# Two questions on $l_{p}$-saturated Banach spaces

Let $1<p<\infty$. Recall that a Banach space $X$ is $l_{p}$-saturated if every infinite-dimensional subspace of $X$ contains a subspace isomorphic to $l_{p}$. I have a seemingly stronger notion of $l_{p}$-saturated Banach spaces. We say that a Banach space $X$ is strongly $l_{p}$-saturated if for every infinite-dimensional subspace $M$ and every $\epsilon>0$, there exists an infinite-dimensional subspace $N$ of $M$ such that $N$ is $(1+\epsilon)$-isomorphic to $l_{p}$. I have the following two questions:

Question 1. If a Banach space $X$ is $l_{p}$-saturated, is $X$ strongly $l_{p}$-saturated?

Question 2. Are there other examples of strongly $l_{p}$-saturated spaces besides $l_{p}$? Is $L_{p}$ strongly $l_{p}$-saturated?

Thank you!

• $L_p$ contains copies of $\ell_2$; it is not $\ell_p$-saturated. – Tomek Kania Dec 8 '16 at 15:12
• The Lorentz and Garling sequence spaces and certain Orlicz sequence spaces are strongly $\ell_p$-saturated. Probably so are the Lorentz-Orlicz and Musielak-Orlicz sequence spaces. Other spaces which I don't remember off the top of my head but are worth checking, include the $(p,q,\xi)$-Schreier spaces, the James spaces $J_p$ and spaces of the form $(\oplus X_n)_p$ for dim$(X_n)<\infty$. Note that the Lorentz sequence spaces are saturated with $(1+\epsilon)$-complemented copies of $\ell_p$, and probably also some Orlicz. It is an open question whether the same goes for Garling sequence spaces. – Ben W Dec 8 '16 at 15:54
• @BenWallis, Could you check that $(\oplus X_{n})_{p}$ for $dim(X_{n})<\infty$ is strongly $l_{p}$-saturated? I can check that the James $p$-space is strongly $l_{p}$-saturated. – Dongyang Chen Dec 9 '16 at 12:08
• For $(\oplus X_n)_p$, as we may assume $1<p<\infty$, this is just a gliding hump argument. – Ben W Dec 9 '16 at 12:31
• One condition which guarantees that an $\ell_p$ saturated space be strongly $\ell_p$ saturated would be that the space has a Schauder basis $(e_i)_{i=1}^\infty$ (or even just an FDD) such that either $(1)$ for any $n\in\mathbb{N}$, $0=k_0<\ldots <k_n$, and $x_i=\sum_{j=k_{i-1}+1}^{k_i} a_je_j$, $$\|\sum_{i=1}^n x_i\|^p \leqslant \sum_{i=1}^n \|x_i\|^p,$$ or $(2)$ for any $n\in\mathbb{N}$, $0=k_0<\ldots <k_n$, and $x_i=\sum_{j=k_{i-1}+1}^{k_i} a_je_j$, $$\|\sum_{i=1}^n x_i\|^p \geqslant \sum_{i=1}^n \|x_i\|^p.$$ It would also be true if we only assumed such an inequality for skipped blocks. – user114263 Jul 31 '18 at 17:55

As for question 1; this is the famous distortion problem. Your question has negative answer already for some renorming of $\ell_p$:
As for your second question, $L_p$ is not $\ell_p$-saturaded as the Rademachers span a copy of $\ell_2$.
• Thanks, Ben and Tomek. $L_{p}$ is not $l_{p}$-saturated. But I have another question on $L_{p}$. Given any infinite-dimensional closed subspace $M$ of $L_{p}(1<p<\infty)$ and any $\epsilon>0$. Do there exist an infinite-dimensional closed subspace $N$ of $M$ and $1<q<\infty$ such that $N$ is $(1+\epsilon)$-isomorphic to $l_{q}$? – Dongyang Chen Dec 9 '16 at 9:10
• I'm not completely sure, but my guess is that probably yes. I would look at chapter 6 from the Albiac/Kalton book to see if the isomorphism constants can be made that small. At least for $L_p$, $2\leq p<\infty$, it "should" be doable. – Ben W Dec 9 '16 at 10:48