniques have found ample applica- tions in mathematical physics, ranging from Connes’ reformulation of the stan- dard model of particle physics [15], to quantum field theory [16], and to solid-state physics. The noncommutative approach to the study of complex solid-state sys- tems was initiated and developed in [5, 6], focusing on the quantum Hall effect and resulting in the computation of topological invariants via pairings between K- theory and cyclic homology. -
A Symplectic Banach Space with No Lagrangian Subspaces
transactions of the american mathematical society Volume 273, Number 1, September 1982 A SYMPLECTIC BANACHSPACE WITH NO LAGRANGIANSUBSPACES BY N. J. KALTON1 AND R. C. SWANSON Abstract. In this paper we construct a symplectic Banach space (X, Ü) which does not split as a direct sum of closed isotropic subspaces. Thus, the question of whether every symplectic Banach space is isomorphic to one of the canonical form Y X Y* is settled in the negative. The proof also shows that £(A") admits a nontrivial continuous homomorphism into £(//) where H is a Hilbert space. 1. Introduction. Given a Banach space E, a linear symplectic form on F is a continuous bilinear map ß: E X E -> R which is alternating and nondegenerate in the (strong) sense that the induced map ß: E — E* given by Û(e)(f) = ü(e, f) is an isomorphism of E onto E*. A Banach space with such a form is called a symplectic Banach space. It can be shown, by essentially the argument of Lemma 2 below, that any symplectic Banach space can be renormed so that ß is an isometry. Any symplectic Banach space is reflexive. Standard examples of symplectic Banach spaces all arise in the following way. Let F be a reflexive Banach space and set E — Y © Y*. Define the linear symplectic form fiyby Qy[(^. y% (z>z*)] = z*(y) ~y*(z)- We define two symplectic spaces (£,, ß,) and (E2, ß2) to be equivalent if there is an isomorphism A : Ex -» E2 such that Q2(Ax, Ay) = ß,(x, y). A. Weinstein [10] has asked the question whether every symplectic Banach space is equivalent to one of the form (Y © Y*, üy). -
Arxiv:1712.01188V2 [Math.AP]
GREEN’S FUNCTION FOR SECOND ORDER ELLIPTIC EQUATIONS WITH SINGULAR LOWER ORDER COEFFICIENTS SEICK KIM AND GEORGIOS SAKELLARIS Abstract. We construct Green’s function for second order elliptic operators of the form Lu = −∇ · (A∇u + bu) + c · ∇u + du in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients A is uniformly elliptic and bounded and the lower order coefficients b, c, and d belong to certain Lebesgue classes and satisfy the condition d −∇· b ≥ 0. In particular, we allow the lower order coefficients to be singular. We also obtain the global pointwise bounds for the gradient of Green’s function in the case when the mean oscillations of the coefficients A and b satisfy the Dini conditions and the domain is C1,Dini. 1. Introduction Let Ω be a domain (i.e., an open connected set) in Rn with n ≥ 3. We consider second order elliptic operators in divergence form n n ij i i Lu = − Di(a (x)Dju + b (x)u) + c (x)Diu + d(x)u, iX,j=1 Xi=1 which hereafter shall be abbreviated as Lu = − div(A∇u + bu) + c · ∇u + du. We assume that the principal coefficients A = (aij) are measurable n × n matrices that are bounded and uniformly elliptic; i.e. there is a constant λ> 0 such that n λ|ξ|2 ≤ A(x)ξ · ξ = aij(x)ξiξj, ∀x ∈ Ω, ∀ξ ∈ Rn. (1.1) iX,j=1 We also assume that the lower order coefficients b = (b1,..., bn), c = (c1,..., cn), and d are such that b ∈ Lq(Ω), c ∈ Lr(Ω), d ∈ Ls(Ω) for some q, r ≥ n, s ≥ n/2. -
Fourier Analysis in Function Space Theory
