Operators on Bourgain-Delbaen's spaces

TL;DR

This paper investigates the properties of operators between classical Hilbert spaces and Bourgain-Delbaen's spaces, showing that under certain conditions, all such operators are necessarily compact, revealing structural insights about these spaces.

Contribution

It establishes that for specific parameters, all operators between  and Bourgain-Delbaen's spaces are compact, highlighting a new operator-theoretic property of these spaces.

Findings

Operators between  and X_{a,b} are compact under certain parameters.

Operators from X_{a,b} to  are also compact under the same conditions.

The result characterizes the operator structure of Bourgain-Delbaen's spaces.

Abstract

We prove that, for a suitable choice of real numbers $a, b$, every operator from $\ell_2$ to $X_{a,b}$ and from $X_{a,b}$ to $\ell_2$ must be compact, where $X_{a,b}$ is the Bourgain- Delbaen's space.