Weakly precompact operators on C b ( X , E ) with the strict topology

Juliusz Stochmal

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2016)

  • Volume: 36, Issue: 1, page 65-77
  • ISSN: 1509-9407

Abstract

top
Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let C b ( X , E ) be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study weakly precompact operators T : C b ( X , E ) → F . In particular, we show that if X is a paracompact k-space and E contains no isomorphic copy of l¹, then every strongly bounded operator T : C b ( X , E ) → F is weakly precompact.

How to cite

top

Juliusz Stochmal. "Weakly precompact operators on $C_{b}(X,E)$ with the strict topology." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 36.1 (2016): 65-77. <http://eudml.org/doc/286883>.

@article{JuliuszStochmal2016,
abstract = {Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_\{b\}(X,E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study weakly precompact operators $T:C_\{b\}(X,E) → F$. In particular, we show that if X is a paracompact k-space and E contains no isomorphic copy of l¹, then every strongly bounded operator $T:C_\{b\}(X,E) → F$ is weakly precompact.},
author = {Juliusz Stochmal},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {spaces of vector-valued continuous functions; strict topologies; operator measures; strongly bounded operators; weakly precompact operators},
language = {eng},
number = {1},
pages = {65-77},
title = {Weakly precompact operators on $C_\{b\}(X,E)$ with the strict topology},
url = {http://eudml.org/doc/286883},
volume = {36},
year = {2016},
}

TY - JOUR
AU - Juliusz Stochmal
TI - Weakly precompact operators on $C_{b}(X,E)$ with the strict topology
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2016
VL - 36
IS - 1
SP - 65
EP - 77
AB - Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_{b}(X,E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study weakly precompact operators $T:C_{b}(X,E) → F$. In particular, we show that if X is a paracompact k-space and E contains no isomorphic copy of l¹, then every strongly bounded operator $T:C_{b}(X,E) → F$ is weakly precompact.
LA - eng
KW - spaces of vector-valued continuous functions; strict topologies; operator measures; strongly bounded operators; weakly precompact operators
UR - http://eudml.org/doc/286883
ER -

References

top
  1. [1] C. Abbott, Weakly precompact and GSP operators on continuous function spaces, Bull. Polish Acad. Sci. Math. 37 (1989), 467-476. Zbl0767.47011
  2. [2] C. Abbott, A. Bator, R. Bilyeu and P. Lewis, Weak precompactness, strong boundedness, and weak complete continuity, Math. Proc. Camb. Phil. Soc. 108 (1990), 325-335. doi: 10.1017/S030500410006919X Zbl0737.46008
  3. [3] E. Bator and P. Lewis, Operators having weakly precompact adjoints, Math. Nachr. 157 (1992), 99-103. doi: 10.1002/mana.19921570109 Zbl0792.47021
  4. [4] J. Bourgain, An averaging result for l₁-sequences and application to weakly conditionallycompact sets in L₁^X, Israel J. Math. 32 (1979), 289-298. doi: 10.1007/BF02760458 
  5. [5] A. Bouziad, Luzin measurability of Carath´eodory type mappings, Topology Appl. 154 (2007), 287-301. doi: 10.1016/j.topol.2006.04.013 Zbl1120.54010
  6. [6] J.B. Cooper, The strict topology and spaces with mixed topology, Proc. Amer. Math. Soc. 30 (1971), 583-592. doi: 10.1090/S0002-9939-1971-0284789-2 Zbl0225.46004
  7. [7] J. Diestel, Sequences and Series in Banach Spaces (Graduate Texts in Math., vol. 92, Springer-Verlag, 1984). 
  8. [8] J. Diestel and J.J. Uhl, Vector Measures (Amer. Math. Soc. Surveys 15, 1977). 
  9. [9] N. Dinculeanu, Vector Measures (Pergamon Press, New York, 1967). doi: 10.1016/b978-1-4831-9762-3.50004-4 
  10. [10] N. Dinculeanu, Vector Integration and Stochastic Integration in Banach Spaces (John Wiley and Sons Inc., 2000). doi: 10.1002/9781118033012 Zbl0974.28006
  11. [11] R. Engelking, General Topology (Heldermann Verlag, Berlin, 1989). 
  12. [12] I. Ghenciu, Weak precompactness in the space of vector-valued measures of bounded variation, J. Function Spaces 2015 (2015), ID 213679. Zbl1319.46023
  13. [13] I. Ghenciu and P. Lewis, Strongly bounded representing measures and converging theorems, Glasgow Math. J. 52 (2010), 435-445. doi: 10.1017/S0017089510000133 Zbl1208.46044
  14. [14] L.V. Kantorovitch and G.P. Akilov, Functional Analysis (Pergamon Press, OxfordElmsford, New York, 1982). 
  15. [15] L.A. Khan, The strict topology on a space of vector-valued functions, Proc. Edinburgh Math. Soc. 22 (1979), 35-41. doi: 10.1017/S0013091500027784 Zbl0429.46023
  16. [16] L.A. Khan and K. Rowlands, On the representation of strictly continuous linear functionals, Proc. Edinburgh Math. Soc. 24 (1981), 123-130. doi: 10.1017/S0013091500006428 Zbl0459.46026
  17. [17] M. Nowak, Operator measures and integration operators, Indag. Math. 24 (2013), 279-290. doi: 10.1016/j.indag.2012.10.002 
  18. [18] M. Nowak, A Riesz representation theory for completely regular Hausdorff spaces and its applications, Open Math (in press). Zbl06632375
  19. [19] M. Nowak, Completely continuous operators and the strict topology, (submitted). 
  20. [20] E. Saab and P. Saab, On unconditionally converging and weakly precompact operators, Illinois J. Math. 35 (1991), 522-531. Zbl0729.46014
  21. [21] H.J. Song, Characterizations of weakly precompactness of operators acting between Banach spaces, Kangweon-Kyungki Math. Jour. 12 (2004), 127-146. 
  22. [22] J. Schmets and J. Zafarani, Strict topologies and (gDF)-spaces, Arch. Math. 49 (1987), 227-231. doi: 10.1007/BF01271662 Zbl0615.46032
  23. [23] M. Talagrand, Weak Cauchy sequences in L₁(E), Amer. J. Math. 106 (1984), 703-724. doi: 10.2307/2374292 Zbl0579.46025
  24. [24] A. Wiweger, Linear spaces with mixed topology, Studia Math. 20 (1961), 47-68. Zbl0097.31301

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.