Wyniki wyszukiwania - Biblioteka Nauki

1

Best approximation in spaces of bounded linear operators

100%

Lewicki G.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Chapter 0...............................................................................................................................................................................5 0.1. Introduction..................................................................................................................................................................5 0.2. Preliminary results.......................................................................................................................................................9 Chapter I..............................................................................................................................................................................16 I.1. Best approximation in finite-dimensional subspaces of ℒ(B,D)....................................................................................16 I.2. Kolmogorov's type criteria for spaces of compact operators; general case.................................................................26 I.3. Criteria for the space $K(C_K(T))$.............................................................................................................................30 I.4. The case of sequence spaces....................................................................................................................................38 Chapter II.............................................................................................................................................................................43 II.1. Extensions of linear operators from hyperplanes of $l^{(n)}_∞$.................................................................................43 II.2. Minimal projections onto hyperplanes of $l^{(n)}_1$...................................................................................................52 II.3. Strongly unique minimal projections onto hyperplanes of $l^{(n)}_∞$ and $l^{(n)}_1$...............................................59 II.4. Minimal projections onto subspaces of $l^{(n)}_∞$ of codimension two......................................................................71 II.5. Uniqueness of minimal projections onto subspace of $l^{(n)}_∞$ of codimension two................................................75 II.6. Strong unicity criterion in some space of operators....................................................................................................79 Chapter III.............................................................................................................................................................................83 III.1. Extensions of linear operators from finite-dimensional subspaces I...........................................................................83 III.2. Extensions of linear operators from finite-dimensional subspaces II..........................................................................90 III.3. Algorithms for seeking the constant $W_m$..............................................................................................................97 References..........................................................................................................................................................................99 Index..................................................................................................................................................................................102 Index of symbols................................................................................................................................................................102

2

Czas Jagiellonów bez zagonówCzas Jagiellonów bez zagonów

93%

Lewicki G.

Res Publica Nowa

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2014

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nr 3

39-41

EN

Even the army of innovative Poles will not help, if Poland cannot create new industries in the next twenty years. It is nice “The Economist” suggests Poland may be entering second Jagiellonian Age, but in fact we are more threatened by technological collapse. Thus we may be compared to Jagiellonians without fields, which were a key to Poland’s economic growth of the past. In a healthy society both disruptive and sustaining innovation drive technological progress. However, in Poland there is not much to sustain. The pathology of post-communist privatization left Poland without strategic industries. Now only new industries may save us.

3

Asymptotic estimate of absolute projection constants

93%

Lewicki G.

|

2013

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tom 51

51-60

EN

In this note we construct a sequence of real, k-dimensional symmetric spaces $ Y^k $ satisfying $ \text{lim} \ \underset{k}{ \text{inf} } \ \lambda_k^S // \sqrt{k} \ \geq \ \text{lim} \ \underset{k}{\text{inf}} \ \lambda(Y^k, l_1) // \sqrt{k} \ \gt \ \underset{w in [0,a_2]}{\text{max}} h(w) \ \gt \ 1//(2 - \sqrt{2//\pi})$, where $\lambda_k^S$ is defined by (4) and $h(w)=a_1^2\sqrt{2//\pi}+2a_1\sqrt{a_2^2-w^2}+w\sqrt{a_2^2 - w^2}$ with $a_1 = 1// (2-\sqrt{2//\pi}) $ and $a_2=1-a_1$. This improves the lower bound obtained in [3], Th. 5.3 by $\text{max}_{w \in [0,a_2]} h(w)$.

4

Minimal projections with respect to various norms

58%

Aksoy A. , Lewicki G.

Studia Mathematica

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2012

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tom 210

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nr 1

1-16

EN

A theorem of Rudin permits us to determine minimal projections not only with respect to the operator norm but with respect to various norms on operator ideals and with respect to numerical radius. We prove a general result about N-minimal projections where N is a convex and lower semicontinuous (with respect to the strong operator topology) function and give specific examples for the cases of norms or seminorms of p-summing, p-integral and p-nuclear operator ideals.

5

Minimal multi-convex projections

58%

Lewicki G. , Prophet M.

Studia Mathematica

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2007

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tom 178

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nr 2

99-124

EN

We say that a function from $X = C^{L}[0,1]$ is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Πₙ ⊂ X, where Πₙ denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n-1 and k = n does such a projection exist. So let us consider instead a more general "shape" to preserve. Let σ = ( σ₀, σ₁, ..., σₙ) be an (n + 1)-tuple with $σ_{i} ∈ {0, 1}$; we say f ∈ X is multi-convex if $f^{(i)} ≥ 0$ for i such that $σ_{i} = 1$. We characterize those σ for which there exists a projection onto V preserving the multi-convex shape. For those shapes able to be preserved via a projection, we construct (in all but one case) a minimal norm multi-convex preserving projection. Out of necessity, we include some results concerning the geometrical structure of $C^{L}[0,1]$.

6

Two-dimensional real symmetric spaces with maximal projection constant

58%

Chalmers B. L. , Lewicki G.

Annales Polonici Mathematici

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2000

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tom 73

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nr 2

119-134

EN

Let V be a two-dimensional real symmetric space with unit ball having 8n extreme points. Let λ(V) denote the absolute projection constant of V. We show that $λ(V) ≤ λ(V_n)$ where $V_n$ is the space whose ball is a regular 8n-polygon. Also we reprove a result of [1] and [5] which states that $4/π = λ(l₂^{(2)}) ≥ λ(V)$ for any two-dimensional real symmetric space V.

7

A proof of the Grünbaum conjecture

58%

Chalmers B. , Lewicki G.

Studia Mathematica

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2010

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tom 200

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nr 2

103-129

EN

Let V be an n-dimensional real Banach space and let λ(V) denote its absolute projection constant. For any N ∈ N with N ≥ n define $λₙ^{N} = sup{λ(V): dim(V) = n,V ⊂ l^{(N)}_{∞}}$, λₙ = sup{λ(V): dim(V) = n}. A well-known Grünbaum conjecture [Trans. Amer. Math. Soc. 95 (1960)] says that λ₂ = 4/3. König and Tomczak-Jaegermann [J. Funct. Anal. 119 (1994)] made an attempt to prove this conjecture. Unfortunately, their Proposition 3.1, used in the proof, is incorrect. In this paper a complete proof of the Grünbaum conjecture is presented

8

Symmetric subspaces of $l_1$ with large projection constants

58%

Chalmers B. , Lewicki G.

Studia Mathematica

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1999

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tom 134

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nr 2

119-133

EN

We construct k-dimensional (k ≥ 3) subspaces $V^k$ of $l_1$, with a very simple structure and with projection constant satisfying $λ(V^k) ≥ λ(V^k,l_1) > λ(l_2^{(k)})$.