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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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##### References:
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