The Grothendieck-Pietsch domination principle for nonlinear summing integral operators

Karl Lermer

Studia Mathematica (1998)

  • Volume: 129, Issue: 2, page 97-112
  • ISSN: 0039-3223

Abstract

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We transform the concept of p-summing operators, 1≤ p < ∞, to the more general setting of nonlinear Banach space operators. For 1-summing operators on B(Σ,X)-spaces having weak integral representations we generalize the Grothendieck-Pietsch domination principle. This is applied for the characterization of 1-summing Hammerstein operators on C(S,X)-spaces. For p-summing Hammerstein operators we derive the existence of control measures and p-summing extensions to B(Σ,X)-spaces.

How to cite

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Lermer, Karl. "The Grothendieck-Pietsch domination principle for nonlinear summing integral operators." Studia Mathematica 129.2 (1998): 97-112. <http://eudml.org/doc/216499>.

@article{Lermer1998,
abstract = {We transform the concept of p-summing operators, 1≤ p < ∞, to the more general setting of nonlinear Banach space operators. For 1-summing operators on B(Σ,X)-spaces having weak integral representations we generalize the Grothendieck-Pietsch domination principle. This is applied for the characterization of 1-summing Hammerstein operators on C(S,X)-spaces. For p-summing Hammerstein operators we derive the existence of control measures and p-summing extensions to B(Σ,X)-spaces.},
author = {Lermer, Karl},
journal = {Studia Mathematica},
keywords = {-summing operators; nonlinear Banach space operators; Grothendieck-Pietsch domination principle; 1-summing Hammerstein operators; control measures; -summing extensions},
language = {eng},
number = {2},
pages = {97-112},
title = {The Grothendieck-Pietsch domination principle for nonlinear summing integral operators},
url = {http://eudml.org/doc/216499},
volume = {129},
year = {1998},
}

TY - JOUR
AU - Lermer, Karl
TI - The Grothendieck-Pietsch domination principle for nonlinear summing integral operators
JO - Studia Mathematica
PY - 1998
VL - 129
IS - 2
SP - 97
EP - 112
AB - We transform the concept of p-summing operators, 1≤ p < ∞, to the more general setting of nonlinear Banach space operators. For 1-summing operators on B(Σ,X)-spaces having weak integral representations we generalize the Grothendieck-Pietsch domination principle. This is applied for the characterization of 1-summing Hammerstein operators on C(S,X)-spaces. For p-summing Hammerstein operators we derive the existence of control measures and p-summing extensions to B(Σ,X)-spaces.
LA - eng
KW - -summing operators; nonlinear Banach space operators; Grothendieck-Pietsch domination principle; 1-summing Hammerstein operators; control measures; -summing extensions
UR - http://eudml.org/doc/216499
ER -

References

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  1. [1] J. Batt, Nonlinear integral operators on C(S,E), Studia Math. 48 (1973), 147-181. 
  2. [2] J. Batt and J. Berg, Linear bounded transformations on the space of continuous functions, J. Funct. Anal. 4 (1969), 215-239. Zbl0183.13502
  3. [3] F. Bombal, Operators on spaces of vector valued continuous functions, Extracta Math. 1 (1986), 103-114. 
  4. [4] F. Bombal and P. Cembranos, Characterizations of some classes of operators on spaces of vector valued continuous functions, Math. Proc. Cambridge Philos. Soc. 97 (1985), 137-146. Zbl0564.47013
  5. [5] J. Diestel, Sequences and Series in Banach Spaces, Springer, Berlin, 1984. 
  6. [6] J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977. 
  7. [7] N. Dinculeanu, Vector Measures, Pergamon Press, Oxford, 1967. 
  8. [8] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955). 
  9. [9] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981. Zbl0466.46001
  10. [10] K. Lermer, Characterizations of weakly compact nonlinear integral operators on C(S)-spaces, Stud. Cerc. Mat. 48 (1996), 365-378. Zbl0859.47035
  11. [11] A. Pietsch, Operator Ideals, North-Holland, 1980. 

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