Non-approximable compact operators
Non-approximable compact operators
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不可近似紧算子
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
A. Pietsch
中科院分区：
文献类型：
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作者：
K. Kürsten;A. Pietsch
摘要Enflo对逼近问题的否定回答意味着在Banach空间之间存在不能用有限秩算子逼近的紧算子。不幸的是，标准方法是间接的，因此没有提供具体的例子。这种令人不快的情况是可以改善的。利用Davie构造的无限矩阵，第二个命名的作者在他的《算子理想》一书中证明了正则因式分解的中心部分CT
$$T：Ell_1 Stackrel{Q_T}{Longright tarrow}，ell_1/Mathcal{N}(T)Stackrel{C_T}{Longright tarrow}，Overline{Mathcal{M}(T)}，Stackrel{J_T}{Longright tarrow}，c_0$$T：ℓ1⟶QTℓ1/N(T)⟶CTM(T)某些运算符的⟶JTc0${T：Ell_1 o c_0}$$T：ℓ1→c0是紧凑的，但不可近似。虽然我们对这些算子$${T：Ell_1 o c_0}$$T：ℓ1→c_0有了很好的理解，但由于它们的生成矩阵是用随机方法得到的，所以它们仍然是非具体的。审视运营商的核心部分有着深远的后果：理想
$$Mathfrak{L}_Q^{
Ment}：=Left{T：sum_{n=1}^inty e_n(T)^q<；inty
Ight}四边形{
M With}quad 2<；q<；inty，$$lqent：=T：∑n=1∞en(T)Q<；∞With 2<；Q<；∞与熵数相关，包含不可逼近的算子。正如第一位作者的(未发表的)论文中已经显示的那样，同样的结论也适用于理想$${mathfrak{L}_q^{
M Gel}$$LqGel和$${mathfrak{L}_q^{
MKol}}$$Lqkol分别由Gelfand数和Kolmogorov数生成。我们在这里提出了一个基于新技术的证明。关键是构造不可逼近的运算符，这些运算符不仅是紧凑的，而且具有规定的(不太强)紧致度。
AbstractEnflo’s negative answer to the approximation problem implies that there exist compact operators between Banach spaces that cannot be approximated by finite rank operators. Unfortunately, the standard approach is indirect and, hence, concrete examples are not provided. This unpleasant situation can be improved. Using infinite matrices constructed by Davie, the second-named author showed in his book “Operator Ideals” (proof of § 10.4.6) that the central part CT of the canonical factorization
$$T : ell_1 stackrel{Q_T}{longrightarrow} , ell_1/ mathcal{N} (T) stackrel{C_T}{longrightarrow} , overline{mathcal {M} (T)}, stackrel{J_T}{longrightarrow}, c_0$$T:ℓ1⟶QTℓ1/N(T)⟶CTM(T)¯⟶JTc0of certain operators $${T: ell_1 o c_0}$$T:ℓ1→c0 is compact, but non-approximable. Though we have a good understanding of those operators $${T: ell_1 o c_0}$$T:ℓ1→c0, they are still non-concrete since their generating matrix is obtained by stochastic methods. Looking at the central part of operators has far-reaching consequences: The ideal
$$mathfrak{L}_q^{
m ent} := left{T : sum_{n = 1}^infty e_n (T)^q < infty
ight} quad {
m with} quad 2 < q < infty,$$Lqent:=T:∑n=1∞en(T)q<∞with2<q<∞,which is associated to the entropy numbers, contains non-approximable operators. As already shown in the (unpublished) thesis of the first-named author, the same conclusions hold for the ideals $${mathfrak{L}_q^{
m gel}}$$Lqgel and $${mathfrak{L}_q^{
m kol}}$$Lqkol generated by the Gelfand and Kolmogorov numbers, respectively. We present here a proof based on the new technique. The crux is the construction of non-approximable operators that are not just compact, but have a prescribed (not too strong) degree of compactness.