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## Studia Mathematica

1997 | 125 | 3 | 201-218
Tytuł artykułu

### On the relation between complex and real methods of interpolation

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We study those compatible couples of Banach spaces for which the complex method interpolation spaces are also described by the K-method of interpolation. As an application we present counter-examples to Cwikel's conjecture that all interpolation spaces of a Banach couple are described by the K-method whenever all complex interpolation spaces have this property.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
201-218
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-05-17
Twórcy
autor
• Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland, mastylo@math.amu.edu.pl
• Mathematical Institute, Polish Academy of Sciences, Poznań Branch, Matejki 48/49, 60-769 Poznań, Poland
• Faculty of Mathematics, Voronezh University, Universitetskaya pl., 1, 394693 Voronezh, Russia
Bibliografia
• [A] S. V. Astashkin, Description of interpolation orbits between $(l_1(w_0),l_1(w_1))$ and $(l_∞ (w_0), l_∞ (w_1))$, Mat. Zametki 35 (1984), 497-503 (in Russian); English transl.: Math. Notes 35 (1984), 261-265.
• [B] J. Bergh, On the relation between the two complex methods of interpolation, Indiana Univ. Math. J. 28 (1979), 775-778.
• [BL] J. Bergh and J. Löfström, Interpolation spaces. An Introduction, Springer, Berlin 1976.
• [BK] Yu. A. Brudnyĭ and N. Ya. Krugljak, Interpolation Functors and Interpolation Spaces. I, North-Holland, Amsterdam, 1991.
• [Ca] A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113-190.
• [Cw1] M. Cwikel, On $(L_p_0(A_0),L_p_1(A_1))_θ,q$, Proc. Amer. Math. Soc. 44 (1974), 286-292.
• [Cw2] M. Cwikel, Monotonicity properties of interpolation spaces II, Ark. Mat. 19 (1981), 123-136.
• [Cw3] M. Cwikel, Real and complex interpolation and extrapolation of compact operators, Duke Math. J. 65 (1992), 333-343.
• [CM1] M. Cwikel and M. Mastyło, The universal right K-property for Banach lattices, preprint.
• [CM2] M. Cwikel and M. Mastyło, On Banach couples which satisfy the condition $(A_0, A_1)_θ, ∞^0 =(A_0, A_1)_θ ,∞$, preprint.
• [CM3] M. Cwikel and M. Mastyło, On interpolation spaces containing copies of $c_0$ and $l_∞$, preprint.
• [CN] M. Cwikel and P. Nilsson, On Calderón-Mityagin couples of Banach lattices, in: Proc. Conf. Constructive Theory of Functions, Varna, 1984, Bulgar. Acad. Sci., 1984, 232-236.
• [CP] M. Cwikel and J. Peetre, Abstract K and J spaces, J. Math. Pures Appl. 60 (1981), 1-50.
• [DKO] V. I. Dmitriev, S. G. Kreĭn and V. I. Ovchinnikov, Fundamentals of the theory of interpolation of linear operators, in: Geometry of Linear Spaces and Operator Theory, Yaroslavl', 1977, 31-74 (in Russian).
• [DO] V. I. Dmitriev and V. I. Ovchinnikov, On the real method spaces, Dokl. Akad. Nauk SSSR 246 (1979), 794-799 (in Russian); English transl.: Soviet Math. Dokl. 20 (1979), 538-542.
• [H] T. Holmstedt, Interpolation of quasi-normed spaces, Math. Scand. 26 (1970), 177-199.
• [J] S. Janson, Minimal and maximal methods of interpolation, J. Funct. Anal. 44 (1981), 50-73.
• [K] N. J. Kalton, Calderón couples of rearrangement invariant spaces, Studia Math. 106 (1993), 233-277.
• [KPS] S. G. Kreĭn, Yu. I. Petunin and E. M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian); English transl.: Amer. Math. Soc., Providence, 1982.
• [M1] M. Mastyło, Interpolation between some symmetric spaces, Arch. Math. (Basel) 52 (1989), 571-579.
• [M2] M. Mastyło, K-monotone spaces between spaces of absolutely summable series, Forum Math. 2 (1990), 73-87.
• [MX] M. Mastyło and Q. Xu, Monotonicity between vector-valued bounded sequence spaces, Bull. Polish Acad. Sci. Math. 41 (1993), 177-187.
• [O1] V. I. Ovchinnikov, Interpolation theorems resulting from Grothendieck's inequality, Funktsional. Anal. i Prilozhen. 10 (4) (1976), 45-54 (in Russian); English transl.: Functional Anal. Appl. 10 (1977), 287-294.
• [O2] V. I. Ovchinnikov, The method of orbits in interpolation theory, Math. Rep. 1 (1984), 349-516.
• [OD] V. I. Ovchinnikov and V. I. Dmitriev, The limits of the applicability of the K-method of interpolation, in: Collection of Articles on Applications of Functional Analysis, Voronezh 1975, 102-108 (in Russian).
• [Pe] J. Peetre, Banach couples, I: Elementary theory, Technical Report, Lund, 1971.
• [Pi] G. Pisier, Interpolation between $H^p$ spaces and non-commutative generalizations. I, Pacific J. Math. 155 (1992), 341-368.
• [R] H. P. Rosenthal, On relatively disjoint families of measures with some applications to Banach space theory, Studia Math. 37 (1970), 13-36.
• [S] V. A. Shestakov, Complex interpolation in Banach spaces of measurable functions, Vestnik Leningrad. Univ. 19 (1974), 64-68 (in Russian); English transl.: Vestnik Leningrad Univ. Math. 7 (1979).
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Bibliografia
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