This is an announcement for the paper Uniform subsequential estimates on weakly null sequences” by M. Brixeyhttps://arxiv.org/search/math?searchtype=author&query=Brixey%2C+M, R.M. Causeyhttps://arxiv.org/search/math?searchtype=author&query=Causey%2C+R+M, P. Frankarthttps://arxiv.org/search/math?searchtype=author&query=Frankart%2C+P.

Abstract: We provide a generalization of two results of Knaust and Odell from \cite{KO2} and \cite{KO}. We prove that if $X$ is a Banach space and $(g_n)_{n=1}^\infty$ is a right dominant Schauder basis such that every normalized, weakly null sequence in $X$ admits a subsequence dominated by a subsequence of $(g_n)_{n=1}^\infty$, then there exists a constant $C$ such that every normalized, weakly null sequence in $X$ admits a subsequence $C$-dominated by a subsequence of $(g_n)_{n=1}^\infty$. We also prove that if every spreading model generated by a normalized, weakly null sequence in $X$ is dominated by some spreading model generated by a subsequence of $(g_n)_{n=1}^\infty$, then there exists $C$ such that every spreading model generated by a normalized, weakly null sequence in $X$ is $C$-dominated by every spreading model generated by a subsequence of $(g_n)_{n=1}^\infty$. We also prove a single, ordinal-quantified result which unifies and interpolates between these two results.

https://arxiv.org/abs/1912.13443