Conjugate norms in ℂⁿ and related geometrical problems

Baran Mirosław

GDML_Books, (1998), p.

Résumé

AbstractWe 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.

CONTENTSIntroduction.........................................................................................................51. Conjugate norms in ℝⁿ..................................................................................102. Conjugate norms in ℂⁿ..................................................................................163. Extremal properties of norms F(f,·) in pluripotential theory............................294. Biholomorphic inequivalence of some convex circular domains....................355. The complex method of interpolation and conjugate norms in ℂⁿ.................436. On tensor products $ℝⁿ ⊗ ℝ^{k}$ and $ℂⁿ ⊗ ℂ^{k}$....................................547. The complex interpolation of a complexification of a real Hilbert space.........62References........................................................................................................65

1991 Mathematics Subject Classification: 32F05, 32M05, 41A17, 46B20, 46C99, 52A43.

Détail

EUDML-ID : urn:eudml:doc:271247