Abstract
Let $\|A\|_{\mathrm{e}}$ be the essential norm of an operator $A$ and
$\|A\|_m$ be the infimum of the norms of restrictions of $A$ to the subspaces
of finite codimension. We show that if $\|A\|_{\mathrm{e}}<M\|A\|_m$ holds
for every bounded noncompact operator $A$ from a Banach space $X$ to every
Banach space $Y$, then the space $X$ has the dual compact approximation
property. This is an analogue of a result by K. Astala and H.-O. Tylli (1987)
concerning the Hausdorff measure of noncompactness and the bounded
compact approximation property.
$\|A\|_m$ be the infimum of the norms of restrictions of $A$ to the subspaces
of finite codimension. We show that if $\|A\|_{\mathrm{e}}<M\|A\|_m$ holds
for every bounded noncompact operator $A$ from a Banach space $X$ to every
Banach space $Y$, then the space $X$ has the dual compact approximation
property. This is an analogue of a result by K. Astala and H.-O. Tylli (1987)
concerning the Hausdorff measure of noncompactness and the bounded
compact approximation property.
| Original language | English |
|---|---|
| Article number | DOI: 10.1007/s00013-021-01679-w |
| Pages (from-to) | 73-77 |
| Number of pages | 5 |
| Journal | ARCHIV DER MATHEMATIK |
| Volume | 118 |
| Issue number | 1 |
| Early online date | 24 Nov 2021 |
| Publication status | Published - 1 Jan 2022 |