Operators on the Kalton–Peck space Z2

Abstract We study operators on the Kalton–Peck Banach space Z 2 from various points of view: matrix representations, examples, spectral properties and operator ideals. For example, we prove that there are non-compact, strictly singular operators acting on Z 2 , but the product of two of them is a compact operator. Among applications, we show that every copy of Z 2 in Z 2 is complemented, and each semi-Fredholm operator on Z 2 has complemented kernel and range, the space Z 2 is Z 2 -automorphic and we give a partial solution to a problem of Johnson, Lindenstrauss and Schechtman about strictly singular perturbations of operators on Z 2 .