DOCSLIB.ORG

247A Notes on Lorentz Spaces Definition 1. for 1 ≤ P < ∞ and F : R D → C We Define (1) Weak(Rd) := Sup Λ∣∣{X

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

247A Notes on Lorentz spaces Definition 1. For 1 ≤ p < 1 and f : Rd ! C we define ∗ 1=p (1) kfkLp ( d) := sup λ fx : jf(x)j > λg weak R λ>0 and the weak Lp space p d ∗ L ( ) := f : kfk p d < 1 : weak R Lweak(R ) p −p Equivalently, f 2 Lweak if and only if jfx : jf(x)j > λgj . λ . Warning. The quantity in (1) does not define a norm. This is the reason we append the asterisk to the usual norm notation. To make a side-by-side comparison with the usual Lp norm, we note that ZZ 1=p p−1 kfkLp = pλ dλ dx 0≤λ<jf(x)j Z 1 dλ1=p = jfjfj > λgj pλp 0 λ 1=p 1=p = p λjfjfj > λgj p dλ L ((0;1); λ ) and, with the convention that p1=1 = 1, ∗ 1=1 1=p kfkLp = p λjfjfj > λgj 1 dλ : weak L ((0;1); λ ) This suggests the following definition. Definition 2. For 1 ≤ p < 1 and 1 ≤ q ≤ 1 we define the Lorentz space Lp;q(Rd) as the space of measurable functions f for which ∗ 1=q 1=p (2) kfkLp;q := p λjfjfj > λgj q dλ < 1: L ( λ ) p;p p p;1 p From the discussion above, we see that L = L and L = Lweak. Again ∗ k · kLp;q is not a norm in general. Nevertheless, it is positively homogeneous: for all a 2 C, ∗ −1 1=p ∗ (3) kafkLp;q = λ fjfj > jaj λg Lq (dλ/λ) = jaj · kfkLp;q (strictly the case a = 0 should receive separate treatment). In lieu of the triangle inequality, we have the following: ∗ 1=p kf + gkLp;q = λ fjf + gj > λg Lq (dλ/λ) 1 1 1=p ≤ λ fjfj > 2 λg + fjgj > 2 λg Lq (dλ/λ) 1 1=p 1 1=p ≤ λ fjfj > 2 λg Lq (dλ/λ) + λ fjgj > 2 λg Lq (dλ/λ) by the subadditivity of fractional powers and the triangle inequality in Lq(dλ/λ). Thus ∗ ∗ ∗ (4) kf + gkLp;q ≤ 2kfkLp;q + 2kgkLp;q : Combining (3), (4), and the fact that kfkLp;q = 0 implies f ≡ 0 almost every- ∗ where, we see that k · kLp;q obeys the axioms of a quasi-norm. When p > 1, this quasi-norm is equivalent to an actual norm (see below). When p = 1 and q 6= 1, there cannot be a norm that is equivalent to our quasi-norm. However there is a metric that generates the same topology. In either case, we obtain a complete metric space. ∗ ∗ Notice that (i) if jfj ≥ jgj then kfkLp;q ≥ kgkLp;q and (ii) The quasi-norms are ∗ ∗ rearrangement invariant, which is to say that kfkLp;q = kf ◦φkLp;q for any measure preserving bijection φ : Rd ! Rd. 1 2 p;q P Proposition 3. Given f 2 L , we write f = fm where fm(x) := f(x)χfx:2m≤|f(x)j<2m+1g: Then ∗ kfk p;q ≈p;q kfmk p d L Lx(R ) q `m(Z) p;q1 p;q2 In particular, L ⊆ L whenever q1 ≤ q2. P m Proof. It suffices to consider f of the form f = 2 χEm with disjoint sets Em (cf. m m+1 Em = f2 ≤ jfj < 2 g). Now Z 1 ∗ q q q=p dλ kfkLp;q = p λ jfjfj > λgj 0 λ Z 2m q=p X q X dλ = p λ jEnj m−1 λ m 2 n≥m 1=p q X m X ≈ 2 jEnj : m n≥m To obtain a lower bound, we keep only the summand n = m; for an upper bound, we use the subadditivity of fractional powers. This yields m 1=p ∗ X m 1=p (5) 2 jE j q kfk 2 jE j : m ` . Lp;q . n m q m≤n `m m m 1=p p As k2 χEm kL = 2 jEmj , we have our desired lower bound. To obtain the upper bound, we use the triangle inequality in `q(Z): 1 1 X −k m+k X −k m RHS(5) = 2 k2 χE kLp ≤ 2 k2 χE kLp m+k m q q `m k=0 `m k=0 This completes the proof of the upper bound. Lemma 4. Given 1 ≤ q < 1 and a finite set A ⊂ 2Z, X X q q X Aq ≤ A ≤ 2 max A ≤ 2q Aq A2A where all sums are over A 2 A. More generally, for any subset A of a geometric series and any 0 < q < 1, q X q X A ≈ A where the implicit constants depend on q and the step size of the geometric series. Proposition 5. For 1 < p < 1 and 1 ≤ q ≤ 1, R ∗ ∗ (6) sup j fgj : kgkLp0;q0 ≤ 1 ≈ kfkLp;q : Indeed, LHS(6) defines a norm on Lp;q. Note that by (6), this norm is equivalent to our quasi-norm. Moreover, under this norm, Lp;q is a Banach space and when 0 0 q 6= 1, the dual Banach space is Lp ;q , under the natural pairing. R Remark. When p = 1 (and q 6= 1), the LHS(6) is typically infinite; indeed, E jfj may well be infinite even for some set E of finite measure. In fact, there there cannot be a norm on Lp;q equivalent to our quasi-norm. For example, the impossibility of finding an equivalent norm for L1;1(R) can be deduced by computing N ∗ N X −1 X −1 ∗ jx − nj ≈ N log(N) and jx − nj L1;1 ≈ N: 1;1 n=0 L n=0 3 Proof. Because the quasi-norm is positively homogeneous, we need only verify (6) in the case that f and g have quasi-norm comparable to one. We may also assume P n P m that f = 2 χFn and g = 2 χEm . By the normalization just mentioned, 0 X n 1=pq X m 1=p0 q (7) 2 jFnj ≈ 1 ≈ 2 jEmj n m Combining the above with Lemma 4, we obtain q X X n 1=p X X n 1=pq (8) 2 A ≈ 2 jFnj ≈ 1: A22Z n:jFn|≈A A22Z n:jFn|≈A and similarly for g. Now we compute: Z X n m jfgj dx = 2 2 jFn \ Emj n;m X X n X m ≤ 2 · min(A; B) · 2 A;B22Z n:jFn|∼A m:jEm|∼B 1 1 0 X X n 1=p A 0 B p X m 1=p ≤ 2 A · min p ; · 2 B : B A A;B22Z n:jFn|∼A m:jEm|∼B Notice that this has the structure of a bilinear form: two vectors (indexed over 2Z) with a matrix sitting between them. Moreover, by Schur's test, the matrix is a bounded operator on `q(2Z). Thus, Z X n 1=p X m 1=p0 jfgj dx . 2 A · 2 B ≈ 1 0 `q (A22Z) `q (B22Z) n:jFn|∼A m:jEm|∼B by (8) and the corresponding statement for g. This completes proof of the . part P n p;q of (6). We turn now to the opposite inequality. Given f = 2 χFn 2 L , we choose q−1 1 − 1 q−p X n p p0 X n(q−1) p g = 2 jFnj jFnj χFn = 2 jFnj χFn : n n Then Z q−1 q q 1 1− 1 1 X n p n p0 X n p ∗ fg = 2 jFnj 2 jFnj = 2 jFnj ≈ kfkLp;q ≈ 1: n n ∗ It remains to show that kgk 0 0 1. By Proposition 3, Lp ;q . q0=p0 q0 0 q−p ∗ X q X n(q−1) p kgk p0;q0 ≈ A jFnj where n 2 N(A) , 2 jFnj ≈ A: L A22Z n2N(A) Notice that for each A, the sum in n is over part of a geometric series; indeed, p −n p(q−1) n 2 N(A) () jFnj ≈ A q−p 2 q−p : Thus Lemma 4 applies and yields 0 ∗ q X q0 X q0=p0 X nq q=p kgkLp0;q0 ≈ A jFnj ≈ 2 jFnj ≈ 1: A22Z n2N(A) n This provides the needed bound on g and so completes the proof of (6). The fact that LHS(6) is indeed a norm is a purely abstract statement about vector spaces and (separating) linear functionals. The proof that Lp;q is complete in this norm differs little from the usual Riesz{Fischer argument. p;q 1=p Let ` be a continuous linear functional on L . By definition, j`(χE)j . jEj and so the measure E 7! `(χE) is absolutely continuous with respect to Lebesgue measure and so is represented by some locally L1 function g. This is the Radon{ Nikodym Theorem. By linearity this representation of the functional extends to 4 simple functions. Boundedness when tested against simple functions suffices to 0 0 show that g 2 Lp ;q . When q 6= 1, the simple functions are dense in Lp;q and so our linear functional admits the desired representation. When q = 1 the simple functions are not dense. For example, one cannot approximate jxj−d=p 2 Lp;1(Rd) by simple functions. Indeed, inspired by the Banach limit linear functionals on `1(Z) we can construct a non-trivial linear functional on Lp;1 that vanishes on simple functions. Let L denote the vector space of f 2 Lp;1 such that `(f) := lim jxjd=pf(x) exists. x!0 Notice that L contains the simple functions and that ` vanishes on these.
Recommended publications
  • Duality for Outer $ L^ P \Mu (\Ell^ R) $ Spaces and Relation to Tent Spaces

