Some applications of the open mapping theorem in locally convex cones

S. Jafarizad; A. Ranjbari

doi:10.37863/umzh.v73i3.222

S. Jafarizad Univ. Tabriz, Iran
A. Ranjbari Univ. Tabriz, Iran

DOI: https://doi.org/10.37863/umzh.v73i3.222

Keywords: Locally convex cones, adjoint, open mapping, uc-cone LATEXstyle

Abstract

UDC 515.12

We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones. Specially, we prove some relations between an open linear operator and its adjoint in $uc$-cones (locally convex cones which their convex quasi-uniform structures are generated by one element).

References

D. Ayaseh, A. Ranjbari, Bornological locally convex cones, Le Mat., 69, № 8 12, 267 – 284 (2014), https://doi.org/10.4418/2014.69.2.23

S. Jafarizad, A. Ranjbari, Openness and continuity in locally convex cones, Filomat, 31, № 16, 5093 – 5103 (2017), https://doi.org/10.2298/fil1716093j DOI: https://doi.org/10.2298/FIL1716093J

G. J. O. Jameson, Topology and normed spaces, Chapman and Hall, London (1974).

K. Keimel, W. Roth, Ordered cones and approximation, Lect. Notes Math., 1517, Springer-Verlag, Berlin (1992). DOI: https://doi.org/10.1007/BFb0089190

M. R. Motallebi, H. Saiflu, Products and direct sums in locally convex cones, Canad. Math. Bull., 55, № 4, 783 – 798 (2012), https://doi.org/10.4153/CMB-2011-161-8 DOI: https://doi.org/10.4153/CMB-2011-161-8

A. Ranjbari, H. Saiflu, Projective and inductive limits in locally convex cones, J. Math. Anal. and Appl., 332, № 2, 1097 – 1108 (2007), https://doi.org/10.1016/j.jmaa.2006.11.001 DOI: https://doi.org/10.1016/j.jmaa.2006.11.001

A. Ranjbari, H. Saiflu, A locally convex quotient cone, Methods Funct. Anal., 12, № 3, 281 – 285 (2006).

W. Roth, Operator-valued measures and integrals for cone-valued functions, Lect. Notes Math., 1964, Springer-Verlag, Berlin (2009).

W. Roth, Locally convex quotient cones, J. Convex Anal., 18, № 4, 903 – 913 (2011).

W. Roth, A uniform boundedness theorem for locally convex cones, Proc. Amer. Math. Soc., 126, № 7, 1973 – 1982 (1998), https://doi.org/10.1090/S0002-9939-98-04699-1 DOI: https://doi.org/10.1090/S0002-9939-98-04699-1