Lifting Non-topological Divisors of Zero modulo the Compact Operators
✍ Scribed by H.O. Tylli
- Publisher
- Elsevier Science
- Year
- 1994
- Tongue
- English
- Weight
- 967 KB
- Volume
- 125
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
This paper provides the first examples of non-semiFredholm operators (S) on a Banach space such that the left or right multiplication operators (R \mapsto S R) or (R \mapsto R S) define linear embeddings of the corresponding Calkin algebra into itself. For instance, if (S) is a bounded linear operator on (C(0,1)) with closed range such that (\operatorname{Ker} S \sim l^{1}), then there is a constant (c>0) with
[
\operatorname{dist}(S R, K(C(0,1))) \geqslant c \operatorname{dist}(R, K(C(0,1)))
]
for all bounded operators (R \in L(C(0,1))). Here (K(C(0,1))) stands for the compact operators on (C(0,1)). Moreover, if (S: L^{1} \rightarrow L^{1}) has closed range and (L^{1} / \operatorname{Im} S) contains no copies of (l^{1}), then there is a constant (c>0) such that