The GL-l.u.st.\ constant and asymmetry of the Kalton-Peck twisted sum in finite dimensions

TL;DR

This paper investigates the growth of the GL-l.u.st. constant in finite-dimensional Kalton-Peck twisted sums, revealing a logarithmic growth pattern and contrasting it with the bounded GL constant, and also explores asymmetry constants.

Contribution

It provides the first explicit example showing different growth orders for GL and GL-l.u.st. constants in finite-dimensional Banach spaces.

Findings

GL-l.u.st. constant grows as log n in Z_2^n

GL constant remains bounded in Z_2^n

Analysis of asymmetry constants in Z_2^n

Abstract

We prove that the Kalton-Peck twisted sum $Z_2^n$ of $n$-dimensional Hilbert spaces has GL-l.u.st.\ constant of order $\log n$ and bounded GL constant. This is the first concrete example which shows different explicit orders of growth in the GL and GL-l.u.st.\ constants. We discuss also the asymmetry constants of $Z_2^n$.