R. C. James's classical paper in the early 1950s introduced and investigated two special classes of bases, shrinking bases and boundedly complete bases. These two special classes of bases are used to characterize the reflexivity of a Banach space with a basis. My concern is to give a quantitative version of this classical result.
Let $X$ be a Banach space and define
$\gamma(X)=\sup\{|\lim\limits_{n}\lim\limits_{m}\langle x^{*}_{m},x_{n}\rangle-\lim\limits_{m}\lim\limits_{n}\langle x^{*}_{m},x_{n}\rangle|:(x_{n})_{n}$ is a sequence in $B_{X}$, $(x^{*}_{m})_{m}$ is a sequence in $B_{X^{*}}$ and all the involved limits exist$\}$.
Obviously, $X$ is reflexive if and only if $\gamma(X)=0$.
Let $(x_{n})_{n=1}^{\infty}$ be a basis for a Banach space $X$ with biorthogonal functionals $(x^{*}_{n})_{n=1}^{\infty}$. We set $$\textrm{sh}_{2}((x_{n})_{n=1}^{\infty})=\sup_{x^{*}\in B_{X^{*}}}\limsup_{n}\|x^{*}-\sum_{i=1}^{n}\langle x^{*},x_{i}\rangle x^{*}_{i}\|.$$ It is clear that $(x_{n})_{n}$ is shrinking if and only if $\textrm{sh}_{2}((x_{n})_{n=1}^{\infty})=0$. I observed that $\textrm{sh}_{2}((x_{n})_{n=1}^{\infty})\geq 1$ if $(x_{n})_{n}$ is not shrinking. My question is the following.
Question. Let $(x_{n})_{n=1}^{\infty}$ be a boundedly complete basis for a Banach space $X$ with biorthogonal functionals $(x^{*}_{n})_{n=1}^{\infty}$. Do there exist two constants $C_{1},C_{2}$ so that $$C_{1}\textrm{sh}_{2}((x_{n})_{n=1}^{\infty})\leq \gamma(X)\leq C_{2}\textrm{sh}_{2}((x_{n})_{n=1}^{\infty}) ?$$
According to my observation and R. C. James's result, the constant $C_{2}$ can be taken to be $2$. Furthermore, the constant $2$ is optimal. Indeed, $\gamma(\ell_{1})=2$ and $\textrm{sh}_{2}((e^{*}_{n})_{n=1}^{\infty})=1$, where $(e^{*}_{n})_{n=1}^{\infty}$ is the unit vector basis of $\ell_{1}$. But I do not know whether the constant $C_{1}$ exists.
Thank you !