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This is an adaptation of a Heinrich proof, but I'm missing a key ingredient. Conjecture. Suppose $(x_n)_{n=1}^\infty$ is a Schauder basis for a Banach space $X$ whose canonical isometric copy in $X^{ …

Bourgain/Casazza/Lindenstrauss/Tzafriri proved in their unconditional basis UTAP book (1985) that $\ell_1$ is the only nontrivial complemented subspace of $(\bigoplus\ell_2^n)_1$, and hence by duality …

I have a quick question which is probably supposed to be obvious, but for some reason I just don't see it: How does one re-norm a quasi-Banach space to produce a $p$-Banach space ($0<p\leq 1$) with t …

Denote $L_1=L_1[0,1]$ The lattice of closed ideals in $\mathcal{L}(L_1)$ includes the chain $$ \{0\}\subsetneq\mathcal{K}(L_1)\subsetneq\mathcal{FS}(L_1) \subsetneq\mathcal{J}_{\ell_1}(L_1)\subsetneq …

Let $d(\textbf{w},p)$, $1\leq p<\infty$, denote the Lorentz sequence space, where $\textbf{w}=(w_n)_{n=1}^\infty\in c_0\setminus\ell_1$ is a normalized decreasing weight. Is there very much known abo …

Note: By "subspace" I always mean an infinite-dimensional closed subspace. Notation. Let us write $$\oplus_p\ell_q^n:=\left(\bigoplus_{n=1}^\infty\ell_q^n\right)_{\ell_p}\;\;\;\text{ and }\;\;\;\op …

Fix any $1\leq p\leq\infty$. If $X$ is a Banach space and $C\in(0,\infty)$, we say that $T\in\mathcal{A}_C(X)$ whenever, for each $(x_n)_{n=1}^\infty\subset B_X$ (where $B_X$ is the closed unit ball …

A few years ago, Schechtman showed that $\ell_p(\ell_q)$ fails to admit a greedy basis whenever $1\leq p\neq q<\infty$. This furnishes an example of a Banach space with an unconditional basis but not …

I have two quick questions: It can be shown without too much trouble (using methods from Altshuler/Casazza/Lin, 1973) that any Lorentz sequence space admits a unique (up to equivalence) subsymmetric …

The short version of my question: Suppose $T\in\mathcal{L}(X,Y)$ is strictly cosingular. Must $T^*$ be strictly singular? The long version. Let $X$ and $Y$ be Banach spaces, and denote by $\mathcal{ …

Definitions and notation. Let $X$ be a complex Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator on $X$. We define the quasi-nilpotent part of $T$ as \begin{equation*}H_0(T):=\left\ …

Let $X$ be an infinite-dimensional Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. A closed subspace $Y$ of $X$ is said to be an almost-invariant halfspace (hereafte …

Throughout, $X$ and $Y$ will denote Banach spaces with $T\in\mathcal{L}(X,Y)$ (the space of continuous linear operators between $X$ and $Y$). We define the operator $\overline{T}\in\mathcal{L}(X^{**} …

Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis wh …

Giorgos Petsoulas, in his paper "A class of $\ell^p$ saturated Banach spaces," has constructed for each $1<p<\infty$ a space $\mathfrak{X}_p$ which is complementably $\ell_p$-saturated but admits no u …