Property \((V)\) still fails the three-space property (Q2839810)

A Banach \(X\) is said to have (Pełczyński's) property (V) if, for every non-weakly compact operator \(T:X\rightarrow Y\), there is a subspace \(M\) of \(X\) isomorphic to \(c_0\) such that the restriction \(T|_M\) has a continuous inverse. In this paper, the authors construct twisted sums of \(C[0,1]\) with itself, providing new examples showing that property (V) is not a three-space property. This fact was shown by \textit{J. M. F. Castillo} and \textit{M. González} [Glasg. Math. J. 36, No. 3, 297--299 (1994; Zbl 0821.46011)] with a somewhat more artificial example. They also prove that every twisted sum of \(c_0(\Gamma)\) and a space with property (V) has property (V).NEWLINENEWLINEThe authors say that \(X\) has Rosenthal's property (V\(^*\)) if, for every operator \(T:X\rightarrow Y\) with \(T^*(Y^*)\) non-separable, there is a subspace \(M\) of \(X\) isomorphic to \(C[0,1]\) such that the restriction \(T|_M\) has a continuous inverse, and show that it is not a three-space by constructing a twisted sum of \(C[0,1]\) with itself failing Rosenthal's property (V\(^*\)).NEWLINENEWLINEA twisted sum of two Banach spaces \(Y\) and \(Z\) is a Banach space \(X\) with a subspace \(M\) isomorphic to \(Y\) such that \(X/M\) is isomorphic to \(Z\). For detailed information on the topic of the paper, we refer to the monograph [\textit{J. M. F. Castillo} and \textit{M. González}, Three-space problems in Banach space theory. Berlin: Springer (1997; Zbl 0914.46015)].