Uniqueness of unconditional basis of $\ell_{2}\oplus \mathcal{T}^{(2)}$

Bases of random unconditional convergence in Banach spaces

We study random unconditional convergence for a basis in a Banach space. The connections between this notion and classical unconditionality are explored. In particular, we analyze duality relations, reflexivity, uniqueness of these bases and existence of unconditional subsequences.

0<p≤1时向量值$H_p$空间之间的同构与无条件结构的唯一性

Isomorphisms between vector-valued $H_p$-spaces for $0<p\le 1$ and uniqueness of unconditional structure

本文通过识别Banach值Hardy空间并发展新格估计技术，证明$0<p<1$时$H_p(\mathbb{D},\ell_2)$具有唯一原子格结构。

Algebras of diagonal operators of the form scalar-plus-compact are Calkin algebras

For every Banach space $X$ with a Schauder basis consider the Banach algebra $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ of all diagonal operators that are of the form $λI + K$. We prove that $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ is a Calkin algbra i.e., there exists a Banach space $\mathcal{Y}_X$ so that the Calkin algebra of $\mathcal{Y}_X$ is isomorphic as a Banach algebra to $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$. Among other applications of this theorem we obtain that certain hereditarily indecomposable spaces and the James spaces $J_p$ and their duals endowed wi…

On a generalization of Bourgain's tree index

For a Banach space $X$, a sequence of Banach spaces $(Y_n)$, and a Banach space $Z$ with an unconditional basis, D. Alspach and B. Sari introduced a generalization of a Bourgain tree called a $(\oplus_n Y_n)_Z$-tree in $X$. These authors also prove that any separable Banach space admitting a $(\oplus_n Y_n)_Z$-tree with order $ω_1$ admits a subspace isomorphic to $(\oplus_n Y_n)_Z$. In this paper we give two new proofs of this result.

Narrow operators and the Daugavet property for ultraproducts

We show that if $T$ is a narrow operator on $X=X_{1}\oplus_{1} X_{2}$ or $X=X_{1}\oplus_{\infty} X_{2}$, then the restrictions to $X_{1}$ and $X_{2}$ are narrow and conversely. We also characterise by a version of the Daugavet property for positive operators on Banach lattices which unconditional sums of Banach spaces inherit the Daugavet property, and we study the Daugavet property for ultraproducts.

Banach空间的无条件结构与少数算子

Unconditional structure of Banach spaces with few operators

本文通过p-凸化Gowers空间，证明其具有唯一无条件基，推翻了关于无条件基空间等距同构的猜想，并揭示了少数算子空间的无条件结构唯一性。

Real hereditarily indecomposable Banach spaces and uniqueness of complex structure

There exists a real hereditarily indecomposable Banach space $X$ such that the quotient space $L(X)/S(X)$ by strictly singular operators is isomorphic to the complex field (resp. to the quaternionic division algebra). Up to isomorphism, the example with complex quotient space has exactly two complex structures, which are conjugate, totally incomparable, and both hereditarily indecomposable. So there exist two Banach spaces which are isometric as real spaces but totally incomparable as complex spaces. This extends results of J. Bourgain and S. Szarek from 1986. The quaternionic example, on the …

Complemented subspaces of spaces obtained by interpolation

If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A_0,A_1) such that A_0 and A_1 are isometric to $X\oplus V$, and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype, having the Radon-Nikodym property, and having the analytic unconditional martingale difference sequence property, do not pass to intermediate spaces.

A $c\_0$-saturated Banach space with no long unconditional basic sequences

We present a Banach space $\mathfrak X$ with a Schauder basis of length $ω\_1$ which is saturated by copies of $c\_0$ and such that for every closed decomposition of a closed subspace $X=X\_0\oplus X\_1$, either $X\_0$ or $X\_1$ has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of $\mathfrak X$ have ``few operators'' in the sense that every bounded operator $T:X \to \mathfrak{X}$ from a subspace $X$ of $\mathfrak{X}$ into $\mathfrak{X}$ is the sum of a multiple of the inclusion and a $ω\_1$-sin…

巴拿赫空间上有界算子代数的商空间的代数范数唯一性

Uniqueness of algebra norm on quotients of the algebra of bounded operators on a Banach space

本研究证明三类巴拿赫空间上有界算子代数的商空间对所有闭理想具有唯一代数范数，等价于到巴拿赫代数的同态均连续且有闭值域，核心是恒等算子的可控范数定量分解结论，具独立研究意义。

自反Calkin代数

Reflexive Calkin algebras

本研究证明Banach空间的Calkin代数可以是无穷维自反空间甚至同构于Hilbert空间，通过构造特定Banach空间及相互零化投影列，得到Calkin代数与对应空间酉化的Banach代数同构关系，并列举了多类满足条件的Banach空间。

Non-existence of greedy bases in direct sums of mixed $\ell_{p}$ spaces

The fact that finite direct sums of two or more mutually different spaces from the family $\{\ell_{p} : 1\le p<\infty\}\cup c_{0}$ fail to have greedy bases is stated in [Dilworth et al., Greedy bases for Besov spaces, Constr. Approx. 34 (2011), no. 2, 281-296]. However, the concise proof that the authors give of this fundamental result in greedy approximation relies on a fallacious argument, namely the alleged uniqueness of unconditional basis up to permutation of the spaces involved. The main goal of this note is to settle the problem by providing a correct proof. For that we first show that…

Factorisation in stopping-time Banach spaces: identifying unique maximal ideals

Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as $L^1$ or $C(Δ)$, but unlike these, they do have unconditional bases. In the present paper we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this …

巴拿赫空间上有界算子构成的巴拿赫代数的极大右理想

Maximal right ideals of the Banach algebra of bounded operators on a Banach space

该研究探讨巴拿赫代数$\boldsymbol{\beta}(E)$的极大右理想，明确有限生成提升理想的算子论障碍，通过相关方法对多类巴拿赫空间构造非有限生成极大右理想，还确定部分空间的有限生成极大右理想均为固定的。