mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, multipliers and centralizers are algebraic objects in the study of

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...

s. They are used, for example, in generalizations of the

Banach–Stone theorem In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone. In brief, the Banach–Stone theorem allows one to recove ...

Definitions

Let (''X'', ‖·‖) be a Banach space over a field K (either the

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

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 fo ...

s), and let Ext(''X'') be the set of

extreme point In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex ...

s of the

closed unit ball In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...

of the

continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...

''X''^∗. A

continuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear o ...

''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 of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

for the

adjoint operator In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where ...

''T''^∗ : ''X''^∗ → ''X''^∗. That is, there exists a function ''a''_''T'' : Ext(''X'') → K such that :

p \circ T = a_ (p) p \; \mbox p \in \mathrm (X),

making

a_ (p)

the eigenvalue corresponding to ''p''. Given two multipliers ''S'' and ''T'' on ''X'', ''S'' is said to be an adjoint for ''T'' if :

a_ = \overline,

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, then) the complex conjugate of a + bi is equal to 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''.

References

* {{Functional analysis Banach spaces Operator theory

Definitions

Properties

See also

References