Series representation of compact linear operators in Banach spaces

Let $p\in (1,\mathrm{\infty })$ and $I=(0,1)$; suppose that $T:L_p(I)\to L_p(I)$ is a~compact linear map with trivial kernel and range dense in $L_p(I)$. It is shown that if the Gelfand numbers of $T$ decay sufficiently quickly, then the action of $T$ is given by a series with calculable coefficients. The special properties of $L_p(I)$ enable this to be established under weaker conditions on the Gelfand numbers than in earlier work set in the context of more general spaces.