On decompositions of Banach spaces into a sum of operator ranges

V. Fonf; V. Shevchik

Studia Mathematica (1999)

  • Volume: 132, Issue: 1, page 91-100
  • ISSN: 0039-3223

Abstract

top
It is proved that a separable Banach space X admits a representation X = X 1 + X 2 as a sum (not necessarily direct) of two infinite-codimensional closed subspaces X 1 and X 2 if and only if it admits a representation X = A 1 ( Y 1 ) + A 2 ( Y 2 ) as a sum (not necessarily direct) of two infinite-codimensional operator ranges. Suppose that a separable Banach space X admits a representation as above. Then it admits a representation X = T 1 ( Z 1 ) + T 2 ( Z 2 ) such that neither of the operator ranges T 1 ( Z 1 ) , T 2 ( Z 2 ) contains an infinite-dimensional closed subspace if and only if X does not contain an isomorphic copy of l 1 .

How to cite

top

Fonf, V., and Shevchik, V.. "On decompositions of Banach spaces into a sum of operator ranges." Studia Mathematica 132.1 (1999): 91-100. <http://eudml.org/doc/216588>.

@article{Fonf1999,
abstract = {It is proved that a separable Banach space X admits a representation $X = X_1 + X_2$ as a sum (not necessarily direct) of two infinite-codimensional closed subspaces $X_1$ and $X_2$ if and only if it admits a representation $X = A_1(Y_1) + A_2(Y_2)$ as a sum (not necessarily direct) of two infinite-codimensional operator ranges. Suppose that a separable Banach space X admits a representation as above. Then it admits a representation $X = T_1(Z_1) + T_2(Z_2)$ such that neither of the operator ranges $T_1(Z_1)$, $T_2(Z_2)$ contains an infinite-dimensional closed subspace if and only if X does not contain an isomorphic copy of $l_1$.},
author = {Fonf, V., Shevchik, V.},
journal = {Studia Mathematica},
keywords = {decomposition of Banach spaces; representation; infinite-codimensional operator ranges},
language = {eng},
number = {1},
pages = {91-100},
title = {On decompositions of Banach spaces into a sum of operator ranges},
url = {http://eudml.org/doc/216588},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Fonf, V.
AU - Shevchik, V.
TI - On decompositions of Banach spaces into a sum of operator ranges
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 1
SP - 91
EP - 100
AB - It is proved that a separable Banach space X admits a representation $X = X_1 + X_2$ as a sum (not necessarily direct) of two infinite-codimensional closed subspaces $X_1$ and $X_2$ if and only if it admits a representation $X = A_1(Y_1) + A_2(Y_2)$ as a sum (not necessarily direct) of two infinite-codimensional operator ranges. Suppose that a separable Banach space X admits a representation as above. Then it admits a representation $X = T_1(Z_1) + T_2(Z_2)$ such that neither of the operator ranges $T_1(Z_1)$, $T_2(Z_2)$ contains an infinite-dimensional closed subspace if and only if X does not contain an isomorphic copy of $l_1$.
LA - eng
KW - decomposition of Banach spaces; representation; infinite-codimensional operator ranges
UR - http://eudml.org/doc/216588
ER -

References

top
  1. [1] Fillmore P. A., and Williams J. P., On operator ranges, Adv. Math. 7 (1971), 254-281. Zbl0224.47009
  2. [2] Fonf V. P., On supportless absorbing convex subsets in normed spaces, Studia Math. 104 (1993), 279-284. Zbl0815.46019
  3. [3] Fonf V. P., and Shevchik V. V., Representing a Banach space as a sum of operator ranges, Funct. Anal. Appl. 29 (1995), no. 3, 220-221 (transl. from the Russian). Zbl0867.47004
  4. [4] Gowers W. T., and Maurey B., The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874. Zbl0827.46008
  5. [5] Johnson W. B., and Rosenthal H. P., On w*-basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77-92. Zbl0213.39301
  6. [6] Lindenstrauss J., and Tzafriri L., Classical Banach Spaces, Vol. 1, Springer, Berlin, 1977. Zbl0362.46013
  7. [7] Pełczyński A., On strictly singular and strictly cosingular operators, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 13 (1965), 31-41. Zbl0138.38604
  8. [8] Pietsch A., Operator Ideals, North-Holland, Amsterdam, 1980. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.