# Complemented subspaces in the James space

Let $J$ be the James space. I have the following questions:

Question 1: Does every infinite-dimensional closed subspace of $J$ contain an infinite-dimensional closed subspace that is $C$-complemented in $J$? where the $C$ is the universal constant.

Question 2: Let $(u_{n})_{n}$ be a normalized skipped block basic sequence of the unit vector basis in $J$. Is the subspace spanned by $(u_{n})_{n}$ $C$-complemented in $J$? where the $C$ is the universal constant.

Obviously, if Question 2 is true, then Question 1 is true. I do not know what is the above universal constant $C$.

Thank you!

• The answer to question 1 is yes. See the comments after Remark 2.11 in arxiv.org/abs/1401.4231 and Proposition 2.4 in acadsci.fi/mathematica/Vol36/… I do not know the answer to question 2. – Ben W Jul 10 '16 at 23:53
• I have not looked carefully yet, but it seems that the answer to question 2 might also be true. Check out section 2.d in The James Forest by Helga Fetter and Berta Gamboa de Buen. Most of it is on Google books. books.google.com/books?id=GQJVVtDwx5wC&pg=PA43 – Ben W Jul 11 '16 at 0:23
• Thanks, Ben. Question 2 may follow from the book you mentioned. But I do not have this book and so I am not sure. Do you have the electronic version of this book? – Dongyang Chen Jul 11 '16 at 0:47
• No, I don't. If it is important you could try to piece it together from what fragments are available on Google books. I believe all of theorem 2.d.2 is visible there. By the way---out of curiosity, what are you studying about the James space? – Ben W Jul 11 '16 at 0:59
• I am studying about compact operators, strictly singular operators and strictly co-singular operators on the James space. – Dongyang Chen Jul 11 '16 at 1:50

The answer to both questions is yes. Indeed, suppose $y_j=\sum_{n=p_j}^{q_j}\alpha_ne_n$ forms a block basic sequence in $J$ satisfying $p_{j+1}-q_j>1$ for all $j$. It is shown in the proof of Theorem 2.d.2 of The James Forest that $[y_j]_{j=1}^\infty$ is complemented by a projection of norm $\leq 2\sqrt{2}$. (It is also shown that if $(y_i)_{i=1}^\infty$ is seminormalized then it is equivalent to the $\ell_2$ basis.)