Complexity of weakly null sequences

Abstract

We introduce an ordinal index which measures the complexity of a weakly null sequence, and show that a construction due to J. Schreier can be iterated to produce for each alpha < omega_1, a weakly null sequence (x^{alpha}_n)_n in C(omega^{omega^{alpha}})) with complexity alpha. As in the Schreier example each of these is a sequence of indicator functions which is a suppression-1 unconditional basic sequence. These sequences are used to construct Tsirelson-like spaces of large index. We also show that this new ordinal index is related to the Lavrentiev index of a Baire-1 function and use the index to sharpen some results of Alspach and Odell on averaging weakly null sequences.

References in corpus (1)

• Banach Spaces Of The Type Of Tsirelson

Cited by in corpus (42)

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• How many operators do there exist on a Banach space?
• Classes of operators determined by ordinal indices
• Distorting Mixed Tsirelson Spaces
• On the hereditary proximity to ${\ell }_1$
• New examples of $c_0$-saturated Banach spaces II
• On the structure of the spreading models of a Banach space
• New examples of $c_0$-saturated Banach spaces
• Higher Order Spreading Models
• Arbitrarily distortable Banach spaces of higher order
• A dichotomy on Schreier sets
• Quantification of Banach-Saks properties of higher orders
• Proximity to ${\ell }_1$ and Distortion in Asymptotic ${\ell }_1$ Spaces
• The strength of Ramsey Theorem for coloring relatively large sets
• Minimality properties of Tsirelson type spaces
• \ell^1-spreading models in subspaces of mixed Tsirelson spaces
• A problem on spreading models
• Concerning the Bourgain $el{l}_1$ index of a Banach space
• Variants of the James Tree space
• \ell ^1-spreading models in mixed Tsirelson space
• Descriptive set theoretic methods applied to strictly singular and strictly cosingular operators
• An Ordinal Index on the Space of Strictly Singular Operators
• Operators on Asymptotic ${\ell }_p$ Spaces which are not Compact Perturbations of a Multiple of the Identity
• A Hereditarily Indecomposable asymptotic ${\ell }_2$ Banach space
• More Mixed Tsirelson Spaces That Are Not Isomorphic To Their Modified Versions
• Types of tightness in spaces with unconditional basis
• Constructing Banach ideals using upper ${\ell }_p$-estimates
• Proximity to ${\ell }_p$ and $c_0$ in Banach spaces
• Operators on (C[0,1]) preserving copies of asymptotic (\ell_1) spaces
• Counting Unions of Schreier Sets
• On the complemented subspaces of the Schreier spaces
• The Bourgain ell ^1-index of mixed Tsirelson space
• The ${\ell }^1$-index of Tsirelson type spaces
• A continuum of totally incomparable hereditarily indecomposable Banach spaces
• Unconditional structures of weakly null sequences
• Some more weak Hilbert spaces
• On certain equivalent norms on Tsirelson's space
• Modified mixed Tsirelson spaces