    Duality for Outer $ L^ P \Mu (\Ell^ R) $ Spaces and Relation to Tent Spaces

    p r DUALITY FOR OUTER Lµpℓ q SPACES AND RELATION TO TENT SPACES MARCO FRACCAROLI p r Abstract. We prove that the outer Lµpℓ q spaces, introduced by Do and Thiele, are isomorphic to Banach spaces, and we show the expected duality properties between them for 1 ă p ď8, 1 ď r ă8 or p “ r P t1, 8u uniformly in the finite setting. In the case p “ 1, 1 ă r ď8, we exhibit a counterexample to uniformity. We show that in the upper half space setting these properties hold true in the full range 1 ď p,r ď8. These results are obtained via greedy p r decompositions of functions in Lµpℓ q. As a consequence, we establish the p p r equivalence between the classical tent spaces Tr and the outer Lµpℓ q spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on Rd. 1. Introduction The Lp theory for outer measure spaces discussed in [13] generalizes the classical product, or iteration, of weighted Lp quasi-norms. Since we are mainly interested in positive objects, we assume every function to be nonnegative unless explicitly stated. We first focus on the finite setting. On the Cartesian product X of two finite sets equipped with strictly positive weights pY,µq, pZ,νq, we can define the classical product, or iterated, L8Lr,LpLr spaces for 0 ă p, r ă8 by the quasi-norms 1 r r kfkL8ppY,µq,LrpZ,νqq “ supp νpzqfpy,zq q yPY zÿPZ ´1 r 1 “ suppµpyq ωpy,zqfpy,zq q r , yPY zÿPZ p 1 r r p kfkLpppY,µq,LrpZ,νqq “ p µpyqp νpzqfpy,zq q q , arXiv:2001.05903v1 [math.CA] 16 Jan 2020 yÿPY zÿPZ where we denote by ω “ µ b ν the induced weight on X.
  • ON SEQUENCE SPACES DEFINED by the DOMAIN of TRIBONACCI MATRIX in C0 and C Taja Yaying and Merve ˙Ilkhan Kara 1. Introduction Th

    ON SEQUENCE SPACES DEFINED by the DOMAIN of TRIBONACCI MATRIX in C0 and C Taja Yaying and Merve ˙Ilkhan Kara 1. Introduction Th

    Korean J. Math. 29 (2021), No. 1, pp. 25{40 http://dx.doi.org/10.11568/kjm.2021.29.1.25 ON SEQUENCE SPACES DEFINED BY THE DOMAIN OF TRIBONACCI MATRIX IN c0 AND c Taja Yaying∗;y and Merve Ilkhan_ Kara Abstract. In this article we introduce tribonacci sequence spaces c0(T ) and c(T ) derived by the domain of a newly defined regular tribonacci matrix T: We give some topological properties, inclusion relations, obtain the Schauder basis and determine α−; β− and γ− duals of the spaces c0(T ) and c(T ): We characterize certain matrix classes (c0(T );Y ) and (c(T );Y ); where Y is any of the spaces c0; c or `1: Finally, using Hausdorff measure of non-compactness we characterize certain class of compact operators on the space c0(T ): 1. Introduction Throughout the paper N = f0; 1; 2; 3;:::g and w is the space of all real valued sequences. By `1; c0 and c; we mean the spaces all bounded, null and convergent sequences, respectively. Also by `p; cs; cs0 and bs; we mean the spaces of absolutely p-summable, convergent, null and bounded series, respectively, where 1 ≤ p < 1: We write φ for the space of all sequences that terminate in zero. Moreover, we denote the space of all sequences of bounded variation by bv: A Banach space X is said to be a BK-space if it has continuous coordinates. The spaces `1; c0 and c are BK- spaces with norm kxk = sup jx j : Here and henceforth, for simplicity in notation, `1 k k the summation without limit runs from 0 to 1: Also, we shall use the notation e = (1; 1; 1;:::) and e(k) to be the sequence whose only non-zero term is 1 in the kth place for each k 2 N: Let X and Y be two sequence spaces and let A = (ank) be an infinite matrix of real th entries.
  • Superadditive and Subadditive Transformations of Integrals and Aggregation Functions

