Abstract: We study operators on the Kalton-Peck Banach space Z2 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 Z2, but the product of two of them is a compact operator. Among applications, we show that every copy of Z2 in Z2 is complemented, and each semi-Fredholm operator on Z2 has complemented kernel and range, the space Z2 is Z2-automorphic and we give a partial solution to a problem of Johnson, Lindenstrauss and Schechtman about strictly singular perturbations of operators on Z2.

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