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 BanachStone 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 a_{T} : Ext(X) → K such that
Given two multipliers S and T on X, S is said to be an adjoint for T if
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.
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.
References
Categories: Banach spaces
 Operator theory
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