Operators on the Kalton-Peck space $Z_2$

TL;DR

This paper investigates operators on the Kalton-Peck space $Z_2$, revealing properties of their spectra, ideals, and the structure of subspaces, including the existence of non-compact strictly singular operators and complemented copies.

Contribution

It provides new insights into the structure of operators on $Z_2$, including examples, spectral analysis, and solutions to open problems regarding strictly singular perturbations.

Findings

Existence of non-compact, strictly singular operators on $Z_2$

Every copy of $Z_2$ in itself is complemented

Semi-Fredholm operators on $Z_2$ have complemented kernel and range

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