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Minimal projections with respect to various norms

100%
Aksoy A., Lewicki G.
Studia Mathematica
|
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.
2
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Some results on metric trees

100%
Aksoy A., Oikhberg T.
Banach Center Publications
|
2010
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tom 91
|
nr 1
9-34
EN
Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree (T, d) is a metric space such that between any two of its points there is a unique arc that is isometric to an interval in ℝ. We begin our investigation by examining isometric embeddings of metric trees into Banach spaces. We then investigate the possible images x₀ = π((x₁ + ... + xₙ)/n), where π is a contractive retraction from the ambient Banach space X onto T (such a π always exists) in order to understand the "metric" barycenter of a family of points x₁,...,xₙ in a tree T. Further, we consider the metric properties of trees such as their type and cotype. We identify various measures of compactness of metric trees (their covering numbers, ϵ-entropy and Kolmogorov widths) and the connections between them. Additionally, we prove that the limit of the sequence of Kolmogorov widths of a metric tree is equal to its ball measure of non-compactness.
3
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Compact widths in metric trees

100%
Aksoy A., Kinneberg K.
Banach Center Publications
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2011
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tom 92
|
nr 1
15-25
EN
The definition of n-width of a bounded subset A in a normed linear space X is based on the existence of n-dimensional subspaces. Although the concept of an n-dimensional subspace is not available for metric trees, in this paper, using the properties of convex and compact subsets, we present a notion of n-widths for a metric tree, called Tn-widths. Later we discuss properties of Tn-widths, and show that the compact width is attained. A relationship between the compact widths and Tn-widths is also obtained.
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