IJPAM: Volume 36, No. 4 (2007)


João de Deus Marques
Department of Mathematics
Faculty of Sciences and Technology
New University of Lisbon
Quinta da Torre, Caparica, 2829-516, PORTUGAL
e-mail: jdm@fct.unl.pt

Abstract.A vectorial norm is a mapping from a linear space into a real ordered vector space with the properties of a usual norm. Here we consider the ordered vector space to be a unitary Archimedean-Riesz space (Yosida space), Dedekind complete and such that the intersection of all its hypermaximal bands is the zeroelement of the space ($\cal B$-regular Yosida space). Let $E$ be a linear space and $X,Y$ $\cal B$-regular Yosida spaces. In Theorem 2.2.1 we define a vectorial norm $G$ on the linear space ${\cal L}(E,Y)$ of all bounded linear operators from $E$ into $Y$ and with range in the partially ordered linear space ${\cal L}(X,Y)$ of all continuous linear operators from $X$ into $Y$. Next, in Theorem 3.1 we establish the following result: If $t$ is a bounded linear operator on a linear subspace $F$ of $E$ into $Y$, then there exists a bounded linear operator $T$ defined on $E$ that is an extension of $t$ and with the same vectorial norm, i.e. $G(T)=G(t)$. We finish with some consequences of this result.

Received: February 8, 2007

AMS Subject Classification: 46A22, 46A40

Key Words and Phrases: Yosida space, vectorial norm, family of seminorms, extension of bounded linear operators

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 36
Issue: 4