    Superadditive and Subadditive Transformations of Integrals and Aggregation Functions

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Portsmouth University Research Portal (Pure) Superadditive and subadditive transformations of integrals and aggregation functions Salvatore Greco⋆12, Radko Mesiar⋆⋆34, Fabio Rindone⋆⋆⋆1, and Ladislav Sipeky†3 1 Department of Economics and Business 95029 Catania, Italy 2 University of Portsmouth, Portsmouth Business School, Centre of Operations Research and Logistics (CORL), Richmond Building, Portland Street, Portsmouth PO1 3DE, United Kingdom 3 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering Slovak University of Technology 810 05 Bratislava, Slovakia 4 Institute of Theory of Information and Automation Czech Academy of Sciences Prague, Czech Republic Abstract. We propose the concepts of superadditive and of subadditive trans- formations of aggregation functions acting on non-negative reals, in particular of integrals with respect to monotone measures. We discuss special properties of the proposed transforms and links between some distinguished integrals. Superaddi- tive transformation of the Choquet integral, as well as of the Shilkret integral, is shown to coincide with the corresponding concave integral recently introduced by Lehrer. Similarly the transformation of the Sugeno integral is studied. Moreover, subadditive transformation of distinguished integrals is also discussed. Keywords: Aggregation function; superadditive and subadditive transformations; fuzzy integrals. 1 Introduction The concepts of subadditivity and superadditivity are very important in economics. For example, consider a production function A : Rn R assigning to each vector of pro- + → + duction factors x = (x1,...,xn) the corresponding output A(x1,...,xn). If one has available resources given by the vector x =(x1,..., xn), then, the production function A assigns the output A(x1,..., xn).
  • The Semi-M Property for Normed Riesz Spaces Compositio Mathematica, Tome 34, No 2 (1977), P

    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.
  • Chapter 4 the Lebesgue Spaces

    Chapter 4 the Lebesgue Spaces

    Chapter 4 The Lebesgue Spaces In this chapter we study Lp-integrable functions as a function space. Knowledge on functional analysis required for our study is briefly reviewed in the first two sections. In Section 1 the notions of normed and inner product spaces and their properties such as completeness, separability, the Heine-Borel property and espe- cially the so-called projection property are discussed. Section 2 is concerned with bounded linear functionals and the dual space of a normed space. The Lp-space is introduced in Section 3, where its completeness and various density assertions by simple or continuous functions are covered. The dual space of the Lp-space is determined in Section 4 where the key notion of uniform convexity is introduced and established for the Lp-spaces. Finally, we study strong and weak conver- gence of Lp-sequences respectively in Sections 5 and 6. Both are important for applications. 4.1 Normed Spaces In this and the next section we review essential elements of functional analysis that are relevant to our study of the Lp-spaces. You may look up any book on functional analysis or my notes on this subject attached in this webpage. Let X be a vector space over R. A norm on X is a map from X ! [0; 1) satisfying the following three \axioms": For 8x; y; z 2 X, (i) kxk ≥ 0 and is equal to 0 if and only if x = 0; (ii) kαxk = jαj kxk, 8α 2 R; and (iii) kx + yk ≤ kxk + kyk. The pair (X; k·k) is called a normed vector space or normed space for short.
  • Baire Category Theorem and Uniform Boundedness Principle)

    Baire Category Theorem and Uniform Boundedness Principle)

