Interpolation of real method spaces via some ideals of operators

Mieczysław Mastyło; Mario Milman

Studia Mathematica (1999)

  • Volume: 136, Issue: 1, page 17-35
  • ISSN: 0039-3223

Abstract

top
Certain operator ideals are used to study interpolation of operators between spaces generated by the real method. Using orbital equivalence a new reiteration formula is proved for certain real interpolation spaces generated by ordered pairs of Banach lattices of the form ( X , L ∞ ( w ) ) . As an application we extend Ovchinnikov’s interpolation theorem from the context of classical Lions-Peetre spaces to a larger class of real interpolation spaces. A description of certain abstract J-method spaces is also presented.

How to cite

top

Mastyło, Mieczysław, and Milman, Mario. "Interpolation of real method spaces via some ideals of operators." Studia Mathematica 136.1 (1999): 17-35. <http://eudml.org/doc/216657>.

@article{Mastyło1999,
abstract = {Certain operator ideals are used to study interpolation of operators between spaces generated by the real method. Using orbital equivalence a new reiteration formula is proved for certain real interpolation spaces generated by ordered pairs of Banach lattices of the form $(X,L_∞(w))$. As an application we extend Ovchinnikov’s interpolation theorem from the context of classical Lions-Peetre spaces to a larger class of real interpolation spaces. A description of certain abstract J-method spaces is also presented.},
author = {Mastyło, Mieczysław, Milman, Mario},
journal = {Studia Mathematica},
keywords = {Lions-Peetre spaces; theorem of Ovchinnikov; -summing operators; abstract real method; quasi-normed lattices; multiplicators; quasi-Banach space; quasi-norm; Banach lattice; factorisation theorem of Graham Bennett},
language = {eng},
number = {1},
pages = {17-35},
title = {Interpolation of real method spaces via some ideals of operators},
url = {http://eudml.org/doc/216657},
volume = {136},
year = {1999},
}

TY - JOUR
AU - Mastyło, Mieczysław
AU - Milman, Mario
TI - Interpolation of real method spaces via some ideals of operators
JO - Studia Mathematica
PY - 1999
VL - 136
IS - 1
SP - 17
EP - 35
AB - Certain operator ideals are used to study interpolation of operators between spaces generated by the real method. Using orbital equivalence a new reiteration formula is proved for certain real interpolation spaces generated by ordered pairs of Banach lattices of the form $(X,L_∞(w))$. As an application we extend Ovchinnikov’s interpolation theorem from the context of classical Lions-Peetre spaces to a larger class of real interpolation spaces. A description of certain abstract J-method spaces is also presented.
LA - eng
KW - Lions-Peetre spaces; theorem of Ovchinnikov; -summing operators; abstract real method; quasi-normed lattices; multiplicators; quasi-Banach space; quasi-norm; Banach lattice; factorisation theorem of Graham Bennett
UR - http://eudml.org/doc/216657
ER -

References

top
  1. [1] G. Bennett, Inclusion mappings between l p spaces, J. Funct. Anal. 13 (1973), 20-27. Zbl0255.47033
  2. [2] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976. Zbl0344.46071
  3. [3] Yu. A. Brudnyĭ and N. Ya. Krugljak, Interpolation Functors and Interpolation Spaces I, North-Holland, Amsterdam, 1991. 
  4. [4] A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113-190. Zbl0204.13703
  5. [5] M. Cwikel and J. Peetre, Abstract K and J spaces, J. Math. Pures Appl. 60 (1981), 1-50. 
  6. [6] M. Cwikel and Y. Sagher, Relation between real and complex interpolation spaces, Indiana Univ. Math. J. 36 (1987), 905-912. 
  7. [7] V. I. Dmitriev, Duality of the interpolation methods of constants and means, Dokl. Akad. Nauk SSSR 214 (1974), 22-24 (in Russian). 
  8. [8] S. Janson, Maximal and minimal methods of interpolation, J. Funct. Anal. 44 (1981), 50-73. Zbl0492.46059
  9. [9] S. G. Kreĭn, Yu. I. Petunin and E. M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian); English transl.: Amer. Math. Soc., Providence, 1982. 
  10. [10] G. Ya. Lozanovskiĭ, On some Banach lattices IV, Sibirsk. Mat. Zh. 14 (1973), 140-155 (in Russian); English transl.: Siberian Math. J. 14 (1973), 97-108. 
  11. [11] L. Maligranda and M. Mastyło, Inclusion mappings between Orlicz sequence spaces, Report No. 090/1998, 12 pp., Faculty of Math. & Comp. Sci., Poznań. Zbl0982.46017
  12. [12] V. I. Ovchinnikov, The methods of orbits in interpolation theory, Math. Rep. 1 (1984), 349-516. 
  13. [13] V. I. Ovchinnikov, Interpolation theorems for L p , q -spaces, Mat. Sb. 136 (1988), 227-240 (in Russian). Zbl0657.46056
  14. [14] A. Pietsch, Operator Ideals, North-Holland, Berlin, 1978. 
  15. [15] Y. Sagher, Interpolation of r-Banach spaces, Studia Math. 41 (1972), 45-70. 
  16. [16] G. Sparr, Interpolation of weighted L p -spaces, ibid. 62 (1978), 229-271. Zbl0393.46029

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.