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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
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A proof of the Grünbaum conjecture

61%
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
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Minimal projections with respect to various norms

61%
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.
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Two-dimensional real symmetric spaces with maximal projection constant

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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.
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Minimal multi-convex projections

61%
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 $C^{L}[0,1]$.
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Symmetric subspaces of $l_1$ with large projection constants

61%
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 $λ(V^k) ≥ λ(V^k,l_1) > λ(l_2^{(k)})$.
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