Numerical index and renorming
HTML articles powered by AMS MathViewer
- by Catherine Finet, Miguel Martín and Rafael Payá PDF
- Proc. Amer. Math. Soc. 131 (2003), 871-877 Request permission
Abstract:
We study the numerical index of a Banach space from the isomorphic point of view, that is, we investigate the values of the numerical index which can be obtained by renorming the space. The set of these values is always an interval which contains $[0,1/3[$ in the real case and $[e^{-1},1/2[$ in the complex case. Moreover, for “most” Banach spaces the least upper bound of this interval is as large as possible, namely $1$.References
- M. D. Acosta, Operadores que alcanzan su radio numérico, Tesis doctoral, Secretariado de publicaciones, Universidad de Granada. 1990.
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras, London Mathematical Society Lecture Note Series, vol. 2, Cambridge University Press, London-New York, 1971. MR 0288583
- F. F. Bonsall and J. Duncan, Numerical ranges. II, London Mathematical Society Lecture Note Series, No. 10, Cambridge University Press, New York-London, 1973. MR 0442682
- Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
- J. Duncan, C. M. McGregor, J. D. Pryce, and A. J. White, The numerical index of a normed space, J. London Math. Soc. (2) 2 (1970), 481–488. MR 264371, DOI 10.1112/jlms/2.Part_{3}.481
- C. Finet and W. Schachermayer, Equivalent norms on separable Asplund spaces, Studia Math. 92 (1989), no. 3, 275–283. MR 985557, DOI 10.4064/sm-92-3-275-283
- B. W. Glickfeld, On an inequality of Banach algebra geometry and semi-inner product space theory, Illinois J. Math. 14 (1970), 76–81. MR 253024
- B. V. Godun and S. L. Troyanski, Renorming Banach spaces with fundamental biorthogonal system, Banach spaces (Mérida, 1992) Contemp. Math., vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 119–126. MR 1209453, DOI 10.1090/conm/144/1209453
- Joram Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139–148. MR 160094, DOI 10.1007/BF02759700
- Ginés López, Miguel Martín, and Rafael Payá, Real Banach spaces with numerical index 1, Bull. London Math. Soc. 31 (1999), no. 2, 207–212. MR 1664125, DOI 10.1112/S002460939800513X
- Miguel Martín, A survey on the numerical index of a Banach space, Extracta Math. 15 (2000), no. 2, 265–276. III Congress on Banach Spaces (Jarandilla de la Vera, 1998). MR 1823892
- Miguel Martín and Rafael Payá, Numerical index of vector-valued function spaces, Studia Math. 142 (2000), no. 3, 269–280. MR 1792610, DOI 10.4064/sm-142-3-269-280
- J. P. Moreno, On geometry of Banach spaces with property $\alpha$, J. Math. Anal. Appl. 201 (1996), no. 2, 600–608. MR 1396921, DOI 10.1006/jmaa.1996.0276
- J. P. Moreno, Geometry of Banach spaces with $(\alpha ,\epsilon )$-property or $(\beta ,\epsilon )$-property, Rocky Mountain J. Math. 27 (1997), no. 1, 241–256. MR 1453101, DOI 10.1216/rmjm/1181071959
- S. Negrepontis, Banach spaces and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1045–1142. MR 776642
- J. R. Partington, Norm attaining operators, Israel J. Math. 43 (1982), no. 3, 273–276. MR 689984, DOI 10.1007/BF02761947
- Walter Schachermayer, Norm attaining operators and renormings of Banach spaces, Israel J. Math. 44 (1983), no. 3, 201–212. MR 693659, DOI 10.1007/BF02760971
- K. Tillekeratne, Spatial numerical range of an operator, Proc. Cambridge Philos. Soc. 76 (1974), 515–520. MR 346556, DOI 10.1017/s0305004100049240
- M. Valdivia, Topological direct sum decompositions of Banach spaces, Israel J. Math. 71 (1990), no. 3, 289–296. MR 1088821, DOI 10.1007/BF02773747
Additional Information
- Catherine Finet
- Affiliation: Institut de Mathématique et d’Informatique, Université de Mons-Hainaut, Avenue du champ de mars 8, B-7000 Mons, Belgium
- Email: catherine.finet@umh.ac.be
- Miguel Martín
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
- MR Author ID: 643000
- ORCID: 0000-0003-4502-798X
- Email: mmartins@ugr.es
- Rafael Payá
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
- Email: rpaya@ugr.es
- Received by editor(s): May 29, 2001
- Received by editor(s) in revised form: October 18, 2001
- Published electronically: August 19, 2002
- Additional Notes: The first author was partially supported by La Banque Nationale de Belgique
The second and third authors were partially supported by Spanish MCYT project no. BFM2000-1467 - Communicated by: N. Tomczak-Jaegermann
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 871-877
- MSC (2000): Primary 46B20, 47A12
- DOI: https://doi.org/10.1090/S0002-9939-02-06576-0
- MathSciNet review: 1937425