    Functional Analysis (WS 19/20), Problem Set 3 (Baire Category Theorem and Uniform Boundedness Principle) Baire Category Theorem B1. Let (X; k·kX ) be an innite dimensional Banach space. Prove that X has uncountable Hamel basis. Note: This is Problem A2 from Problem Set 1. B2. Consider subset of bounded sequences 1 A = fx 2 l : only nitely many xk are nonzerog : Can one dene a norm on A so that it becomes a Banach space? Consider the same question with the set of polynomials dened on interval [0; 1]. B3. Prove that the set L2(0; 1) has empty interior as the subset of Banach space L1(0; 1). B4. Let f : [0; 1) ! [0; 1) be a continuous function such that for every x 2 [0; 1), f(kx) ! 0 as k ! 1. Prove that f(x) ! 0 as x ! 1. B5.( Uniform Boundedness Principle) Let (X; k · kX ) be a Banach space and (Y; k · kY ) be a normed space. Let fTαgα2A be a family of bounded linear operators between X and Y . Suppose that for any x 2 X, sup kTαxkY < 1: α2A Prove that . supα2A kTαk < 1 Uniform Boundedness Principle U1. Let F be a normed space C[0; 1] with L2(0; 1) norm. Check that the formula 1 Z n 'n(f) = n f(t) dt 0 denes a bounded linear functional on . Verify that for every , F f 2 F supn2N j'n(f)j < 1 but . Why Uniform Boundedness Principle is not satised in this case? supn2N k'nk = 1 U2.( pointwise convergence of operators) Let (X; k · kX ) be a Banach space and (Y; k · kY ) be a normed space.
  • Burkholder's Inequalities in Noncommutative Lorentz

    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 .
  • 3 Bilinear Forms & Euclidean/Hermitian Spaces

    3 Bilinear Forms & Euclidean/Hermitian Spaces

    3 Bilinear Forms & Euclidean/Hermitian Spaces Bilinear forms are a natural generalisation of linear forms and appear in many areas of mathematics. Just as linear algebra can be considered as the study of `degree one' mathematics, bilinear forms arise when we are considering `degree two' (or quadratic) mathematics. For example, an inner product is an example of a bilinear form and it is through inner products that we define the notion of length in n p 2 2 analytic geometry - recall that the length of a vector x 2 R is defined to be x1 + ... + xn and that this formula holds as a consequence of Pythagoras' Theorem. In addition, the `Hessian' matrix that is introduced in multivariable calculus can be considered as defining a bilinear form on tangent spaces and allows us to give well-defined notions of length and angle in tangent spaces to geometric objects. Through considering the properties of this bilinear form we are able to deduce geometric information - for example, the local nature of critical points of a geometric surface. In this final chapter we will give an introduction to arbitrary bilinear forms on K-vector spaces and then specialise to the case K 2 fR, Cg. By restricting our attention to thse number fields we can deduce some particularly nice classification theorems. We will also give an introduction to Euclidean spaces: these are R-vector spaces that are equipped with an inner product and for which we can `do Euclidean geometry', that is, all of the geometric Theorems of Euclid will hold true in any arbitrary Euclidean space.
  • A Mini-Introduction to Information Theory

    A Mini-Introduction to Information Theory

    A Mini-Introduction To Information Theory Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract This article consists of a very short introduction to classical and quantum information theory. Basic properties of the classical Shannon entropy and the quantum von Neumann entropy are described, along with related concepts such as classical and quantum relative entropy, conditional entropy, and mutual information. A few more detailed topics are considered in the quantum case. arXiv:1805.11965v5 [hep-th] 4 Oct 2019 Contents 1 Introduction 2 2 Classical Information Theory 2 2.1 ShannonEntropy ................................... .... 2 2.2 ConditionalEntropy ................................. .... 4 2.3 RelativeEntropy .................................... ... 6 2.4 Monotonicity of Relative Entropy . ...... 7 3 Quantum Information Theory: Basic Ingredients 10 3.1 DensityMatrices .................................... ... 10 3.2 QuantumEntropy................................... .... 14 3.3 Concavity ......................................... .. 16 3.4 Conditional and Relative Quantum Entropy . ....... 17 3.5 Monotonicity of Relative Entropy . ...... 20 3.6 GeneralizedMeasurements . ...... 22 3.7 QuantumChannels ................................... ... 24 3.8 Thermodynamics And Quantum Channels . ...... 26 4 More On Quantum Information Theory 27 4.1 Quantum Teleportation and Conditional Entropy . ......... 28 4.2 Quantum Relative Entropy And Hypothesis Testing . ......... 32 4.3 Encoding
  • Representation of Bilinear Forms in Non-Archimedean Hilbert Space by Linear Operators

    Representation of Bilinear Forms in Non-Archimedean Hilbert Space by Linear Operators

