×

Equimodular and linearity in modular spaces. (English) Zbl 1035.46003

Let \(X_\rho\) and \(Y_\rho\) be two modular spaces generated by modulars \(\rho_X\) and \(\rho_Y\), respectively. The authors consider the question when \(\rho\) is an inverting modular for the norm (\(F\)-norm) in \(X\), and the problem of sufficient conditions in order that a mapping \(T: X_\rho\to Y_\rho\) be affine. The key role is played by the notion of a \(\delta\)-equimodular mapping \(T\), i.e., a map \(T\) such that for every \(x,y\in X_\rho\) with \(\rho_X(x- y)< \delta\), \(\rho_Y(Tx- Ty)= \rho_X(x- y)\) holds. It is proved that if \((X,\rho_X)\), \((Y,\rho_Y)\) are \(\beta\)-positive homogeneous \((0< \beta\leq 1)\) and a surjection \(T: X_\rho\to Y_\rho\) is \(\delta\)-equimodular, then \(T\) is affine in \(XJ_\rho\).

MSC:

46A80 Modular spaces
47A99 General theory of linear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Day, M. M., Normed Linear Spaces (1958), Springer-Verlag: Springer-Verlag Berlin · Zbl 0082.10603
[2] Kozlowski, W. M., Modular Function Spaces. Modular Function Spaces, Pure Appl. Math. (1988), Dekker · Zbl 0661.46023
[3] Y.-M. Ma, Dissertation, Department of Mathematics in Nankai University, 1999 (in Chinese); Y.-M. Ma, Dissertation, Department of Mathematics in Nankai University, 1999 (in Chinese)
[4] Mazur, S.; Ulam, S., Sur les transformations isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris, 194, 946-948 (1932) · Zbl 0004.02103
[5] Musielak, J., Orlicz Spaces and Modular Spaces. Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034 (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0557.46020
[6] Rassias, Th. M., Properties of isometric mappings, J. Math. Anal. Appl., 235, 108-121 (1999) · Zbl 0936.46009
[7] Rolewicz, S., Metric Linear Spaces (1972), PWN: PWN Warsaw · Zbl 0226.46001
[8] Wang, J.; Wang, J.-Y., The relations between Orlicz \(F\)-norm ‖·‖\(_ρ\) and its parental modular \(ρ\) and inverting problem, Acta Math. Sci., 20, 58-62 (2000), (in Chinese) · Zbl 0958.46002
[9] Wang, J., On generalizations of Mazur-Ulam’s isometric theorem, J. Math. Anal. Appl., 263, 510-521 (2001) · Zbl 1039.46002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.