Course No. 401-4463-62L Fourier Analysis in Function Space Theory Dozent: Tristan Rivi`ere Assistant: Alessandro Pigati Contents 1 The Fourier transform of tempered distributions 1 1.1 The Fourier transforms of L1 functions . 1 1.2 The Schwartz Space S(Rn)........................... 4 1.3 Frechet Spaces . 6 1.4 The space of tempered distributions S0(Rn).................. 12 1.5 Convolutions in S0(Rn)............................. 21 2 The Hardy-Littlewood Maximal Function 26 2.1 Definition and elementary properties. 26 2.2 Hardy-Littlewood Lp−theorem for the Maximal Function. 27 2.3 The limiting case p =1. ............................ 30 3 Quasi-normed vector spaces 32 3.1 The Metrizability of quasi-normed vector spaces . 32 3.2 The Lorentz spaces Lp;1 ............................ 36 3.3 Decreasing rearrangement . 37 3.4 The Lorentz spaces Lp;q ............................ 39 3.5 Functional inequalities for Lorentz spaces . 44 3.6 Dyadic characterization of some Lorentz spaces and another proof of Lorentz{ Sobolev embedding (optional) . 49 4 The Lp−theory of Calder´on-Zygmund convolution operators. 52 4.1 Calder´on-Zygmund decompositions. 52 4.2 An application of Calder´on-Zygmund decomposition . 54 4.3 The Marcinkiewicz Interpolation Theorem - The Lp case . 56 4.4 Calderon Zygmund Convolution Operators over Lp ............. 58 4.4.1 A \primitive" formulation . 60 4.4.2 A singular integral type formulation . 64 4.4.3 The case of homogeneous kernels . 69 4.4.4 A multiplier type formulation . 71 4.4.5 Applications: The Lp theory of the Riesz Transform and the Laplace and Bessel Operators . 74 4.4.6 The limiting case p =1........................ -
MAT641-Inner Product Spaces
MAT641- MSC Mathematics, MNIT Jaipur Functional Analysis-Inner Product Space and orthogonality Dr. Ritu Agarwal Department of Mathematics, Malaviya National Institute of Technology Jaipur April 11, 2017 1 Inner Product Space Inner product on linear space X over field K is defined as a function h:i : X × X −! K which satisfies following axioms: For x; y; z 2 X, α 2 K, i. hx + y; zi = hx; zi + hy; zi; ii. hαx; yi = αhx; yi; iii. hx; yi = hy; xi; iv. hx; xi ≥ 0, hx; xi = 0 iff x = 0. An inner product defines norm as: kxk = phx; xi and the induced metric on X is given by d(x; y) = kx − yk = phx − y; x − yi Further, following results can be derived from these axioms: hαx + βy; zi = αhx; zi + βhy; zi hx; αyi = αhx; yi hx; αy + βzi = αhx; yi + βhx; zi α, β are scalars and bar denotes complex conjugation. Remark 1.1. Inner product is linear w.r.to first factor and conjugate linear (semilin- ear) w.r.to second factor. Together, this property is called `Sesquilinear' (means one and half-linear). Date: April 11, 2017 Email:[email protected] Web: drrituagarwal.wordpress.com MAT641-Inner Product Space and orthogonality 2 Definition 1.2 (Hilbert space). A complete inner product space is known as Hilbert Space. Example 1. Examples of Inner product spaces n n i. Euclidean Space R , hx; yi = ξ1η1 + ::: + ξnηn, x = (ξ1; :::; ξn), y = (η1; :::; ηn) 2 R . n n ii. Unitary Space C , hx; yi = ξ1η1 + ::: + ξnηn, x = (ξ1; :::; ξn), y = (η1; :::; ηn) 2 C . -
Ideal Convergence and Completeness of a Normed Space
mathematics Article Ideal Convergence and Completeness of a Normed Space Fernando León-Saavedra 1 , Francisco Javier Pérez-Fernández 2, María del Pilar Romero de la Rosa 3,∗ and Antonio Sala 4 1 Department of Mathematics, Faculty of Social Sciences and Communication, University of Cádiz, 11405 Jerez de la Frontera, Spain; [email protected] 2 Department of Mathematics, Faculty of Science, University of Cádiz, 1510 Puerto Real, Spain; [email protected] 3 Department of Mathematics, CASEM, University of Cádiz, 11510 Puerto Real, Spain 4 Department of Mathematics, Escuela Superior de Ingeniería, University of Cádiz, 11510 Puerto Real, Spain; [email protected] * Correspondence: [email protected] Received: 23 July 2019; Accepted: 21 September 2019; Published: 25 September 2019 Abstract: We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each wuc series, then the underlying space is complete. In the process the existing proofs are simplified and some unanswered questions are solved. This research line was originated in the PhD thesis of the second author. Since then, it has been possible to characterize the completeness of a normed spaces through different convergence subspaces (which are be defined using different kinds of convergence) associated to an unconditionally Cauchy sequence. Keywords: ideal convergence; unconditionally Cauchy series; completeness ; barrelledness MSC: 40H05; 40A35 1. Introduction A sequence (xn) in a Banach space X is said to be statistically convergent to a vector L if for any # > 0 the subset fn : kxn − Lk > #g has density 0. -
Weak-Type Weights and Normable Lorentz Spaces 1
