  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 Schetchman about strictly singular
perturbations of operators on $Z_2$.
