Function theoretic results for complex interpolation families of Banach spaces

Rochberg, Richard

doi:10.1090/S0002-9947-1984-0743742-6

Function theoretic results for complex interpolation families of Banach spaces
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Trans. Amer. Math. Soc. 284 (1984), 745-758 Request permission

Abstract:

The theory of complex interpolation of Banach spaces is viewed as a branch of the theory of vector valued holomorphic functions. Versions of the Schwarz lemma, Liouville’s theorem, the identity theorem and the reflection principle are proved and are interpreted from the point of view of interpolation theory.

References

Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York, Inc., New York, 1967. MR 0230022
R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss, Complex interpolation for families of Banach spaces, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 269–282. MR 545314
R. R. Coifman, R. Rochberg, G. Weiss, M. Cwikel, and Y. Sagher, The complex method for interpolation of operators acting on families of Banach spaces, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 123–153. MR 576042
R. R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss, A theory of complex interpolation for families of Banach spaces, Adv. in Math. 43 (1982), no. 3, 203–229. MR 648799, DOI 10.1016/0001-8708(82)90034-2
S. G. Kreĭn and Ju. I. Petunin, Scales of Banach spaces, Uspehi Mat. Nauk 21 (1966), no. 2 (128), 89–168 (Russian). MR 0193499
Aleksander Pełczyński, Geometry of finite-dimensional Banach spaces and operator ideals, Notes in Banach spaces, Univ. Texas Press, Austin, Tex., 1980, pp. 81–181. MR 606222
Richard Rochberg, Interpolation of Banach spaces and negatively curved vector bundles, Pacific J. Math. 110 (1984), no. 2, 355–376. MR 726495
Richard Rochberg and Guido Weiss, Derivatives of analytic families of Banach spaces, Ann. of Math. (2) 118 (1983), no. 2, 315–347. MR 717826, DOI 10.2307/2007031
Thomas H. Wolff, A note on interpolation spaces, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 199–204. MR 654187
Misha Zafran, Spectral theory and interpolation of operators, J. Functional Analysis 36 (1980), no. 2, 185–204. MR 569253, DOI 10.1016/0022-1236(80)90099-3