Freeman, Daniel B. (2009-08). Upper Estimates for Banach Spaces. Doctoral Dissertation. Thesis uri icon

abstract

  • We study the relationship of dominance for sequences and trees in Banach spaces. In the context of sequences, we prove that domination of weakly null sequences is a uniform property. More precisely, if $(v_i)$ is a normalized basic sequence and $X$ is a Banach space such that every normalized weakly null sequence in $X$ has a subsequence that is dominated by $(v_i)$, then there exists a uniform constant $C\geq1$ such that every normalized weakly null sequence in $X$ has a subsequence that is $C$-dominated by $(v_i)$. We prove as well that if $V=(v_i)_{i=1}^\infty$ satisfies some general conditions, then a Banach space $X$ with separable dual has subsequential $V$ upper tree estimates if and only if it embeds into a Banach space with a shrinking FDD which satisfies subsequential $V$ upper block estimates. We apply this theorem to Tsirelson spaces to prove that for all countable ordinals $\alpha$ there exists a Banach space $X$ with Szlenk index at most $\omega^{\alpha \omega +1}$ which is universal for all Banach spaces with Szlenk index at most $\omega^{\alpha\omega}$.
  • We study the relationship of dominance for
    sequences and trees in Banach spaces. In the context of sequences,
    we prove that domination of weakly null sequences is a uniform
    property. More precisely, if $(v_i)$ is a normalized basic sequence
    and $X$ is a Banach space such that every normalized weakly null
    sequence in $X$ has a subsequence that is dominated by $(v_i)$, then
    there exists a uniform constant $Cgeq1$ such that every normalized
    weakly null sequence in $X$ has a subsequence that is $C$-dominated
    by $(v_i)$. We prove as well that if $V=(v_i)_{i=1}^infty$
    satisfies some general conditions, then a Banach space $X$ with
    separable dual has subsequential $V$ upper tree estimates if and
    only if it embeds into a Banach space with a shrinking FDD which
    satisfies subsequential $V$ upper block estimates. We apply this
    theorem to Tsirelson spaces to prove that for all countable ordinals
    $alpha$ there exists a Banach space $X$ with Szlenk index at most
    $omega^{alpha omega +1}$ which is universal for all Banach spaces
    with Szlenk index at most $omega^{alphaomega}$.

publication date

  • August 2009