Sequence-singular operators (Q302075)

Given a seminormalized basis \(e = (e_n)\) for a Banach space \(E\), a continuous linear operator \(T: X \rightarrow Y\), where \(X\) and \(Y\) are Banach spaces, is called \((e_n)\)-singular if \((Tx_n)\) fails to dominate \((e_n)\), for every normalized basic sequence \((x_n)\) in \(X\). The class of all \((e_n)\)-singular operators from \(X\) to \(Y\) is denoted by \(\mathcal{WS}_{e,w_1}(X;Y)\). The definition of this new class appears in the paper [Extr. Math. 26, No. 2, 173--194 (2011; Zbl 1264.46011)], due to \textit{K. Beanland} and \textit{D. Freeman}, that also proves interesting results. For instance, taking \(E = c_0\) and \(E = \ell_1\), with \(e = (e_n)\) the canonical basis for \(E\), then \(\mathcal{WS}_{e,w_1} = \mathcal K\) and \(\mathcal{WS}_{e,w_1} = \mathcal R\), respectively, where \(\mathcal K\) and \(\mathcal R\) are the (closed ideals of) compact and Rosenthal operators.NEWLINENEWLINENEWLINEIn this nice paper, the authors prove that, contrary to the expectation generated by the above example, the class \(\mathcal{WS}_{e,w_1}\) does not form an operator ideal in general. They show that, for \(E = \ell_p\), \(1 <p<\infty\), and \(e = (e_n)\) the canonical basis for \(E\), there are Banach spaces \(X\) and \(Y\) such that \(\mathcal{WS}_{e,w_1}(X;Y)\) is not closed under addition. Even so, they use the class \(\mathcal{WS}_{e,w_1}\) to answer a longstanding open question proposed by A. Pietsch concerning the closed ideal structures of \(\mathcal L(\ell_p \oplus \ell_q)\), \(1 \leq p < q< \infty\).