I am interseted in the following problem. By Bessaga and Pelczynski paper On bases and unconditional convergence of series in Banach spaces we know that if $X$ is a Banach space with a basis $(x_n)$ and $Y$ is its closed subspace with infinite dimension, then there is $Z$ - subspace of $Y$ with a basis which is equivalent to a block basis of $(x_n)$. I was wondering however about some modification and start from space $X$ with unconditional basis.
Namely, suppose that $X$ is a Banach space with unconditional basis and let $Y$ be a closed subspace in $X$ of infinite dimension. Is it true that there is $Z$ - subspace of $Y$ that also has an unconditional basis?
Intuition tells me that this statement should be true, although I have realized that in Banach space theory many things turn out to be contrary to what I think.
I will be grateful for any (sketch of) proof (or counterexamples), and for some references if it is not trivial.