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Conjugate norms in ℂⁿ and related geometrical problems

Rozprawy Matematyczne tom/nr w serii: 377 wydano: 1998
Warianty tytułu
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.
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
Miejsce publikacji
Rozprawy Matematyczne tom/nr w serii: 377
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Dissertationes Mathematicae, Tom CCCLXXVII
  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland,
  • [A] D. Amir, Characterizations of Inner Product Spaces, Birkhäuser, 1986.
  • [A-B] R. Aron, B. Beauzamy and P. Enflo, Polynomials in many variables: real vs complex norms, J. Approx. Theory 74 (1993), 181-198.
  • [B1] M. Baran, Siciak's extremal function of convex sets in ℂⁿ, Ann. Polon. Math. 48 (1988), 275-280.
  • [B2] M. Baran, Plurisubharmonic extremal function and complex foliation for the complement of a convex subset of ℝⁿ, Michigan Math. J. 39 (1992), 395-404.
  • [B3] M. Baran, Bernstein type theorems for compact sets in ℝⁿ revisited, J. Approx. Theory 79 (1994), 190-198.
  • [B4] M. Baran, Complex equilibrium measure and Bernstein type theorems for compact sets in ℝⁿ, Proc. Amer. Math. Soc. 123 (1995), 485-494.
  • [B5] M. Baran, Two applications of the complex interpolation method, Rend. Circ. Mat. Palermo 40 (1996), 57-62.
  • [B6] M. Baran, Homogeneous extremal function for a ball in ℝ², Ann. Polon. Math. (to appear).
  • [B7] M. Baran, Polynomial inequalities in Banach spaces (I), preprint, 1997.
  • [B-P] E. Bedford and S. Pinchuk, Convex domains with noncompact automorphism group, Mat. Sb. 185 (1994), 3-26.
  • [B-T] E. Bedford and B. A. Taylor, The complex equilibrium measure of a symmetric convex set in ℝⁿ, Trans. Amer. Math. Soc. 294 (1986), 705-717.
  • [B-L] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976.
  • [BO] J. Bochnak, Analytic functions in Banach spaces, Studia Math. 35 (1970), 273-292.
  • [B-S] J. Bochnak and J. Siciak, Polynomials and multilinear mappings in topological vector spaces, ibid. 39 (1971), 59-76.
  • [B-K] R. Braun, W. Kaup and H. Upmeier, On the automorphisms of circular and Reinhardt domains in complex Banach spaces, Manuscripta Math. 25 (1978), 97-133.
  • [CA] A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 133-190.
  • [CI] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990.
  • [CL] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396-414.
  • [DIE] J. Diestel, Geometry of Banach Spaces--Selected Topics, Lecture Notes in Math. 485, Springer, 1975.
  • [DI] J. Dieudonné, La Géométrie des Groupes Classiques, Springer, Berlin, 1971.
  • [D] L. M. Drużkowski, Effective formula for the crossnorm in complexified unitary spaces, Univ. Iagell. Acta Math. 16 (1974), 47-53.
  • [G-L] I. Glazman et Y. Liubitch, Analyse linéaire dans les espaces de dimensions finies, Mir, Moscou, 1972.
  • [G] R. Grząślewicz, Finite dimensional Orlicz spaces, Bull. Polish Acad. Sci. Math. 33 (1985), 277-283.
  • [G-H] R. Grząślewicz and H. Hudzik, Smooth points of Orlicz spaces equipped with Luxemburg norm, Math. Nachr. 155 (1992), 31-45.
  • [H-P] K. T. Hahn and P. Pflug, On a minimal complex norm that extends the real Euclidean norm, Monatsh. Math. 105 (1988), 107-112.
  • [HE] S. Heinrich, Strongly exposed and conical points in projective tensor products, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 22 (1975), 146-154 (in Russian).
  • [H] L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Amer. Math. Soc., Providence, R.I., 1963.
  • [J-P] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, de Gruyter Exp. Math. 9, de Gruyter, Berlin, 1993.
  • [KI1] K. T. Kim, Complete localization of domains with noncompact automorphism group, Trans. Amer. Math. Soc. 319 (1990), 139-153.
  • [KI2] K. T. Kim, Automorphism groups of certain domains in ℂⁿ with a singular boundary, Pacific J. Math. 131 (1991), 57-64.
  • [KL] M. Klimek, Pluripotential Theory, Oxford Univ. Press, 1991.
  • [KO1] O. Kouba, Sur l'interpolation des produits tensoriels projectifs ou injectifs d'espaces de Banach, C. R. Acad. Sci. Paris 309 (1989), 683-686.
