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Conjugate norms in ℂⁿ and related geometrical problems

Seria
Rozprawy Matematyczne tom/nr w serii: 377 wydano: 1998
Zawartość
Warianty tytułu
Abstrakty
EN
Abstract
We consider ℂⁿ as a normed space equipped with a complex norm F and we investigate some geometrical problems related with the notion of a conjugate norm F*. A crucial role in our considerations is played by the classical Shmul'yan theorem on exposed points in dual spaces. Many applications of this theorem are given for different problems including characterization of linear (biholomorphic) equivalence for a class of balls in ℂⁿ, calculation of the group of linear automorphisms (Section 4) and for problems related to the complex method of interpolation (Sections 5-7). The main result is an effective formula for interpolating norms for the couple (ℝⁿ ⊗̆ ℂ ,ℝⁿ ⊗̂ ℂ) (Section 5) and, more generally, for the couple (H ⊗̆ ℂ,H ⊗̂ ℂ), where H is a real Hilbert space. In Section 3 we present connections of conjugate norms with problems of pluripotential theory and approximation theory. Here a special role is played by a class of complex norms that are natural complexifications of norms in ℝⁿ. In Section 2 we consider some properties of such norms, in particular we prove an essential generalization of a result by Hahn and Pflug.
EN
CONTENTS
Introduction.........................................................................................................5
1. Conjugate norms in ℝⁿ..................................................................................10
2. Conjugate norms in ℂⁿ..................................................................................16
3. Extremal properties of norms F(f,·) in pluripotential theory............................29
4. Biholomorphic inequivalence of some convex circular domains....................35
5. The complex method of interpolation and conjugate norms in ℂⁿ.................43
6. On tensor products $ℝⁿ ⊗ ℝ^{k}$ and $ℂⁿ ⊗ ℂ^{k}$....................................54
7. The complex interpolation of a complexification of a real Hilbert space.........62
References........................................................................................................65
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 377
Liczba stron
67
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXXVII
Daty
wydano
1998
otrzymano
1998-01-27
poprawiono
1998-05-22
Twórcy
  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland, baran@im.uj.edu.pl
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Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: 32F05, 32M05, 41A17, 46B20, 46C99, 52A43.
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