Multipliers and centralizers (Banach spaces)

In mathematics, multipliers and centralizers are algebraic objects in the study of Banach spaces. They are used, for example, in generalizations of the Banach–Stone theorem.

Definitions
Let (X, ‖·‖) be a Banach space over a field K (either the real or complex numbers), 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 → X is said to be a multiplier if every point p in Ext(X) is an eigenvector for the adjoint operator T∗ : X∗ → X∗. That is, there exists a function aT : Ext(X) → K such that


 * $$p \circ T = a_{T} (p) p \; \mbox{ for all } p \in \mathrm{Ext} (X),$$

making $$a_{T} (p)$$ the eigenvalue corresponding to p. Given two multipliers S and T on X, S is said to be an adjoint for T if


 * $$a_{S} = \overline{a_{T}},$$

i.e. aS agrees with aT in the real case, and with the complex conjugate of aT in the complex case.

The centralizer (or commutant) of X, denoted Z(X), is the set of all multipliers on X for which an adjoint exists.

Properties

 * The multiplier adjoint of a multiplier T, if it exists, is unique; the unique adjoint of T is denoted T∗.
 * If the field K is the real numbers, then every multiplier on X lies in the centralizer of X.