## Lions–Schechter’s methods of complex interpolation and some quantitative estimates.(English)Zbl 1110.46015

After reviewing variants of the Lions–Schechter method of complex interpolation with derivatives, the author formulates some basic inequalities and then discusses the behaviour under Lions–Schechter interpolation of estimates for multidimensional rotundity of Banach spaces, measures of weak non-compactness, and spectral inequalities for bounded linear operators.

### MSC:

 46B70 Interpolation between normed linear spaces 46M35 Abstract interpolation of topological vector spaces 46A03 General theory of locally convex spaces 46B50 Compactness in Banach (or normed) spaces 47B07 Linear operators defined by compactness properties
Full Text:

### References:

 [1] E. Albrecht, Spectral interpolation, in: Operator Theory: Advances and Applications, vol. 14, Birkhäuser, Basel, 1984, pp. 13-37. [2] Albrecht, E.; Müller, V., Spectrum of interpolated operators, Proc. amer. math. soc., 129, 807-814, (2001) · Zbl 0967.46021 [3] J. Bergh, J. Löfström, Interpolation Spaces, Grundlehren der mathematischen Wissenschaften, vol. 223, Springer, Heidelberg, New York, 1976. [4] Casini, E.; Vignati, M., The uniform non-squareness for the complex interpolation spaces, J. math. anal. appl., 104, 518-523, (1992) · Zbl 0773.46009 [5] Fan, M., Complex interpolation functors with a family of quasi-power function parameters, Studia math., 111, 283-305, (1994) · Zbl 0805.46075 [6] M. Fan, On interpolation of vector-valued Banach lattices and Calderón-Lozanovskii construction, Math. Nachr. 227 (2001) 63-80; M. Fan, Erratum, Math. Nachr. 278 (2005) 735-737. · Zbl 1001.46009 [7] Fan, M.; Kaijser, S., Complex interpolation with derivatives of analytic functions, J. funct. anal., 120, 380-402, (1994) · Zbl 0806.46077 [8] Kaijser, S., Interpolation of Banach algebras and open sets, Integral equations operor. theory, 41, 189-222, (2001) · Zbl 1022.46046 [9] Kryczka, A.; Prus, S., Measure of weak noncompactness under complex interpolation, Studia math., 147, 89-102, (2001) · Zbl 0995.46018 [10] S. Prus, Geometrical background of metric fixed point theory, in: Handbook of metric fixed point theory, Kluwer Academic Publishers, Dordrecht, 2001, pp. 93-132. · Zbl 1018.46010 [11] Salvatori, M.; Vignati, M., On the stability of multi-dimensional convexity under interpolation, Math. nachr., 164, 299-308, (1993) · Zbl 0810.46021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.