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In [[mathematics]], '''multipliers and centralizers''' are algebraic objects in the study of [[Banach space]]s. They are used, for example, in generalizations of the [[Banach-Stone theorem]].

==Definitions==

Let (''X'',&nbsp;||·||) be a Banach space over a field '''K''' (either the [[real number|real]] or [[complex number]]s), and let Ext(''X'') be the set of [[extreme point]]s of the [[closed unit ball]] of the [[continuous dual space]] ''X''<sup>∗</sup>.

A [[continuous linear operator]] ''T''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''X'' is said to be a '''multiplier''' if every point ''p'' in Ext(''X'') is an [[eigenvector]] for the [[adjoint operator]] ''T''<sup>∗</sup>&nbsp;:&nbsp;''X''<sup>∗</sup>&nbsp;→&nbsp;''X''<sup>∗</sup>. That is, there exists a function ''a''<sub>''T''</sub>&nbsp;:&nbsp;Ext(''X'')&nbsp;→&nbsp;'''K''' such that

:<math>p \circ T = a_{T} (p) p \mbox{ for all } p \in \mathrm{Ext} (X),</math>

making <math>a_{T} (p)</math> the eigenvalue corresponding to ''p''. Given two multipliers ''S'' and ''T'' on ''X'', ''S'' is said to be an '''adjoint''' for ''T'' if

:<math>a_{S} = \overline{a_{T}},</math>

i.e. ''a''<sub>''S''</sub> agrees with ''a''<sub>''T''</sub> in the real case, and with the [[complex conjugate]] of ''a''<sub>''T''</sub> 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''<sup>∗</sup>.
* If the field '''K''' is the real numbers, then every multiplier on ''X'' lies in the centralizer of ''X''.

==References==

* {{cite journal
|     last = Araujo
|    first = Jesús
|    title = The noncompact Banach-Stone theorem
|  journal = J. Operator Theory
|   volume = 55
|     year = 2006
|    issue = 2
|    pages = 285&ndash;294
|     issn = 0379-4024
}} {{MathSciNet|id=2242851}}

[[Category:Banach spaces]]
[[Category:Operator theory]]