Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-17T12:58:46.139Z Has data issue: false hasContentIssue false

Additive Functionals on Lorentz Spaces

Published online by Cambridge University Press:  20 November 2018

Pratibha G. Ghatage*
The Cleveland, State University Department Of Mathematics ClevelandOhio 44115
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If (X, β, μ) is a σ-finite, non-atomic measure space, and ϕ is an increasing non-negative concave function defined on the positive real numbers, we give a set of necessary and sufficient conditions for an additive functional T on the Lorentz space Nϕ to have an integral representation with a Caratheodory kernel. In the special case when T is statistical we classify the functional properties (enjoyed by the kernels) in terms of the Lorentz norm on the space.


Research Article
Copyright © Canadian Mathematical Society 1984


1. Batt, L. and Gruber, M., Additive operators on Lϕ and their Caratheodory kernels, Comment. Math. Special issue 2 (1979), 1-15.Google Scholar
2. Dunford, N. and Schwartz, J. T., Linear Operators I, General Theory, Pure and Applied Math., Vol. 7, Interscience, New York, 1958.Google Scholar
3. Mizel, V. J., Characterization of non-linear transformations possessing kernels, Can. J. Math., Vol. 22, No. 3, 1970, 449-471.Google Scholar
4. Mizel, V. J. and Sundaresan, K., Representation of vector-valued non-linear functions, Trans. AMS, Vol. 159, 1971, 11-127.Google Scholar
5. Steigerwalt, M. S. and White, A. J., Some function spaces related to Lp spaces, Proc. London Math. Soc. (3) Vol. 22, 1971, 137-163.Google Scholar
6. Sundaresan, K., Additive junctionals on Orlicz spaces, Studia Math., Vol. 32, 1969, 269-276.Google Scholar
7. Frewnowski, L. and Orlicz, W., Continuity and representation of orthogonally additive junctionals, Bull. Acad. Polon. Sci. Deri. Sci-Math-Astronom-Phys. 17 (1969), 647-653.Google Scholar