## How to characterize an operator $T$ that factors through a special space?

Let $T\in \mathcal{L}(X,Y)$ and $1<p<\infty$. My question is: Is there a convienent and useful characterization of the operator $T$ factoring through a space $Z$ satisfying $\mathcal{L}(Z,l_{p})=\mathcal{K}(Z,l_{p})$? where $\mathcal{L}$ denotes linear bounded operator space, $\mathcal{K}$ denotes the compact operator space. This question seems open-ended. But it is useful. Thank you! Answer AttributionSource : Link , Question … Read more

## Weakenings of the Bounded Approximation Property

Let $X$ be a Banach space, and let $F(X)$ be the finite-rank operators on $X$. Then $X$ has the $\lambda$-BAP if there exists a net $(S_i)_i$ in $F(X)$ with $\sup_i\|S_i\|\leq\lambda$ such that $S_i\to\mathrm{id}_X$ pointwise, that is, $\|S_i(x)-x\|\to 0$ for every $x\in X$. What is the connection to the following variants of this property: (Explanation of … Read more

## On the weakly sequential completeness of the dual of the James space JJ

Let me first introduce some definitions. Let 1≤p≤∞. A sequence (xn)n in a Banach space X is said to be weakly p-convergent to x∈X if the sequence (xn−x)n is weakly p-summable in X. We say that a sequence (xn)n in a Banach space X is weakly p-Cauchy if for each pair of strictly increasing sequences … Read more

## Good reference for noncommutative LpL^p spaces

I’m looking for good references to learn about Lp spaces associated with von Neumann algebras. I already know about Uffe Haagerup’s paper “Lp–spaces associated with an arbitrary von Neumann algebra“; but I’m looking for further references. Can you please recommend me some other documents? Answer AttributionSource : Link , Question Author : Screwdriver , Answer … Read more

## quasi-nilpotent part of a dual operator

Definitions and notation. Let X be a complex Banach space and T∈L(X) a continuous linear operator on X. We define the quasi-nilpotent part of T as H0(T):={x∈X:lim Note that H_0(T) is a T-invariant linear subspace of X, although perhaps not a closed subspace. Trivially, we have the inclusion \begin{equation*}\mathcal{N}^\infty(T):=\bigcup_{n=1}^\infty\mathcal{N}(T^n)\subseteq H_0(T),\end{equation*} where \mathcal{N}(\cdot) denotes the null … Read more

## Does every separable Banach space have a Markushevich–Auerbach basis?

Let X be a separable Banach space and X∗ be its dual, let {xi} be a sequence in X with dense linear span and such that there exists a sequence {x∗i} in X∗ satisfying x∗i(xj)=δi,j (Kronecker delta). It is clear that {x∗i} is uniquely determined. Let us call {xi} a Markushevich–Auerbach basis if the linear … Read more

## A vector that makes all norms grow

Suppose that E is a finite dimensional subspace of a Banach space X. A classical theorem (that I think is from Mazur) says that for every ϵ>0 we can find a vector x such that for all y∈E we have ‖. This (to me at least) generalizes the notion of an orthonormal vector from Hilbert … Read more

## Banach space admitting a unique subsymmetric basis but not a symmetric one

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 basis (which is also symmetric). Question 1. Are there any other known examples of a Banach space admitting a unique subsymmetric basis? In 2004, Sari … Read more

## Embedding of ℓ2\ell_2 in Lp([0,1])L^p([0,1])

Let (gn)n≥1 be a sequence of i.i.d. complex Gaussian random variables on [0,1]. Then it is easy to see that the map j:ℓ2→Lp([0,1]) defined as jen=[E(gpn)]1pgn,n≥1 is an isometry and j(ℓ2) is a closed subset of Lp([0,1]) which is also complemented. Let P:Lp([0,1])→j(ℓ2) be the projection map. Then what can be said about the norm … Read more

## Weak to weak$^*$ continuity of the duality mapping

Let $X$ be a uniformly convex and uniformly smooth Banach space. We consider the duality mapping $J_p^X$ defined as the sub-gradient $\partial (\frac1p\|\cdot\|^p)$. Is there a characterisation of the Banach spaces $X$ for which the mapping $J_p^X$ is weak-weak$^*$-continuous? Let me summarise some statements I could find in the literature. In Xu, Roach: Characteristic inequalities … Read more