On the bounded compact approximation property and measures of noncompactness (Q1820354)

Let E, F be Banach spaces, \(B_ E\) the closed unit ball and T:E\(\to F\) linear. Then the measure of noncompactness \(\gamma (T)=\inf \{r>0:TB_ E\) can be covered by a finite number of balls of radius \(r\}\) induces a natural norm in the Calkin algebra \(C(E)=L(E)/K(E)\) and in the quotient space \(C(E,F)=L(E,F)/K(E,F)\). Here K(E,F) denotes the space of compact operators. The paper studies the following problem: When is (C(E,F),\(\gamma)\) complete, i.e. when is \(\gamma\) equivalent to the essential norm \(\| T\|_ e=\inf \{\| t-K\|:K\in K(E,F)\}.\) It is shown that (C(E,F),\(\gamma)\) is complete for every E if and only if F has the bounded compact approximation property. Hence, by the work of Per Enflo, there exists a noncomplete Calkin algebra (C(E),\(\gamma)\). The question of the completeness has arisen, in particular, in the study of the algebraic properties of the semi Fredholm operators \(\Phi_+(E)\), \(\Phi_-(E)\). As an application the paper answers e.g. in the negative the previous question whether the image of \(\Phi_{\pm}(E)\) in C(E) is equal to the set of those elements of C(E) which are not left (right) topological divisors of zero. Further information on the algebraic properties of semi-Fredholm operators can also be found in the recent paper of the authors [J. Lond. Math. Soc., II. Ser. 34, 541-551 (1986; Zbl 0585.47005)].