On operator ideals related to (p,σ)-absolutely continuous operators

J. López Molina; E. Sánchez Pérez

Studia Mathematica (2000)

  • Volume: 138, Issue: 1, page 25-40
  • ISSN: 0039-3223

Abstract

top
We study tensor norms and operator ideals related to the ideal P p , σ , 1 < p < ∞, 0 < σ < 1, of (p,σ)-absolutely continuous operators of Matter. If α is the tensor norm associated with P p , σ (in the sense of Defant and Floret), we characterize the ( α ' ) t -nuclear and ( α ' ) t - integral operators by factorizations by means of the composition of the inclusion map L r ( μ ) → L 1 ( μ ) + L p ( μ ) with a diagonal operator B w : L ∞ ( μ ) → L r ( μ ) , where r is the conjugate exponent of p’/(1-σ). As an application we study the reflexivity of the components of the ideal P p , σ .

How to cite

top

López Molina, J., and Sánchez Pérez, E.. "On operator ideals related to (p,σ)-absolutely continuous operators." Studia Mathematica 138.1 (2000): 25-40. <http://eudml.org/doc/216688>.

@article{LópezMolina2000,
abstract = {We study tensor norms and operator ideals related to the ideal $P_\{p,σ\}$, 1 < p < ∞, 0 < σ < 1, of (p,σ)-absolutely continuous operators of Matter. If α is the tensor norm associated with $P_\{p,σ\}$ (in the sense of Defant and Floret), we characterize the $(α^\{\prime \})^t$-nuclear and $(α^\{\prime \})^t$- integral operators by factorizations by means of the composition of the inclusion map $L^r(μ) → L^1(μ) + L^p(μ)$ with a diagonal operator $B_w:L^\{∞\}(μ) → L^r(μ)$, where r is the conjugate exponent of p’/(1-σ). As an application we study the reflexivity of the components of the ideal $P_\{p,σ\}$.},
author = {López Molina, J., Sánchez Pérez, E.},
journal = {Studia Mathematica},
keywords = {tensor norms; operator ideals; (p,σ)-absolutely continuous operators; $g_\{p,σ\}$-nuclear and $g_\{p,σ\}$ integral operators; Matter's ideal; -absolutely continuous operators; factorization properties; inclusion maps; diagonal operators},
language = {eng},
number = {1},
pages = {25-40},
title = {On operator ideals related to (p,σ)-absolutely continuous operators},
url = {http://eudml.org/doc/216688},
volume = {138},
year = {2000},
}

TY - JOUR
AU - López Molina, J.
AU - Sánchez Pérez, E.
TI - On operator ideals related to (p,σ)-absolutely continuous operators
JO - Studia Mathematica
PY - 2000
VL - 138
IS - 1
SP - 25
EP - 40
AB - We study tensor norms and operator ideals related to the ideal $P_{p,σ}$, 1 < p < ∞, 0 < σ < 1, of (p,σ)-absolutely continuous operators of Matter. If α is the tensor norm associated with $P_{p,σ}$ (in the sense of Defant and Floret), we characterize the $(α^{\prime })^t$-nuclear and $(α^{\prime })^t$- integral operators by factorizations by means of the composition of the inclusion map $L^r(μ) → L^1(μ) + L^p(μ)$ with a diagonal operator $B_w:L^{∞}(μ) → L^r(μ)$, where r is the conjugate exponent of p’/(1-σ). As an application we study the reflexivity of the components of the ideal $P_{p,σ}$.
LA - eng
KW - tensor norms; operator ideals; (p,σ)-absolutely continuous operators; $g_{p,σ}$-nuclear and $g_{p,σ}$ integral operators; Matter's ideal; -absolutely continuous operators; factorization properties; inclusion maps; diagonal operators
UR - http://eudml.org/doc/216688
ER -

References

top
  1. [1] Aliprantis C. D. and Burkinshaw, O., Positive Operators, Academic Press, New York, 1985. Zbl0608.47039
  2. [2] Beauzamy, B., Espaces d'Interpolation Réels: Topologie et Géométrie, Lecture Notes in Math. 666, Springer, Berlin, 1978. Zbl0382.46021
  3. [3] Defant, A. and Floret, K., Tensor Norms and Operator Ideals, North-Holland Math. Stud. 176, North-Holland, Amsterdam, 1993. Zbl0774.46018
  4. [4] Gilbert, J. E. and Leih, T. J., Factorization, tensor products and bilinear forms in Banach space theory, in: Notes in Banach Spaces, Univ. of Texas Press, Austin, 1980, 182-305. Zbl0471.46053
  5. [5] Köthe, G., Topological Vector Spaces II, Springer, New York, 1979. Zbl0417.46001
  6. [6] López Molina, J. A. and Sánchez Pérez, E. A., Ideales de operadores absolutamente continuos, Rev. Real Acad. Cienc. Madrid 87 (1993), 349-378. 
  7. [7] López Molina, J. A. and Sánchez Pérez, E. A., On ultraproducts of compositions of certain operators between some finite dimensional ℓ p -spaces, preprint, 1999. 
  8. [8] Matter, U., Absolutely continuous operators and super-reflexivity, Math. Nachr. 130 (1987), 193-216. Zbl0622.47045
  9. [9] Matter, Factoring through interpolation spaces and super-reflexive Banach spaces, Rev. Roumaine Math. Pures Appl. 34 (1989), 147-156. Zbl0675.46032
  10. [10] Niculescu, C. P., Absolute continuity in Banach space theory, ibid. 24 (1979), 413-422. Zbl0406.47010
  11. [11] Pietsch, A., Operator Ideals, North-Holland, Amsterdam, 1980. 
  12. [12] Saphar, P., Produits tensoriels d'espaces de Banach et classes d'applications linéaires, Studia Math. 38 (1970), 71-100. Zbl0213.14201

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.