On intersections of ranges of projections of norm one in Banach spaces

Author:
T. S. S. R. K. Rao

Journal:
Proc. Amer. Math. Soc. **141** (2013), 3579-3586

MSC (2010):
Primary 47L05; Secondary 46B20, 46E15

Published electronically:
July 9, 2013

MathSciNet review:
3080180

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Abstract: In this short note we are interested in studying Banach spaces in which the range of a projection of norm one whose kernel is of finite dimension is the intersection of ranges of finitely many projections of norm one whose kernels are of dimension one. We show that for a certain class of Banach spaces , the natural duality between and can be exploited when the range of the projection is of finite codimension. We show that if is isometric to , then any central subspace of finite codimension is an intersection of central subspaces of codimension one. These results extend a recent result of Bandyopadhyay and Dutta which was proved for ranges of projections of norm one with finite dimensional kernel in continuous function spaces and unifies some earlier work of Baronti and Papini.

**1.**Pradipta Bandyopadhyay and T. S. S. R. K. Rao,*Central subspaces of Banach spaces*, J. Approx. Theory**103**(2000), no. 2, 206–222. MR**1749962**, 10.1006/jath.1999.3420**2.**Pradipta Bandyopadhyay and S. Dutta,*Almost constrained subspaces of Banach spaces. II*, Houston J. Math.**35**(2009), no. 3, 945–957. MR**2534290****3.**Marco Baronti,*A note on norm-one projections onto subspaces of finite codimension of 𝑙^{∞}*, Arch. Math. (Basel)**54**(1990), no. 4, 384–388. MR**1042133**, 10.1007/BF01189587**4.**M. Baronti and P. Papini,*Norm-one projections onto subspaces of finite codimension in 𝑙₁ and 𝑐₀*, Period. Math. Hungar.**22**(1991), no. 3, 161–174. MR**1142506**, 10.1007/BF01960506**5.**J. Blatter and E. W. Cheney,*Minimal projections on hyperplanes in sequence spaces*, Ann. Mat. Pura Appl. (4)**101**(1974), 215–227. MR**0358179****6.**Á. P. Bosznay and B. M. Garay,*On norms of projections*, Acta Sci. Math. (Szeged)**50**(1986), no. 1-2, 87–92. MR**862183****7.**P. Harmand, D. Werner, and W. Werner,*𝑀-ideals in Banach spaces and Banach algebras*, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR**1238713****8.**J. E. Jamison, A. Kamińska, and G. Lewicki,*One-complemented subspaces of Musielak-Orlicz sequence spaces*, J. Approx. Theory**130**(2004), no. 1, 1–37. MR**2086807**, 10.1016/j.jat.2004.07.001**9.**Anna Kamińska and Grzegorz Lewicki,*Contractive and optimal sets in modular spaces*, Math. Nachr.**268**(2004), 74–95. MR**2054533**, 10.1002/mana.200310160**10.**Anna Kamińska, Han Ju Lee, and Grzegorz Lewicki,*Extreme and smooth points in Lorentz and Marcinkiewicz spaces with applications to contractive projections*, Rocky Mountain J. Math.**39**(2009), no. 5, 1533–1572. MR**2546654**, 10.1216/RMJ-2009-39-5-1533**11.**H. Elton Lacey,*The isometric theory of classical Banach spaces*, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 208. MR**0493279****12.**Grzegorz Lewicki and Giulio Trombetta,*Optimal and one-complemented subspaces*, Monatsh. Math.**153**(2008), no. 2, 115–132. MR**2373365**, 10.1007/s00605-007-0510-4**13.**Joram Lindenstrauss,*Extension of compact operators*, Mem. Amer. Math. Soc. No.**48**(1964), 112. MR**0179580**

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Additional Information

**T. S. S. R. K. Rao**

Affiliation:
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, R. V. College P. O., Bangalore 560059, India

Email:
tss@isibang.ac.in

DOI:
http://dx.doi.org/10.1090/S0002-9939-2013-11639-4

Keywords:
Projections of norm one,
finite codimensional subspaces,
spaces of continuous functions

Received by editor(s):
July 11, 2011

Received by editor(s) in revised form:
January 5, 2012

Published electronically:
July 9, 2013

Communicated by:
Thomas Schlumprecht

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.