Local approach to monotonicity properties in \(\psi\)-direct sums of Köthe spaces. (English) Zbl 1420.46017
Summary: The points of upper and lower monotonicity of absolute norms in \(\mathbb{R} ^{2}\) are characterized. These results are used to calculate the characteristic of monotonicity of the space \(\langle \mathbb {R} ^{2},\| \cdot \| _{\psi}\rangle\). Moreover, criteria for points of upper and lower monotonicity as well as points of upper and lower local monotonicity in direct sums of Köthe spaces are given. Some global monotonicity properties of direct sums of Köthe spaces are established.
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |
Keywords:
\(\psi \)-direct sums; Köthe spaces; upper monotonicity points; lower monotonicity points; monotonicity properties; characteristic of monotonicityReferences:
| [1] | Akcoglu, M.A., Sucheston, L.: On uniform monotonicity of norms and ergodic theorems in function spaces. Rend. Circ. Mat. Palermo II Ser. Suppl 8, 325-335 (1985) · Zbl 0626.46010 |
| [2] | Bonsall, F.F., Duncan, J.: Numerical Ranges II. London Mathematical Society Lecture Note Series, vol. 10 (1973) · Zbl 0262.47001 |
| [3] | Birkhoff, G.: Lattice Theory. American Mathematical Society, Providence (1979) · Zbl 0505.06001 |
| [4] | Cerdá, J., Hudzik, H., Mastyło, M.: On the geometry of some Calderón-Lozanovskiĭ. Indag. Math. N.S. 6(1), 35-49 (1995) · Zbl 0831.46016 |
| [5] | Chen, S., He, X., Hudzik, H., Kamińska, A.: Monotonicity and best approximation in Orlicz-Sobolev spaces with the Luxemburg norm. J. Math. Anal. Appl. 344, 687-698 (2008) · Zbl 1153.46014 |
| [6] | Ciesielski, M., Kolwicz, P., Płuciennik, R.: Local approach to Kadec-Klee properties in symmetric function spaces. J. Math. Anal. Appl. 426, 700-726 (2015) · Zbl 1408.46023 |
| [7] | Foralewski, P., Kolwicz, P.: Local uniform rotundity in Calderón-Lozanovskiĭ spaces. J. Convex Anal. 14(2), 395-412 (2007) · Zbl 1159.46018 |
| [8] | Hudzik, H., Kamińska, A., Mastyło, M.: Monotonicity and rotundity properties in Banach lattices. Rocky Mountain J. Math. 30(3), 933-949 (2000) · Zbl 0979.46012 |
| [9] | Hudzik, H., Kurc, W.: Monotonicity properties of Musielak-Orlicz spaces and dominated best approximation in Banach latices. J. Approx. Theory 95, 353-368 (1998) · Zbl 0921.41015 |
| [10] | Hudzik, H., Narloch, A.: Local monotonicity structure of Calderón-Lozanovskiĭ spaces. Indag. Math. N.S. 15(1), 1-12 (2004) · Zbl 1074.46020 |
| [11] | Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Nauka, Moscow (1984). (in Russian) · Zbl 0555.46001 |
| [12] | Kato, M., Saito, K.-S., Tamura, T.: Uniform non-\[ l_1^n\] l1n-ness of \[\psi\] ψ-direct sums of Banach spaces. Math. Inequal. Appl. 7, 429-437 (2004) · Zbl 1057.46019 |
| [13] | Kolwicz, P.: Rotundity properties in Calderón-Lozanovskiĭ spaces. Houston J. Math. 31(3), 883-912 (2005) · Zbl 1084.46011 |
| [14] | Kolwicz, P., Płuciennik, R.: Local \[\Delta_2\left( x\right)\] Δ2x condition as a crucial tool for local structure of Calderón-Lozanovskiĭ spaces. J. Math. Anal. Appl. 356, 605-614 (2009) · Zbl 1211.46012 |
| [15] | Kolwicz, P., Płuciennik, R.: Points of upper local uniform monotonicity in Calderón-Lozanovskiĭ spaces. J. Convex Anal. 17.1, 111-130 (2010) · Zbl 1197.46011 |
| [16] | Kurc, W.: Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation. J. Approx. Theory 69(2), 173-187 (1992) · Zbl 0749.41031 |
| [17] | Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979) · Zbl 0403.46022 |
| [18] | Petrot, N., Suantai, S.: The criteria of strict monotonicity and rotundity points in generalized Calderón-Lozanovski ĭ spaces. Nonlinear Anal. 70, 2206-2215 (2009) · Zbl 1175.46014 |
| [19] | Takahashi, Y., Kato, M., Saito, K.-S.: Strict convexity of absolute norms on \[\mathbb{C}^2\] C2 and direct sums of Banach spaces. J. Inequal. Appl. 7, 179-186 (2002) · Zbl 1032.46022 |
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