Let $X, Y$ be Banach spaces and let $T\colon X\to Y$ be a bounded linear operator. Assume that $X$ admits a Markushevic basis. Does $\overline{T(X)}\subseteq Y$ admit a Markushevic basis as well?
What if $(x_\alpha, x^*_\alpha)$ is a Markushevic of $X$. Let $M$ be a maximal linearly independent subset of $\{Tx_\alpha\colon \alpha\in A\}$. Is there any chance that there is a Markushevic basis for $\overline{T(X)}$ of this form $\{ (Tx_{\alpha}, ?)\colon\; Tx_\alpha\in M \} $?