Hereditarily finitely decomposable Banach spaces

V. Perenczi

Studia Mathematica (1997)

  • Volume: 123, Issue: 2, page 135-149
  • ISSN: 0039-3223

Abstract

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A Banach space is said to be H D n if the maximal number of subspaces of X forming a direct sum is finite and equal to n. We study some properties of H D n spaces, and their links with hereditarily indecomposable spaces; in particular, we show that if X is complex H D n , then dim ( ℒ ( X ) / S ( X ) ) ≤ n 2 , where S(X) denotes the space of strictly singular operators on X. It follows that if X is a real hereditarily indecomposable space, then ℒ(X)/S(X) is a division ring isomorphic either to ℝ, ℂ, or ℍ, the quaternionic division ring.

How to cite

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Perenczi, V.. "Hereditarily finitely decomposable Banach spaces." Studia Mathematica 123.2 (1997): 135-149. <http://eudml.org/doc/216383>.

@article{Perenczi1997,
abstract = {A Banach space is said to be $HD_n$ if the maximal number of subspaces of X forming a direct sum is finite and equal to n. We study some properties of $HD_n$ spaces, and their links with hereditarily indecomposable spaces; in particular, we show that if X is complex $HD_n$, then dim $(ℒ(X)/S(X)) ≤ n^2$, where S(X) denotes the space of strictly singular operators on X. It follows that if X is a real hereditarily indecomposable space, then ℒ(X)/S(X) is a division ring isomorphic either to ℝ, ℂ, or ℍ, the quaternionic division ring.},
author = {Perenczi, V.},
journal = {Studia Mathematica},
keywords = {hereditarily indecomposable spaces; strictly singular operators; quaternionic division ring},
language = {eng},
number = {2},
pages = {135-149},
title = {Hereditarily finitely decomposable Banach spaces},
url = {http://eudml.org/doc/216383},
volume = {123},
year = {1997},
}

TY - JOUR
AU - Perenczi, V.
TI - Hereditarily finitely decomposable Banach spaces
JO - Studia Mathematica
PY - 1997
VL - 123
IS - 2
SP - 135
EP - 149
AB - A Banach space is said to be $HD_n$ if the maximal number of subspaces of X forming a direct sum is finite and equal to n. We study some properties of $HD_n$ spaces, and their links with hereditarily indecomposable spaces; in particular, we show that if X is complex $HD_n$, then dim $(ℒ(X)/S(X)) ≤ n^2$, where S(X) denotes the space of strictly singular operators on X. It follows that if X is a real hereditarily indecomposable space, then ℒ(X)/S(X) is a division ring isomorphic either to ℝ, ℂ, or ℍ, the quaternionic division ring.
LA - eng
KW - hereditarily indecomposable spaces; strictly singular operators; quaternionic division ring
UR - http://eudml.org/doc/216383
ER -

References

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  1. [E] P. Enflo, A counterexample to the approximation property in Banach spaces, Acta Math. 130 (1973), 309-317. Zbl0267.46012
  2. [F1] V. Ferenczi, Operators on subspaces of hereditarily indecomposable Banach spaces, Bull. London Math. Soc., to appear. 
  3. [F2] V. Ferenczi, Quotient hereditarily indecomposable Banach spaces, preprint. 
  4. [G1] W. T. Gowers, A new dichotomy for Banach spaces, preprint. 
  5. [G2] W. T. Gowers, Analytic sets and games in Banach spaces, preprint. 
  6. [GM] W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874. Zbl0827.46008
  7. [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, New York, 1977. Zbl0362.46013
  8. [R] C. E. Richart, General Theory of Banach Algebras, D. Van Nostrand, Princeton, N.J., 1960. 

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