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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.
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
52A41 Convex functions and convex programs in convex geometry
46B07 Local theory of Banach spaces
Full Text: DOI
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