Multipliers and centralizers (Banach spaces)
Encyclopedia
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
.
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 aT : Ext(X) → K such that
Given two multipliers S and T on X, S is said to be an adjoint for T if
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, multipliers and centralizers are algebraic objects in the study of Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s. They are used, for example, in generalizations of the Banach-Stone theorem
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....
.
Definitions
Let (X, ||·||) be a Banach space over a field K (either the realReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
or complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s), and let Ext(X) be the set of extreme point
Extreme point
In mathematics, an extreme point of a convex set S in a real vector space is a point in S which does not lie in any open line segment joining two points of S...
s of the closed unit ball of the continuous dual space X∗.
A continuous linear operator
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....
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
Given two multipliers S and T on X, S is said to be an adjoint for T if
i.e. aS agrees with aT in the real case, and with the complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
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.