# On the Structure of a Certain Class of Mixed Tsirelson Spaces

On the Structure of a Certain Class of Mixed Tsirelson Spaces We study Banach spaces of the form $${X = {T\left[ {\left( {{\theta_i} ,{{\mathcal{A}}_{n_i}}} \right)_{{i = 1}^\infty}} \right]}}$$ We call such a space a p-space, p∈[1,∞), if for every k the space $$T\left[ {\left( {{\theta_i} ,{\mathcal{A}}_{n_i}} \right)_{{i = 1}^k} } \right]$$ is isomorphic to ℓpk and the sequence (pk) strictly decreases to p. We examine the finite block representability of the spaces ℓr in a p-space proving that it depends not only on p but also on the sequences (pk) and (nk). Assuming that θi ni 1/q decreases to 0, where q is the conjugate exponent of p, we prove the existence of an asymptotic biorthogonal system in X and also that c 0 is finitely representable in X. Moreover we investigate the modified versions of p-spaces proving that, if nkm1/pkm-1/pkm-1 increases to infinity for a subsequence (nkm) , then ℓ1 embeds into X. We also investigate complemented minimality for the class of spaces $${T\left[ {\left( {{\theta_i} ,{\mathcal{M}}_{_i}} \right)_{{i = 1}^\infty} } \right]}$$ where $${\left( {\mathcal{M}_{_i}} \right)}$$ is either a subsequence of the sequence of Schreier classes ( $$\mathcal{S}$$ n)n ∈ N or a subsequence of ( $$\mathcal{A}$$ n)n ∈ N. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# On the Structure of a Certain Class of Mixed Tsirelson Spaces

, Volume 5 (3) – Oct 19, 2004
46 pages

/lp/springer_journal/on-the-structure-of-a-certain-class-of-mixed-tsirelson-spaces-4P4JxhSnFp
Publisher
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1011456204116
Publisher site
See Article on Publisher Site

### Abstract

We study Banach spaces of the form $${X = {T\left[ {\left( {{\theta_i} ,{{\mathcal{A}}_{n_i}}} \right)_{{i = 1}^\infty}} \right]}}$$ We call such a space a p-space, p∈[1,∞), if for every k the space $$T\left[ {\left( {{\theta_i} ,{\mathcal{A}}_{n_i}} \right)_{{i = 1}^k} } \right]$$ is isomorphic to ℓpk and the sequence (pk) strictly decreases to p. We examine the finite block representability of the spaces ℓr in a p-space proving that it depends not only on p but also on the sequences (pk) and (nk). Assuming that θi ni 1/q decreases to 0, where q is the conjugate exponent of p, we prove the existence of an asymptotic biorthogonal system in X and also that c 0 is finitely representable in X. Moreover we investigate the modified versions of p-spaces proving that, if nkm1/pkm-1/pkm-1 increases to infinity for a subsequence (nkm) , then ℓ1 embeds into X. We also investigate complemented minimality for the class of spaces $${T\left[ {\left( {{\theta_i} ,{\mathcal{M}}_{_i}} \right)_{{i = 1}^\infty} } \right]}$$ where $${\left( {\mathcal{M}_{_i}} \right)}$$ is either a subsequence of the sequence of Schreier classes ( $$\mathcal{S}$$ n)n ∈ N or a subsequence of ( $$\mathcal{A}$$ n)n ∈ N.

### Journal

PositivitySpringer Journals

Published: Oct 19, 2004

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