Isometric stability property of Banach spaces

Let $X$ be a separable $L_1$ or a separable $C(K)$-space, and let $Y$ be any Banach space. $I(X, Y )$ denotes the set of all isometries from $X$ to $Y$ . showed that for any finite measure space $(\Omega,\mu)$ and any $1 < p <\infty$, every isometry $T : X\rightarrow L_p(\Omega, Y )$ has the form $$T x(t) = h(t)U(t)x ,$$ where $h\in L_p$ with $\parallel h\parallel_p = 1$ and $U$ is a strongly measurable function from $\Omega$ into $I(X, Y )$. In this article, we extend this result to the Köthe-Bochner function spaces $E(Y )$ when $E$ is strictly convex. We also show that every isometry from $\ell^n_\infty$ into $E(Y $) has the above form if $n\geq 3$ and $E$ is a strictly monotone Köthe function space.