Abstract. We prove that, under suitable hypothesis on the dual space ${\cal A}$, the dual space generated by ${\cal A}$ and a finite rank operator $R$ has property $({\msbm A}_{1,\chi_{0}})$ without having any property $E^r_{0,\gamma }$. We also completely discuss in terms of rank $R$, properties $({\msbm A}_{m,n})$ for our example.

AMS Subject Classification (1991): 47A55, 47D25, 47D15

Recu le 31 juillet 1992, sans forme revue le 2 février 1993. (Registered under 5559/2009.)