The cardinality of the sublattice of closed ideals of operators between certain classical sequence spaces

Small operator ideals on the Schlumprecht and Schreier spaces

We present a method of building operators on a Banach space $X$ that generate distinct operator ideals in the algebra $\mathscr{B}(X)$ of bounded linear operators on $X$. We show that there are exactly $2^\mathfrak{c}$ distinct small closed operator ideals on the Schlumprecht space and there is a chain of cardinality $\mathfrak{c}$ of small closed operator ideals on any Schreier space of finite order.

Closed ideals of operators acting on some families of sequence spaces

We study the lattice of closed ideals in the algebra of continuous linear operators acting on $p$th Tandori and $p'$th Cesàro sequence spaces, $1\leqslant p<\infty$, which we show are isomorphic to the classical sequence spaces $(\oplus_{n=1}^\infty\ell_\infty^n)_p$ and $(\oplus_{n=1}^\infty\ell_1^n)_{p'}$, respectively. We also show that Tandori sequence spaces are complemented in certain Lorentz sequence spaces, and that the lattice of closed ideals for certain other Lorentz and Garling sequence spaces has infinite cardinality.

B(H) lattices, density and arithmetic mean ideals

This part of a multi-paper project studies the lattice properties of the arithmetic mean ideals of B(H) introduced by Dykema, Figiel, Weiss, and Wodzicki. We prove: the lattices of all principal ideals, of arithmetic mean or arithmetic mean at infinity stable principal ideals or of principal ideals with a generator that satisfies the Delta_1/2 condition, are all both upper and lower dense in the lattice of general ideals. That is, between any ideal and an ideal (nested above or below respectively) in one of these sublattices, lies another ideal in that sublattice. Among the applications: a pri…

Haar-smallest sets

In this paper we are interested in the following notions of smallness: a subset $A$ of an abelian Polish group $X$ is called Haar-countable/Haar-finite/Haar-$n$ if there are a Borel hull $B\supseteq A$ and a copy $C$ of $2^ω$ such that $(C+x)\cap B$ is countable/finite/of cardinality at most $n$, for all $x\in X$. Recently, Banakh et al. have unified the notions of Haar-null and Haar-meager sets by introducing Haar-$\mathcal{I}$ sets, where $\mathcal{I}$ is a collection of subsets of $2^ω$. It turns out that if $\mathcal{I}$ is the $σ$-ideal of countable sets, the ideal of finite sets or the c…

可分巴拿赫格的波兰空间

Polish spaces of separable Banach lattices

研究可分巴拿赫格类的描述复杂性，引入两种波兰空间编码，证明相关空间是波兰子空间，还证明理想编码上的弗雷姆林射影张量积运算的可测性及图的性质。

A survey on subideals of operators and an introduction to subideal-traces

Operator ideals in B(H) are well understood and exploited but ideals inside them have only recently been studied starting with the 1983 seminal work of Fong and Radjavi and continuing with two recent articles by the authors of this survey. This article surveys this study embodied in these three articles. A subideal is a two-sided ideal of J (for specificity also called a J-ideal) for J an arbitrary ideal of B(H). In this terminology we alternatively call J a B(H)-ideal. This surveys these three articles in which we developed a complete characterization of all J-ideals generated by sets of card…

A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem

The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals, and we explored their role in a multipaper project which we survey in this article. We start by presenting the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sublattice structure of ideals, arithmetic mean ideal cancellation properties of first and second order, and softness properties - a term that we intr…

Properties of simple density ideals

Let $G$ consist of all functions $g \colon ω\to [0,\infty)$ with $g(n) \to \infty$ and $\frac{n}{g(n)} \nrightarrow 0$. Then for each $g\in G$ the family $\mathcal{Z}_g=\{A\subseteqω:\ \lim_{n\to\infty}\frac{\text{card}(A\cap n)}{g(n)}=0\}$ is an ideal associated to the notion of so-called upper density of weight $g$. Although those ideals have recently been extensively studied, they do not have their own name. In this paper, for Reader's convenience, we propose to call them simple density ideals. We show that there are $\mathfrak{c}$ many non-isomorphic (in fact even incomparable with respect…

Closed ideals of operators on the Tsirelson and Schreier spaces

Let $\mathscr{B}(X)$ denote the Banach algebra of bounded operators on $X$, where~$X$ is either Tsirelson's Banach space or the Schreier space of order $n$ for some $n\in\mathbb N$. We show that the lattice of closed ideals of~$\mathscr{B}(X)$ has a very rich structure; in particular $\mathscr{B}(X)$ contains at least continuum many maximal ideals. Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection $P_N\in\mathscr{B}(X)$ corresponding to each non-empt…

Subideals of operators II

A subideal (also called a J-ideal) is an ideal of a B(H)-ideal J. This paper is the sequel to Subideals of operators where a complete characterization of principal and then finitely generated J-ideals were obtained by first generalizing the 1983 work of Fong and Radjavi who determined which principal K(H)-ideals are also B(H)-ideals. Here we determine which countably generated J-ideals are B(H)-ideals, and in the absence of the continuum hypothesis which J-ideals with generating sets of cardinality less than the continuum are B(H)-ideals. These and some other results herein are based on the di…

关于 $\mathcal I$-null 序列空间的补性

On the complementation of spaces of $\mathcal I$-null sequences

本文研究了$c_{0,\mathcal{I}}$空间在$\ell_\infty$中的补性，证明其与理想是否为$\omega$-极大理想相关，并刻画了商空间为有限维的条件。

Amalgamation and injectivity in Banach lattices

We study distinguished objects in the category $\mathcal{BL}$ of Banach lattices and lattice homomorphisms. The free Banach lattice construction introduced by de Pagter and Wickstead generates push-outs, and combining this with an old result of Kellerer on marginal measures, the amalgamation property of Banach lattices is established. This will be the key tool to prove that $L_1([0,1]^{\mathfrak{c}})$ is separably $\mathcal{BL}$-injective, as well as to give more abstract examples of Banach lattices of universal disposition for separable sublattices. Finally, an analysis of the ideals on $C(Δ,…

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。

Interplay between Algebraic Groups, Lie Algebras and Operator Ideals

In the framework of operator theory, we investigate a close Lie theoretic relationship between all operator ideals and certain classical groups of invertible operators that can be described as the solution sets of certain algebraic equations, hence can be regarded as infinite-dimensional linear algebraic groups. Historically, this has already been done for only the complete-norm ideals; in that case one can work within the framework of the well-known Lie theory for Banach-Lie groups. That kind of Lie theory is not applicable for arbitrary operator ideals, so we needed to find a new approach fo…