bounded linear extension of an operator

0.1 Bounded Linear Extension

Let X and Y be normed vector spacesPlanetmathPlanetmath and denote by X~ and Y~ their completions.

Theorem 1 - Every bounded linear operator T:XY can be extended to a bounded linear operator T~:X~Y~. Moreover, this extensionPlanetmathPlanetmathPlanetmath is unique and T=T~.

In particular, if Y is a Banach spaceMathworldPlanetmath and SX is a (not necessarily closed ( subspaceMathworldPlanetmath of X, an operator T:SY has an extension T~:S¯Y to S¯ (the closureMathworldPlanetmathPlanetmath ( of S), which is unique and such that T=T~.

0.2 Functorial Property of the Extension

The extension of bounded linear operators between two normed vector spaces to their completions is functorial. More precisely, let 𝐍𝐕𝐞𝐜 be the categoryMathworldPlanetmath of normed vector spaces (whose morphismsMathworldPlanetmath ( are the bounded linear operators) and 𝐁𝐚𝐧 the categroy of Banach spaces (whose are also the bounded linear operators). We have that

Theorem 2 - The completion ~:𝐍𝐕𝐞𝐜𝐁𝐚𝐧, which associates each normed vector space X with its completion X~ and each bounded linear operator T with its extension T~, is a covariant functorMathworldPlanetmath.

This, in particular, implies that T1T2~=T1~T2~.

0.3 Extensions in Spaces with Additional Structure

When the normed vector spaces X and Y have some additional structureMathworldPlanetmath (for example, when X and Y are normed algebras) it is interesting to know if the (unique) extension of a morphism T:XY preserves the additional structure. The following theorem states that this indeed the case for normed algebras or normed *-algebrasPlanetmathPlanetmath.

Theorem 3 - If X and Y be normed vector spaces that are also normed algebras (normed *-algebras) and T:XY is a boundedPlanetmathPlanetmathPlanetmathPlanetmath homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (bounded *-homomorphism), then the unique bounded linear extension T~ of T is also an homomorphism (*-homomorphism).

Thus, completion is also a covariant functor from the category of normed algebras (normed *-algebras) to category of Banach algebrasMathworldPlanetmath (Banach *-algebras).

Title bounded linear extension of an operator
Canonical name BoundedLinearExtensionOfAnOperator
Date of creation 2013-03-22 17:35:17
Last modified on 2013-03-22 17:35:17
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 11
Author asteroid (17536)
Entry type Theorem
Classification msc 46B99
Classification msc 47A05
Synonym continuousMathworldPlanetmathPlanetmath extension of an operator
Defines completion of normed spaces is a covariant functor
Defines continuous extension of a normed algebra homomorphism