On the geometry of the multiplier space of $\ell_A^p$

TL;DR

This paper investigates the geometric properties of the multiplier space on $\, ext{ell}_A^p$ for $p$ not equal to 2, revealing differences from Hilbert space behavior and characterizing extremal multipliers.

Contribution

It establishes the failure of parallelogram laws and Pythagorean inequalities in the multiplier space and characterizes extremal multipliers as monomials.

Findings

Failure of weak parallelogram laws in $\, ext{ell}_A^p$

Failure of Pythagorean inequalities in $\, ext{ell}_A^p$

Extremal multipliers are monomials for $p eq 2$

Abstract

For $p \in (1,\infty)\setminus \{2\}$, some properties of the space $\mathscr{M}_p$ of multipliers on $\ell^p_A$ are derived. In particular, the failure of the weak parallelogram laws and the Pythagorean inequalities is demonstrated for $\mathscr{M}_p$. It is also shown that the extremal multipliers on the $\ell^p_A$ spaces are exactly the monomials, in stark contrast to the $p=2$ case.