In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, multipliers and centralizers are algebraic objects in the study of
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
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 or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s), and let Ext(''X'') be the set of
extreme point
In mathematics, an extreme point of a convex set S in a Real number, real or Complex number, complex vector space is a point in S that does not lie in any open line segment joining two points of S. The extreme points of a line segment are calle ...
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
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...
for the
adjoint operator ''T''
∗ : ''X''
∗ → ''X''
∗. That is, there exists a function ''a''
''T'' : Ext(''X'') → K such that
:
making
the eigenvalue corresponding to ''p''. 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
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
of ''a''
''T'' in the complex case.
The centralizer (or commutant) 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''.
See also
*
Centralizer and normalizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...
References
*
{{Functional analysis
Banach spaces
Operator theory