bounded linear extension of an operator

0.1 Bounded Linear Extension

Let $X$ and $Y$ be normed vector spaces and denote by $\tilde{X}$ and $\tilde{Y}$ their completions.

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Theorem 1 - Every bounded linear operator $T:X⟶Y$ can be extended to a bounded linear operator $\tilde{T}:\tilde{X}⟶\tilde{Y}$. Moreover, this extension is unique and $\parallel T\parallel =\parallel \tilde{T}\parallel$.

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In particular, if $Y$ is a Banach space and $S\subseteq X$ is a (not necessarily closed (http://planetmath.org/ClosedSet)) subspace of $X$, an operator $T:S⟶Y$ has an extension $\tilde{T}:\overline{S}⟶Y$ to $\overline{S}$ (the closure (http://planetmath.org/Closure) of $S$), which is unique and such that $\parallel T\parallel =\parallel \tilde{T}\parallel$.

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 $\mathrm{𝐍𝐕𝐞𝐜}$ be the category of normed vector spaces (whose morphisms (http://planetmath.org/Category) are the bounded linear operators) and $\mathrm{𝐁𝐚𝐧}$ the categroy of Banach spaces (whose are also the bounded linear operators). We have that

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Theorem 2 - The completion $\stackrel{~}{}:\mathrm{𝐍𝐕𝐞𝐜}⟶\mathrm{𝐁𝐚𝐧}$, which associates each normed vector space $X$ with its completion $\tilde{X}$ and each bounded linear operator $T$ with its extension $\tilde{T}$, is a covariant functor.

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This, in particular, implies that $\widetilde{T_1{T}_2}=\widetilde{T_1}\widetilde{T_2}$.

0.3 Extensions in Spaces with Additional Structure

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

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Theorem 3 - If $X$ and $Y$ be normed vector spaces that are also normed algebras (normed *-algebras) and $T:X⟶Y$ is a bounded homomorphism (bounded *-homomorphism), then the unique bounded linear extension $\tilde{T}$ of $T$ is also an homomorphism (*-homomorphism).

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Thus, completion is also a covariant functor from the category of normed algebras (normed *-algebras) to category of Banach algebras (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	continuous extension of an operator
Defines	completion of normed spaces is a covariant functor
Defines	continuous extension of a normed algebra homomorphism