Wiki states the following definition of a centralizer of a Banach space.
Let $(X,\|\cdot\|)$ be a Banach space over a field $\mathbb{K}$ and let Ext$(X)$ be the set of extreme points of the closed unit ball of the continuous dual space $X^*.$ A continuous linear operator $T:X\to X$ is said to be a multiplier if every point in Ext$(X)$ is an eigenvector for the adjoint operator $T^*:X\to X,$ that is, there exists a function $a_T:Ext(X) \to \mathbb{K}$ such that $$p \circ T = a_T(p)p$$ for all $p\in Ext(X).$ Given two multipliers $S$ and $T$ on $X,$ $S$ is said to be an adjoint for $T$ if $$a_s = a_T.$$ (I only consider real case) The centralizer of $X,$ denoted $Z(X),$ is the set of all multipliers on $X$ for which an adjoint exists.
Question: Suppose that $X = \mathbb{R}$ with Euclidean norm $\|\cdot\|_2.$ What are some examples of centralizer of $\mathbb{R}?$ I have trouble giving example for centralizer. Any hint would be appreciated.
EDITED: Based on comment given by Nate,it seems that $X=\mathbb{R}$ is quite 'trivial'. I would like to ask what about $X=\mathbb{R}^2$ with $\|\cdot\|_1$, that is, $\|(x,y)\|_1=|x|+|y|$?