    Comment.Math.Univ.Carolin. 47,4 (2006)695–705 695 Representation of bilinear forms in non-Archimedean Hilbert space by linear operators Toka Diagana This paper is dedicated to the memory of Tosio Kato. Abstract. The paper considers representing symmetric, non-degenerate, bilinear forms on some non-Archimedean Hilbert spaces by linear operators. Namely, upon making some assumptions it will be shown that if φ is a symmetric, non-degenerate bilinear form on a non-Archimedean Hilbert space, then φ is representable by a unique self-adjoint (possibly unbounded) operator A. Keywords: non-Archimedean Hilbert space, non-Archimedean bilinear form, unbounded operator, unbounded bilinear form, bounded bilinear form, self-adjoint operator Classification: 47S10, 46S10 1. Introduction Representing bounded or unbounded, symmetric, bilinear forms by linear op- erators is among the most attractive topics in representation theory due to its significance and its possible applications. Applications include those arising in quantum mechanics through the study of the form sum associated with the Hamil- tonians, mathematical physics, symplectic geometry, variational methods through the study of weak solutions to some partial differential equations, and many oth- ers, see, e.g., [3], [7], [10], [11]. This paper considers representing symmetric, non- degenerate, bilinear forms defined over the so-called non-Archimedean Hilbert spaces Eω by linear operators as it had been done for closed, positive, symmetric, bilinear forms in the classical setting, see, e.g., Kato [11, Chapter VI, Theo- rem 2.23, p. 331]. Namely, upon making some assumptions it will be shown that if φ : D(φ) × D(φ) ⊂ Eω × Eω 7→ K (K being the ground field) is a symmetric, non-degenerate, bilinear form, then there exists a unique self-adjoint (possibly unbounded) operator A such that (1.1) φ(u, v)= hAu, vi, ∀ u ∈ D(A), v ∈ D(φ) where D(A) and D(φ) denote the domains of A and φ, respectively.
  • Linear Operators and Adjoints

    Linear Operators and Adjoints

    Chapter 6 Linear operators and adjoints Contents Introduction .......................................... ............. 6.1 Fundamentals ......................................... ............. 6.2 Spaces of bounded linear operators . ................... 6.3 Inverseoperators.................................... ................. 6.5 Linearityofinverses.................................... ............... 6.5 Banachinversetheorem ................................. ................ 6.5 Equivalenceofspaces ................................. ................. 6.5 Isomorphicspaces.................................... ................ 6.6 Isometricspaces...................................... ............... 6.6 Unitaryequivalence .................................... ............... 6.7 Adjoints in Hilbert spaces . .............. 6.11 Unitaryoperators ...................................... .............. 6.13 Relations between the four spaces . ................. 6.14 Duality relations for convex cones . ................. 6.15 Geometric interpretation of adjoints . ............... 6.15 Optimization in Hilbert spaces . .............. 6.16 Thenormalequations ................................... ............... 6.16 Thedualproblem ...................................... .............. 6.17 Pseudo-inverseoperators . .................. 6.18 AnalysisoftheDTFT ..................................... ............. 6.21 6.1 Introduction The field of optimization uses linear operators and their adjoints extensively. Example. differentiation, convolution, Fourier transform,
  • Functional Analysis 1 Winter Semester 2013-14

    Functional Analysis 1 Winter Semester 2013-14

    Functional analysis 1 Winter semester 2013-14 1. Topological vector spaces Basic notions. Notation. (a) The symbol F stands for the set of all reals or for the set of all complex numbers. (b) Let (X; τ) be a topological space and x 2 X. An open set G containing x is called neigh- borhood of x. We denote τ(x) = fG 2 τ; x 2 Gg. Definition. Suppose that τ is a topology on a vector space X over F such that • (X; τ) is T1, i.e., fxg is a closed set for every x 2 X, and • the vector space operations are continuous with respect to τ, i.e., +: X × X ! X and ·: F × X ! X are continuous. Under these conditions, τ is said to be a vector topology on X and (X; +; ·; τ) is a topological vector space (TVS). Remark. Let X be a TVS. (a) For every a 2 X the mapping x 7! x + a is a homeomorphism of X onto X. (b) For every λ 2 F n f0g the mapping x 7! λx is a homeomorphism of X onto X. Definition. Let X be a vector space over F. We say that A ⊂ X is • balanced if for every α 2 F, jαj ≤ 1, we have αA ⊂ A, • absorbing if for every x 2 X there exists t 2 R; t > 0; such that x 2 tA, • symmetric if A = −A. Definition. Let X be a TVS and A ⊂ X. We say that A is bounded if for every V 2 τ(0) there exists s > 0 such that for every t > s we have A ⊂ tV .