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More on convexity and smoothness of operators. (English) Zbl 1200.47001
Authors’ abstract: Let \(X\) and \(Y\) be Banach spaces and \(T:Y \to X\) be a bounded operator. In this note, we show first some operator versions of the dual relation between \(q\)-convexity and \(p\)-smoothness of Banach spaces case. Making use of them, we prove then the main result of this note, namely, that the two notions of uniform \(q\)-convexity and uniform \(p\)-smoothness of an operator \(T\) introduced by J. Wenzel [J. Convex Anal. 12, No. 1, 159–171 (2005; Zbl 1087.46007)] are actually equivalent to that the corresponding \(T\)-modulus \(\delta _T\) of convexity and the \(T\)-modulus \(\rho _T\) of smoothness introduced by G. Pisier [Isr. J. Math. 20, 326–350 (1975; Zbl 0344.46030)] are of power type \(q\) and of power type \(p\), respectively. This is also an operator version of a combination of a Hoffman’s theorem and a Figiel-Pisier’s theorem. As their application, we show, finally, that a recent theorem of J. Borwein, A. J. Guirao, P. Hájek and J. Vanderwerff [Proc. Am. Math. Soc. 137, No. 3, 1081–1091 (2009; Zbl 1184.52009)] about \(q\)-convexity of Banach spaces is again valid for \(q\)-convexity of operators.
MSC:
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
52A41 Convex functions and convex programs in convex geometry
46B07 Local theory of Banach spaces
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[1] Beauzamy, B., Opérateurs uniformément convexifiants, Studia math., 57, 2, 103-139, (1976) · Zbl 0372.46016
[2] Beauzamy, B., Introduction to Banach spaces and their geometry, (1985), North-Holland Publishing Co. Amsterdam/New York · Zbl 0585.46009
[3] Borwein, J.; Guirao, A.J.; Hajek, P.; Vanderwerff, J., Uniformly convex functions on Banach spaces, Proc. amer. math. soc., 137, 3, 1081-1091, (2009) · Zbl 1184.52009
[4] A. Dvoretzky, Some results on convex bodies and Banach spaces, in: Proc. Internat. Sympos. Linear Spaces, 1961 · Zbl 0119.31803
[5] Enflo, P., Banach spaces which can be given an equivalent uniformly convex norm, Israel J. math., 13, 281-288, (1972)
[6] Figiel, T.; Pisier, G., Séries aléatoires dans LES espaces uniformément convexes ou uniformément lisses, C. R. acad. sci. Paris Sér. A, 279, 611-614, (1974), (in French) · Zbl 0326.46007
[7] J. Hoffman, On the modulus of smoothness and the \(G_\ast\)-conditions in B-spaces, Preprint Series, Aarhus Universitet, Matematisk Inst., 1974
[8] Guirao, A.J.; Hajek, P., On the moduli of convexity, Proc. amer. math. soc., 135, 10, 3233-3240, (2007) · Zbl 1129.46004
[9] Lindenstrauss, J., On the modulus of smoothness and divergent series in Banach spaces, Michigan math. J., 10, 241-252, (1963)
[10] Nordlander, G., The modulus of convexity in normed linear spaces, Ark. mat., 4, 15-17, (1960) · Zbl 0092.11402
[11] Pisier, G., Martingales with values in uniformly convex spaces, Israel J. math., 20, 3-4, 326-350, (1975) · Zbl 0344.46030
[12] Royden, H.I., Real analysis, (1968), Macmillan New York
[13] Wenzel, J., Uniformly convex operators and martingale type, Rev. mat. iberoamericana, 18, 1, 211-230, (2002) · Zbl 1021.46006
[14] Wenzel, J., Strong martingale type and uniform smoothness, J. convex anal., 12, 1, 159-171, (2005) · Zbl 1087.46007
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