M.I. Kadec and A. Pełczyński proved that if $E$ is a subspace of $L\_{p}(p>2)$ isomorphic to $l\_{2}$, then $E$ is complemented in $L\_{p}$. My question is:


Is there a constant $C\_{p}$ depending only on $p$ such that every subspace of $L\_{p}(p>2)$ isomorphic to $l\_{2}$ is $C\_{p}$-complemented in $L\_{p}$?


Thank you!



On the complemented subspaces of $L_{p}(p>2)$
The answer to the question, as it is stated, is "No", and can be shown as follows: It is known (follows, for example, from results of Sobczyk, Duke Math. J., 1941; or from the Lindenstrauss-Tzafriri characterization of $\ell\_2$) that we can find a sequence of finite-dimensional subspaces $R\_n\subset L\_p$ with going to $\infty$ relative projection constants $\lambda(R\_n, L\_p)$, $p>2$.


Now, we have $L\_p=L\_p\oplus\_p L\_p$. We introduce the following subspaces isomorphic to $\ell\_2$ in $L\_p$. Let $K$ be any subspace of $L\_p$ isomorphic to $\ell\_2$. Consider the subspaces $B\_n=R\_n\oplus\_p K$ as subspaces of $L\_p\oplus\_p L\_p$. It is easy to see that (1) These subspaces are isomorphic to $\ell\_2$. (2) Their relative projection constants in $L\_p$ tend to $\infty$.


On the other hand, the norm of the projection onto a subspace of $L\_p$ isomorphic to $\ell\_2$ can be bounded in terms of the Banach-Mazur distance between the subspace and $\ell\_2$. This can be derived from the result of Maurey (Un théorème de prolongement. C. R. Acad. Sci. Paris Sér. A 279 (1974), 329-332). 


