Operators on L (2 &lt;p&lt; ∞) which factor through X

Zheng, Bentuo

doi:10.1016/j.jfa.2017.04.007

Cited by 2 publications

(4 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…So an operator is strictly singular if and only if it is Z - strictly singular for every infinite-dimensional space Z . (See [2–3, 16, 17] for more details on this topic. )…”

Section: Operators Onmentioning

confidence: 99%

The Maximal Ideal in the Space of Operators On

CADAVID

Monika

Zheng

2022

Bull. Aust. Math. Soc.

Self Cite

View full text Add to dashboard Cite

We study the isomorphic structure of $(\sum {\ell }_{q})_{c_{0}}\ (1< q<\infty )$ and prove that these spaces are complementably homogeneous. We also show that for any operator T from $(\sum {\ell }_{q})_{c_{0}}$ into ${\ell }_{q}$ , there is a subspace X of $(\sum {\ell }_{q})_{c_{0}}$ that is isometric to $(\sum {\ell }_{q})_{c_{0}}$ and the restriction of T on X has small norm. If T is a bounded linear operator on $(\sum {\ell }_{q})_{c_{0}}$ which is $(\sum {\ell }_{q})_{c_{0}}$ -strictly singular, then for any $\epsilon>0$ , there is a subspace X of $(\sum {\ell }_{q})_{c_{0}}$ which is isometric to $(\sum {\ell }_{q})_{c_{0}}$ with $\|T|_{X}\|<\epsilon $ . As an application, we show that the set of all $(\sum {\ell }_{q})_{c_{0}}$ -strictly singular operators on $(\sum {\ell }_{q})_{c_{0}}$ forms the unique maximal ideal of $\mathcal {L}((\sum {\ell }_{q})_{c_{0}})$ .

show abstract

“…So an operator is strictly singular if and only if it is Z - strictly singular for every infinite-dimensional space Z . (See [2–3, 16, 17] for more details on this topic. )…”

Section: Operators Onmentioning

confidence: 99%

The Maximal Ideal in the Space of Operators On

CADAVID

Monika

Zheng

2022

Bull. Aust. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…在相当长的时间里, 作者一直都认为答案是肯定的, 因为之前所有的可通过 ℓ p 子空间分解的定理都是在类似的条件之下, 而且定理 3.11 也告诉我们上 ℓ q ⊕ ℓ 2 树估计是必要条件. 这个问题的答案直到 2017 年才由 Zheng [42] 给出了第一个反例. 这个反例的构造需要下面的定理: 定理 3.12 [42] 记 (e n,i…”

unclassified

“…定理 3.13 [42] [43] 在 1960 年证明了当 X 等于 ℓ p (1 p < ∞) 或者 c 0 时, 严格奇异算子构成 L(X) 中的唯一极大理想. Chen 等 [44] 在 2011 年证明了 Z p -严格奇异算子构成 L(Z p ) 中的唯一极大理想, Dosev 等 [45] 在 2013 年证明了 L p (1 p < ∞)-严格奇异算子构成了 L(L p ) 中的唯一极大理想.…”

unclassified

“…因为所有具有条件弥散基的空间都是拟自反的 (参见文献 [52]), 所以有下面进一步的问题: 问题 4.8 如果拟自反的 Banach 空间 X 具有一组无条件跳跃基, 那么 X 是否可以线性嵌入到具有条件弥散基的空间? 前面提到 Zheng[42] 在 2017 年证明了 L p (2 < p < ∞) 上闭的 Z p -严格奇异算子可以通过 X p 进行分解, 但是存在非闭的 Z p -严格奇异算子不能通过 X p 进行分解. 那么, 是否可以用可通过 X p 进行分解的算子来逼近 Z p -严格奇异算子?…”

unclassified

See 1 more Smart Citation

Linear embedding and factorizations of operators on Banach spaces

Zheng¹

2020

Sci. Sin.-Math.

View full text Add to dashboard Cite

Embedding theory is one of the most important topics in the geometry of Banach spaces and factorizations of operators are the natural extensions. In this paper, we will first systematically introduce the historical results on isomorphic theory and present some of the recent progress in this direction. Then we will discuss related results about factorizations of operators. Interesting open problems will be listed at the end of the paper.

show abstract

Operators on L (2 <p< ∞) which factor through X

Cited by 2 publications

References 20 publications

The Maximal Ideal in the Space of Operators On

The Maximal Ideal in the Space of Operators On

Linear embedding and factorizations of operators on Banach spaces

Contact Info

Product

Resources

About