Functional analysis. Complemented subspaces of the Lorentz sequence spaces?

Leave $ d ( textbf {w}, p) $, $ 1 leq p < infty $, denotes the Lorentz sequence space, where $ textbf {w} = (w_n) _ {n = 1} ^ infty in c_0 setminus ell_1 $ It is a normalized decreasing weight.

Is much known about the subspaces complemented by $ d ( textbf {w}, p) $? In general (that is, without any restriction of $ textbf {w} $ or $ p $), I can only find two: $ ell_p $ Y $ d ( textbf {w}, p) $ itself.

Question 1. What complements the subspaces of $ d ( textbf {w}, p) $in addition to $ ell_p $ And in itself, they already know each other?

A third distinct complementary subspace has been found in the case $ textbf {w} $ meets a certain condition (NUC), namely that $ inf_k ( sum_ {i = 1} ^ {2k} w_n) / ( sum_ {i = 1} ^ kw_n) = 1 $. In this case $ d ( textbf {w}, p) $ contains an isomorphic subspace complemented to 1 $ oplus_p ( ell_ infty ^ n) $.

But can we find other subspaces complemented in the general case?

The first thing to try is to look for basic sequences of constant coefficient blocks. If the length of the blocks is delimited, they include another copy of $ d ( textbf {w}, p) $. However, if your lengths tend to infinity, then your interval will be different from $ d ( textbf {w}, p) $. In this case, the problem is to show that the resulting space is not isomorphic for $ ell_p $.

This is pretty easy when $ p = 1 $ or $ p = 2 $. In these cases, from $ ell_1 $ Y $ ell_2 $ each admits a unique unconditional base, it is enough to make sure that the constant coefficient blocks in $ d ( textbf {w}, p) $ They are not equivalent to those bases. This can be done by taking constant coefficient block sequences $ (d_i ^ {(k)}) _ {i = 1} ^ {N_k} $ fixed length $ k $picking out $ N_k $ Large enough for them to fail more and more to dominate. $ ell_p ^ {N_k} $. Then paste those sequences together for all $ k in mathbb {N} $, pushing them far enough so that they separate.

I suspect this will work for everyone. $ 1 leq p < infty $, but trying it is not as simple as the previous one. However, if it could be shown that $ ell_p $ does not contain uniformly complemented copies of $ text {span} (d_n) _ {n = 1} ^ N $, where $ (d_n) _ {n = 1} ^ infty $ It's the basis for a Lorentz sequence space, so that would do the trick. A) Yes:

Question 2 Make $ ell_p $ contain uniformly supplemented copies of $ text {span} (d_n) _ {n = 1} ^ N $?

Thanks guys!