  • [KO2] O. Kouba, Interpolation of injective or projective tensor products of Banach spaces, J. Funct. Anal. 96 (1991), 38-61.
  • [KR] S. G. Krantz, Function Theory of Several Complex Variables, Wiley-Interscience, New York, 1982.
  • [K-M] M. Krein and D. Milman, On extreme points of regularly convex sets, Studia Math. 9 (1940), 133-138.
  • [K-P] S. G. Krein, Yu. I. Petunin and E. M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian).
  • [KU] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, PWN, Warszawa, 1985.
  • [L] R. Leichtweiss, Konvexe Mengen, Deutscher Verlag Wiss., Berlin, 1980.
  • [L-C] W. A. Light and E. W. Cheney, Approximation Theory in Tensor Product Spaces, Lecture Notes in Math. 1169, Springer, 1985.
  • [L-P] J. Lindenstrauss and M. Perles, On extreme operators in finite dimensional spaces, Duke Math. J. 36 (1969), 301-314.
  • [LI] J. L. Lions, Une construction d'espaces d'interpolation, C. R. Acad. Sci. Paris 251 (1960), 1853-1855.
  • [LU1] M. Lundin, The extremal plurisubharmonic function for convex symmetric subsets of ℝⁿ, Michigan Math. J. 32 (1985), 197-201.
  • [LU2] M. Lundin, An explicit solution to the complex Monge-Ampère equation, preprint, 1985.
  • [M-P1] L. Maligranda and L. E. Persson, On Clarkson's inequalities and interpolation, Math. Nachr. 155 (1992), 187-207.
  • [M-P2] L. Maligranda and L. E. Persson, Inequalities and interpolation, Collect. Math. 44 (1993), 181-199.
  • [M-O] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, 1979.
  • [MA] S. Mazur, Quelques propriétés caractéristiques des espaces euclidiens, C. R. Acad. Sci. Paris 207 (1938), 761-764.
  • [M-P] V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: Lecture Notes in Math. 1376, Springer, 1989, 84-104.
  • [N] J. von Neumann, Some matrix inequalities and metrization of matrix-space, Tomsk Univ. Rev. 1 (1937), 286-300.
  • [P1] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986.
  • [P2] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math. 94, Cambridge Univ. Press, 1989.
  • [P3] G. Pisier, The operator Hilbert space OH, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 585 (1996).
  • [R-R] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Dekker, New York, 1991.
  • [R-T] J. Rauch and B. A. Taylor, The Dirichlet problem for the multidimensional Monge-Ampère equation, Rocky Mountain J. Math. 7 (1977), 345-364.
  • [RO] R. T. Rockafellar, Convex Analysis, Princeton Math. Ser. 28, Princeton Univ. Press, Princeton, N.J., 1970.
  • [R] J. P. Rosay, Sur une caractérisation de la boule parmi les domaines de ℂⁿ par son groupe d'automorphismes, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, 91-97.
  • [RU] W. Ruess, Duality and geometry of spaces of compact operators, in: Functional Analysis: Surveys and Recent Results, III (Paderborn, 1983), North-Holland Math. Stud. 90, North-Holland, Amsterdam, 1984, 59-78.
  • [R-S] W. Ruess and C. Stegall, Exposed and denting points in duals of operator spaces, Israel J. Math. 53 (1986), 163-190.
  • [SA] L. A. Santaló, Un invariante afin para los cuerpos convexos del espacio de n dimensiones, Portugal. Math. 8 (1949), 155-161.
  • [SH] V. M. Shmul'yan, Sur la dérivabilité de la norme dans l'espace de Banach, Dokl. Akad. Nauk SSSR (N.S.) 27 (1940), 643-648.
  • [SI1] J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), 322-357.
  • [SI2] J. Siciak, Extremal plurisubharmonic functions in ℂⁿ, Ann. Polon. Math. 39 (1981), 175-211.
  • [SI3] J. Siciak, Wiener's type sufficient conditions in ℂⁿ, Univ. Iagell. Acta Math. 35 (1997), 47-74.
  • [SG] C. L. Siegel, Automorphic Functions of Several Complex Variables, Izdat. Inostr. Literat., Moscow, 1954 (in Russian).
  • [ST] S. Straszewicz, Über exponierte Punkte abgeschlossener Punktmengen, Fund. Math. 24 (1935), 139-143.
  • [T] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite Dimensional Operator Ideals, Wiley, New York, 1993.
  • [W] B. Wong, Characterization of the unit ball in ℂⁿ by its automorphism group, Invent. Math. 41 (1977), 253-257.
Języki publikacji
1991 Mathematics Subject Classification: 32F05, 32M05, 41A17, 46B20, 46C99, 52A43.
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