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 a_T : Ext(X) → K such that

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

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

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

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.

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

• Araujo, Jesús (2006). "The noncompact Banach-Stone theorem". J. Operator Theory. 55 (2): 285–294. ISSN 0379-4024. MR 2242851