Characterizations of elements of a double dual Banach space and their canonical reproductions

For every element x** in the double dual of a separable Banach space X there exists the sequence ${\displaystyle \left(x^{\left(2n\right)}\right)}$ of the canonical reproductions of x** in the even-order duals of X. In this paper we prove that every such sequence defines a spreading model for X. Using this result we characterize the elements of X**╲ X which belong to the class ${\displaystyle B_1\left(X\right)╲B_{\frac{1}{2}}\left(X\right)}$ (resp. to the class ${\displaystyle B_{\frac{1}{4}}\left(X\right)}$) as the elements with the sequence ${\displaystyle \left(x^{\left(2n\right)}\right)}$ equivalent to the usual basis of ${\displaystyle {\ell }^1}$ (resp. as the elements with the sequence ${\displaystyle (x^{(4n-2)}-x^{\left(4n\right)})}$ equivalent to the usual basis of ${\displaystyle c_0}$). Also, by analogous conditions but of isometric nature, we characterize the embeddability of ${\displaystyle {\ell }^1}$ (resp. ${\displaystyle c_0}$) in X.