# 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

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

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 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.  MR2242851

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Centralizer and normalizer — Normalizer redirects here. For the process of increasing audio amplitude, see Audio normalization. Centralizer redirects here. For centralizers of Banach spaces, see Multipliers and centralizers (Banach spaces). In group theory, the centralizer… …   Wikipedia

• List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia