Ordinal ranks on weakly compact and Rosenthal operators (Q2906822)

Let \(X, Y\) be Banach spaces. A (bounded linear) operator \(T:X\to Y\) is Rosenthal if for each bounded sequence \((x_n)_1^{\infty}\) in \(X\), \((Tx_n)\) has a weakly Cauchy subsequence. The paper starts with the followingNEWLINENEWLINEFact 1. An operator \(T:X\to Y\) is weakly compact (Rosenthal) if and only if, for every normalized basic sequence \((x_n)\) in \(X\) and \(\varepsilon>0\), there are scalars \((a_i)_1^l\) and a finite set \((n_i)_1^l\subset \mathbb{N}\) such that NEWLINE\[NEWLINE\left\|\sum\nolimits_{i=1}^la_iTx_{n_i}\right\| < \varepsilon\max_{1\leq k\leq l}\left|\sum\nolimits_{i=k}^la_i\right|,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\left(\text{resp.,}\;\;\left\|\sum\nolimits_{i=1}^la_iTx_{n_i}\right\|< \varepsilon\sum\nolimits_{i=1}^l|a_i|\right).NEWLINE\]NEWLINE Roughly speaking, Fact 1 states that showing that an operator is weakly compact or Rosenthal reduces to finding, for each \(\varepsilon>0\) and for each normalized basic sequence \((x_n)\), a finite subset \((n_i)_1^l\) satisfying certain properties. So, we may define subclasses of weakly compact or Rosenthal operators by requiring that these finite subsets can always be chosen as elements of some prescribed collections of finite subsets of \(\mathbb{N}\). The authors choose the Schreier families \(S_{\xi}\), \(\xi<\omega_1\), obtain the classes of \(S_{\xi}\)-weakly compact and \(S_{\xi}\)-Rosenthal operators, and prove several results concerning the analytic properties of these classes in \(L(X,Y)\).