Arkiv för Matematik

  • Ark. Mat.
  • Volume 31, Number 2 (1993), 307-338.

Interpolation of subcouples and quotient couples

Svante Janson

Full-text: Open access


We extend recent results by Pisier on K-subcouples, i.e. subcouples of an interpolation couple that preserve the K-functional (up to constants) and corresponding notions for quotient couples. Examples include interpolation (in the pointwise sense) and a reinterpretation of the Adamyan-Arov-Krein theorem for Hankel operators.


This work was done at the Mittag-Leffler Institute. I am particularly grateful to Richard Rochberg for helpful discussions.

Article information

Ark. Mat., Volume 31, Number 2 (1993), 307-338.

Received: 18 December 1991
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

1993 © Institut Mittag-Leffler


Janson, Svante. Interpolation of subcouples and quotient couples. Ark. Mat. 31 (1993), no. 2, 307--338. doi:10.1007/BF02559489.

Export citation


  • Ball, J. A. and Helton, J. W., A Beurling-Lax theorem for the Lie group U (m, n) which contains most classical interpolation theory, J. Operator Theory 9 (1983), 107–142.
  • Bergh, J. and Löfström, L., Interpolation Spaces, Springer-Verlag, Berlin, 1976.
  • Bourgain, J., Some consequences of Pisier's approach to interpolation, Israel J. Math. 77 (1992), 165–185.
  • Brudnyî, Yu. A. and Krugljak, N. Y.Interpolation Functors and Interpolation Spaces, North-Holland, Amsterdam, 1991.
  • Bennett, C. and Sharpley, R., Interpolation of Operators, Academic Press, Orlando, 1988.
  • Cotlar, M. and Sadosky, C., Weighted and two-dimensional Adamjan-Arov-Krein theorems and analogues for Sarason commutants, Mittag-Leffler Report 24 (1990/91).
  • DeVore, R. A. and Scherer, K., Interpolation of operators on Sobolev spaces., Ann. of Math. 109 (1979), 583–599.
  • Garnett, J., Bounded Analytic Functions, Academic Press, New York, 1981.
  • Hernandez, E., Rochberg, R. and Weiss, G., Interpolation of subspaces and quotient spaces by the complex method, in Function Spaces and Applications, Proceedings, Lund 1986 (M. Cwikel, J. Peetre, Y. Sagher, H. Wallin, eds), Lecture Notes in Math. 1302, pp. 253–289, Springer-Verlag, Berlin, 1988.
  • Holmstedt, T., Interpolation of quasi-normed spaces, Math. Scand. 26 (1970), 177–199.
  • Holmstedt, T. and Peetre, J., On certain functionals arising in the theory of interpolation spaces, J. Funct. Anal. 4 (1969), 88–94.
  • Janson, S., Minimal and maximal methods of interpolation, J. Funct. Anal. 14 (1981), 50–72.
  • Kaftal, V., Larson, D. and Weiss, G., Quasitriangular subalgebras of semifinite von Neumann algebras are closed, J. Funct. Anal. 107 (1992), 387–401.
  • Kaijser, S. and Pelletier, J. W., Interpolation Functors and Duality, Lecture Notes in Math. 1208, Springer-Verlag, Berlin, 1986.
  • Miyashi, A., Some Littlewood-Paley type inequalities and their application to the Fefferman-Stein decomposition of BMO, Indiana Univ. Math. J. 39 (1990), 563–583.
  • Nikolskiî, N. K., Treatise on the Shift Operator, Springer-Verlag, Berlin, 1986.
  • Peetre, J., Interpolation functors and Banach couples, in Actes Congrès Intern. Math. 1970, vol. 2, pp. 373–378, Gauthier-Villars, Paris, 1971.
  • Peller, V. V., Hankel operators of classGp and their applications (rational approximation, Gaussian processes, the majorization problems for operators), Mat. Sb. (N.S.)113 (1980), 538–581 (Russian); English transl., Math. USSR-Sb. 41 (1982), 443–479.
  • Peller, V. V., A description of Hankel operators of classGp for p>0 an investigation of the rate of rational approximation, and other applications, Mat. Sb. (N.S.)122 (1983), 481–510 (Russian); English transl., Math. USSR-Sb. 50 (1985), 465–494.
  • Peller, V. V., A remark on interpolation in spaces of vector-valued functions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 141 (1985), 162–164. (Russian.)
  • Pisier, G., Interpolation between Hp spaces and non-commutative generalizations I, Pacific J. Math. 155 (1992), 341–368.
  • Pisier, G., Interpolation between Hp spaces and non-commutative generalizations II, to appear.
  • Pisier, G., A simple proof of a theorem of Jean Bourgain, Michigan Math. J. 39 (1992), 475–484.
  • Shapiro, H. S. and Shields, A. L., On some interpolation problems for annlytic functions, Amer. J. Math. 83 (1961), 513–532.
  • Treil, S. R., The theorem of Adamyan-Arov-Krein: vector variant, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 141 (1985), 56–71. (Russian.)
  • Triebel, H., Allgemeine Legendresche Differentialoperatoren II, Ann. Scuola Norm. Sup. Pisa Cl. Sci (3)24 (1970), 1–35.
  • Wallstén, R., Remarks on interpolation of subspaces, in Function Spaces and Applications, Proceedings, Lund 1986 (M. Cwikel, J. Peetre, Y. Sagher, H. Wallin, eds.), Lecture Notes in Math. 1302, pp. 410–419, Springer-Verlag, Berlin, 1988.