Certain classes of sets in Banach spaces and a topological characterization of operators of type~$RN$

O. I. Reinov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 224-228

Résumé

We study properties of bounded sets in Banach spaces, connected with the concept of equimeasurability introduced by A. Grothendieck. We introduce corresponding ideals of operators and find characterizations of them in terms of continuity of operators in certain topologies. The following result (Corollary 9) follows from the basic theorems: Let $T$ be a continuous linear operator from a Banach space $X$ to a Banach space $Y$. The following assertions are equivalent: 1) $T$ is an operator of type $RN$; 2) for any Banach space $Z$, for any number $p$, $p>0$, and any $p$-absolutely summing operator $U:Z\to X$ the operator $YU$ is approximately $p$-Radonifying; 3) for any Banach space $Z$ and any absolutely summing operator $U:Z\to X$ the operator $YU$ is approximately $I$-Radonifying. We note that the implication $1)\Longrightarrow2)$, is apparently new even if the operator $T$ is weakly compact.