Geometrical Background of Metric Fixed Point Theory

Some quantitative estimates concerning multi-dimensional rotundity, weak noncompactness, and certain spectral inequalities are formulated for Lions-Schechter's complex methods of interpolation with derivatives.The theory of the complex interpolation methods owes its origin to the famous interpolation theorem of Riesz-Thorin [3, Theorem 1.1.1]. Some fundamental inequalities due to Calderón [3, Lemma 4.3.2] imply that the boundedness of linear operators can be interpolated between Banach spaces with a logarithmically convex estimate for the norms of the interpolated operators. This motivated several authors to investigate the behavior of other properties of Banach spaces and linear operators under interpolation in both qualitative and quantitative way. For instance, Salvatori and Vignati obtained the estimate for multi-dimensional rotundity [11], Kryczka and Prus studied the measure of weak noncompactness [9], and Albrecht and Müller formulated some spectral inequalities under the complex interpolation methods [2].In the present paper, we treat the similar problems for the more general Lions-Schechter's methods of complex interpolation with derivatives. In the first section, we review the different variants of these methods and formulate some basic inequalities. Section 2 includes the estimate

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“…Clearly, the condition N (E) > 1 characterizes spaces E with UNS. It is well known that all uniformly convex Banach spaces possess UNS [10,Theorem 5.12]. It is difficult to calculate the normal structure coefficient in an arbitrary Banach space.…”

Section: This Number Is Called the Chebyshev Radius Of Amentioning

confidence: 99%