Multipliers and centralizers (Banach spaces)

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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.

[edit] 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 a_T : Ext(X) → K such that

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

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. a_S agrees with a_T in the real case, and with the complex conjugate of a_T in the complex case.

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

[edit] 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.