Examples of improjective operators (Q1974147)

The authors show that a certain linear operator on a certain indecomposable Banach space is improjective but it is not inessential. They give a characterization of indecomposable Banach spaces in terms of improjective operators, and they use a rightshift operator in an indecomposable Banach space constructed by \textit{W. T. Gowers} and \textit{B. Maurey} [Math. Ann. 307, No. 4, 543-568 (1997; Zbl 0876.46006)] to give their example. The results of the paper give negative answers to several questions on improjective operators.