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 3, March 1996 WEAK-TYPE WEIGHTS AND NORMABLE LORENTZ SPACES MAR´IA J. CARRO, ALEJANDRO GARC´IA DEL AMO, AND JAVIER SORIA (Communicated by J. Marshall Ash) Abstract. We show that the Lorentz space Λ1(w) is a Banach space if and only if the Hardy-Littlewood maximal operator M satisfies a certain weak-type estimate. We also consider the case of general measures. Finally, we study some properties of several indices associated to these spaces. 1. Introduction We are going to study weighted Lorentz spaces of functions defined in Rn as follows (for standard notation we refer to [BS] and [GR]): If u is a weight in Rn, + w is a weight in R , fu∗ denotes the decreasing rearrangement of f with respect to the measure u(x) dx and 0 <p< , we define ∞ 1/p ∞ p p p Λu(w)= f; f Λu(w) = fu∗(t) w(t)dt < . k k 0 ∞ Z If u 1, we will only write Λp(w). Classical examples are obtained by choosing ≡ (p/q) 1 p q,p w(t)=t − .InthiscaseΛ(w)=L . A classical result of G.G. Lorentz (see [Lo]) shows that Λ1(w) is a norm, if and only if, w is a decreasing function. The problem of findingk·k conditions on w so that Λp(w) is a Banach space (that is, there exists a norm equivalent to Λp(w)) was solved, for p>1, by E. Sawyer ([Sa]). This condition is that the Hardy-Littlewoodk·k maximal operator is bounded on Λp(w). -
On Some Bk Spaces
IJMMS 2003:28, 1783–1801 PII. S0161171203204324 http://ijmms.hindawi.com © Hindawi Publishing Corp. ON SOME BK SPACES BRUNO DE MALAFOSSE Received 25 April 2002 ◦ (c) We characterize the spaces sα(∆), sα(∆),andsα (∆) and we deal with some sets generalizing the well-known sets w0(λ), w∞(λ), w(λ), c0(λ), c∞(λ),andc(λ). 2000 Mathematics Subject Classification: 46A45, 40C05. 1. Notations and preliminary results. For a given infinite matrix A = (anm)n,m≥1, the operators An, for any integer n ≥ 1, are defined by ∞ An(X) = anmxm, (1.1) m=1 where X = (xn)n≥1 is the series intervening in the second member being con- vergent. So, we are led to the study of the infinite linear system An(X) = bn,n= 1,2,..., (1.2) where B=(b n)n≥1 is a one-column matrix and X the unknown, see [2, 3, 5, 6, 7, 9]. Equation (1.2) can be written in the form AX = B, where AX = (An(X))n≥1.In this paper, we will also consider A as an operator from a sequence space into another sequence space. A Banach space E of complex sequences with the norm E is a BK space if P X → P X E each projection n : n is continuous. A BK space is said to have AK B = (b ) B = ∞ b e (see [8]) if for every n n≥1, n=1 m m, that is, ∞ bmem → 0 (n →∞). (1.3) m=N+1 E We shall write s, c, c0,andl∞ for the sets of all complex, convergent sequences, sequences convergent to zero, and bounded sequences, respectively. -
Sufficient Generalities About Topological Vector Spaces
(November 28, 2016) Topological vector spaces Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/fun/notes 2016-17/tvss.pdf] 1. Banach spaces Ck[a; b] 2. Non-Banach limit C1[a; b] of Banach spaces Ck[a; b] 3. Sufficient notion of topological vector space 4. Unique vectorspace topology on Cn 5. Non-Fr´echet colimit C1 of Cn, quasi-completeness 6. Seminorms and locally convex topologies 7. Quasi-completeness theorem 1. Banach spaces Ck[a; b] We give the vector space Ck[a; b] of k-times continuously differentiable functions on an interval [a; b] a metric which makes it complete. Mere pointwise limits of continuous functions easily fail to be continuous. First recall the standard [1.1] Claim: The set Co(K) of complex-valued continuous functions on a compact set K is complete with o the metric jf − gjCo , with the C -norm jfjCo = supx2K jf(x)j. Proof: This is a typical three-epsilon argument. To show that a Cauchy sequence ffig of continuous functions has a pointwise limit which is a continuous function, first argue that fi has a pointwise limit at every x 2 K. Given " > 0, choose N large enough such that jfi − fjj < " for all i; j ≥ N. Then jfi(x) − fj(x)j < " for any x in K. Thus, the sequence of values fi(x) is a Cauchy sequence of complex numbers, so has a limit 0 0 f(x). Further, given " > 0 choose j ≥ N sufficiently large such that jfj(x) − f(x)j < " .