Coincidence of topologies on tensor products of Köthe echelon spaces

J. Bonet; A. Defant; A. Peris; M. Ramanujan

Studia Mathematica (1994)

  • Volume: 111, Issue: 3, page 263-281
  • ISSN: 0039-3223

Abstract

top
We investigate conditions under which the projective and the injective topologies coincide on the tensor product of two Köthe echelon or coechelon spaces. A major tool in the proof is the characterization of the επ-continuity of the tensor product of two diagonal operators from l p to l q . Several sharp forms of this result are also included.

How to cite

top

Bonet, J., et al. "Coincidence of topologies on tensor products of Köthe echelon spaces." Studia Mathematica 111.3 (1994): 263-281. <http://eudml.org/doc/216132>.

@article{Bonet1994,
abstract = {We investigate conditions under which the projective and the injective topologies coincide on the tensor product of two Köthe echelon or coechelon spaces. A major tool in the proof is the characterization of the επ-continuity of the tensor product of two diagonal operators from $l_p$ to $l_q$. Several sharp forms of this result are also included.},
author = {Bonet, J., Defant, A., Peris, A., Ramanujan, M.},
journal = {Studia Mathematica},
keywords = {Köthe echelon spaces; topological tensor products; injective and projective topologies; tensor products of diagonal operators; projective and injective topologies; tensor product of two Köthe echelon or coechelon spaces; -continuity of the tensor product of two diagonal operators},
language = {eng},
number = {3},
pages = {263-281},
title = {Coincidence of topologies on tensor products of Köthe echelon spaces},
url = {http://eudml.org/doc/216132},
volume = {111},
year = {1994},
}

TY - JOUR
AU - Bonet, J.
AU - Defant, A.
AU - Peris, A.
AU - Ramanujan, M.
TI - Coincidence of topologies on tensor products of Köthe echelon spaces
JO - Studia Mathematica
PY - 1994
VL - 111
IS - 3
SP - 263
EP - 281
AB - We investigate conditions under which the projective and the injective topologies coincide on the tensor product of two Köthe echelon or coechelon spaces. A major tool in the proof is the characterization of the επ-continuity of the tensor product of two diagonal operators from $l_p$ to $l_q$. Several sharp forms of this result are also included.
LA - eng
KW - Köthe echelon spaces; topological tensor products; injective and projective topologies; tensor products of diagonal operators; projective and injective topologies; tensor product of two Köthe echelon or coechelon spaces; -continuity of the tensor product of two diagonal operators
UR - http://eudml.org/doc/216132
ER -

References

top
  1. [1] V. Bartík, K. John and J. Korbaš, Tensor products of operators and Horn's inequality, in: Oper. Theory: Adv. Appl. 14, Birkhäuser, Basel, 1984, 39-46. Zbl0553.47018
  2. [2] K. D. Bierstedt und R. Meise, Induktive Limiten gewichteter Räume stetiger und holomorpher Funktionen, J. Reine Angew. Math. 282 (1976), 186-220. Zbl0318.46034
  3. [3] K. D. Bierstedt, R. Meise and W. H. Summers, A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 272 (1982), 107-160. Zbl0599.46026
  4. [4] K. D. Bierstedt, R. Meise and W. H. Summers, Köthe sets and Köthe sequence spaces, in: Functional Analysis, Holomorphy and Approximation Theory, North-Holland Math. Stud. 71, North-Holland, 1982, 27-91. Zbl0504.46007
  5. [5] B. Carl, B. Maurey und J. Puhl, Grenzordnungen von absolut-(r, p)-summierenden Operatoren, Math. Nachr. 82 (1978), 205-218. Zbl0407.47010
  6. [6] A. Defant, Barrelledness and nuclearity, Arch. Math. (Basel) 44 (1985), 168-170. Zbl0536.46001
  7. [7] A. Defant and K. Floret, Topological tensor products and the approximation property of locally convex spaces, Bull Soc. Roy. Sci. Liège 58 (1989), 29-51. Zbl0677.46057
  8. [8] A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Stud. 176, North-Holland, 1993. Zbl0774.46018
  9. [9] A. Defant and V. Mascioni, Limit orders of ε-π continuous operators, Math. Nachr. 145 (1990), 337-344. Zbl0726.47011
  10. [10] A. Defant and L. A. Moraes, On unconditional bases in spaces of holomorphic functions in infinite dimensions, Arch. Math. (Basel) 56 (1991), 163-173. Zbl0706.46028
  11. [11] S. Dineen and R. M. Timoney, Absolute bases, tensor products and a theorem of Bohr, Studia Math. 94 (1989), 227-234. Zbl0713.46005
  12. [12] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955). 
  13. [13] H. Jarchow, Locally Convex Spaces, Teubner, 1981. Zbl0466.46001
  14. [14] H. Jarchow and K. John, On the equality of injective and projective tensor products, Czechoslovak Math. J. 38 (1988), 464-472. Zbl0674.46044
  15. [15] H. Jarchow and K. John, Bilinear forms and nuclearity, preprint, 1992. Zbl0854.46002
  16. [16] K. John, Counterexample to a conjecture of Grothendieck, Math. Ann. 265 (1983), 169-179. Zbl0506.46053
  17. [17] K. John, On the compact non-nuclear operator problem, ibid. 287 (1990), 509-514. Zbl0687.47012
  18. [18] G. Köthe, Topological Vector Spaces I, II, Springer, 1969, 1979. Zbl0179.17001
  19. [19] P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, North-Holland Math. Stud. 131, North-Holland, 1987. Zbl0614.46001
  20. [20] A. Pietsch, Nuclear Locally Convex Spaces, Springer, 1972. 
  21. [21] A. Pietsch, Operator Ideals, Deutscher Verlag Wiss., 1978. 
  22. [22] A. Pietsch, Eigenvalues and s-Numbers, Cambridge University Press, 1987. 
  23. [23] G. Pisier, Counterexamples to a conjecture of Grothendieck, Acta Math. 151 (1983), 180-208. 
  24. [24] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986. 

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.