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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'', ||·||) 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''∗.

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

:<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''''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==

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

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

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