Isometric classification of norms in rearrangement-invariant function spaces

Beata Randrianantoanina

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 1, page 73-90
  • ISSN: 0010-2628

Abstract

top
Suppose that a real nonatomic function space on [ 0 , 1 ] is equipped with two rearrangement-invariant norms ∥ · ∥ and | | | · | | | . We study the question whether or not the fact that ( X , ∥ · ∥ ) is isometric to ( X , | | | · | | | ) implies that ∥ f ∥ = | | | f | | | for all f in X . We show that in strictly monotone Orlicz and Lorentz spaces this is equivalent to asking whether or not the norms are defined by equal Orlicz functions, respĿorentz weights. We show that the above implication holds true in most rearrangement-invariant spaces, but we also identify a class of Orlicz spaces where it fails. We provide a complete description of Orlicz functions ϕ ≠ ψ with the property that L ϕ and L ψ are isometric.

How to cite

top

Randrianantoanina, Beata. "Isometric classification of norms in rearrangement-invariant function spaces." Commentationes Mathematicae Universitatis Carolinae 38.1 (1997): 73-90. <http://eudml.org/doc/248048>.

@article{Randrianantoanina1997,
abstract = {Suppose that a real nonatomic function space on $[0,1]$ is equipped with two rearrangement-invariant norms $\Vert \cdot \Vert $ and $|\hspace\{-0.5pt\}|\hspace\{-0.5pt\}|\cdot |\hspace\{-0.5pt\}|\hspace\{-0.5pt\}|$. We study the question whether or not the fact that $(X,\Vert \cdot \Vert )$ is isometric to $(X,|\hspace\{-0.5pt\}|\hspace\{-0.5pt\}|\cdot |\hspace\{-0.5pt\}|\hspace\{-0.5pt\}|)$ implies that $\Vert f\Vert = |\hspace\{-0.5pt\}|\hspace\{-0.5pt\}|f|\hspace\{-0.5pt\}|\hspace\{-0.5pt\}|$ for all $f$ in $X$. We show that in strictly monotone Orlicz and Lorentz spaces this is equivalent to asking whether or not the norms are defined by equal Orlicz functions, respĿorentz weights. We show that the above implication holds true in most rearrangement-invariant spaces, but we also identify a class of Orlicz spaces where it fails. We provide a complete description of Orlicz functions $\varphi \ne \psi $ with the property that $L_\varphi $ and $L_\psi $ are isometric.},
author = {Randrianantoanina, Beata},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {isometries; rearrangement-invariant function spaces; Orlicz spaces; Lorentz spaces; isometries; rearrangement-invariant function spaces; Orlicz spaces; Lorentz spaces},
language = {eng},
number = {1},
pages = {73-90},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Isometric classification of norms in rearrangement-invariant function spaces},
url = {http://eudml.org/doc/248048},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Randrianantoanina, Beata
TI - Isometric classification of norms in rearrangement-invariant function spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 1
SP - 73
EP - 90
AB - Suppose that a real nonatomic function space on $[0,1]$ is equipped with two rearrangement-invariant norms $\Vert \cdot \Vert $ and $|\hspace{-0.5pt}|\hspace{-0.5pt}|\cdot |\hspace{-0.5pt}|\hspace{-0.5pt}|$. We study the question whether or not the fact that $(X,\Vert \cdot \Vert )$ is isometric to $(X,|\hspace{-0.5pt}|\hspace{-0.5pt}|\cdot |\hspace{-0.5pt}|\hspace{-0.5pt}|)$ implies that $\Vert f\Vert = |\hspace{-0.5pt}|\hspace{-0.5pt}|f|\hspace{-0.5pt}|\hspace{-0.5pt}|$ for all $f$ in $X$. We show that in strictly monotone Orlicz and Lorentz spaces this is equivalent to asking whether or not the norms are defined by equal Orlicz functions, respĿorentz weights. We show that the above implication holds true in most rearrangement-invariant spaces, but we also identify a class of Orlicz spaces where it fails. We provide a complete description of Orlicz functions $\varphi \ne \psi $ with the property that $L_\varphi $ and $L_\psi $ are isometric.
LA - eng
KW - isometries; rearrangement-invariant function spaces; Orlicz spaces; Lorentz spaces; isometries; rearrangement-invariant function spaces; Orlicz spaces; Lorentz spaces
UR - http://eudml.org/doc/248048
ER -

References

top
  1. Abramovich Y.A., Zaidenberg M., Rearrangement invariant spaces, Notices Amer. Math. Soc. 26(2):A (1979), 207. (1979) 
  2. Abramovich Y.A., Zaidenberg M., A rearrangement invariant space isometric to L p coincides with L p , in: N.J. Kalton and E. Saab, editors, Interaction between Functional Analysis, Harmonic Analysis, and Probability, pp. 13-18, Lect. Notes in Pure Appl. Math. 175, Marcel Dekker, New York, 1995. MR1358141
  3. Baron K., Hudzik H., Orlicz spaces which are L p -spaces, Aequationes Math. 48 (1994), 254-261. (1994) MR1295094
  4. Carothers N.L., Haydon R.G., Lin P.K., On the isometries of the Lorentz space L w , p , Israel J. Math. 84 (1993), 265-287. (1993) MR1244671
  5. Hudzik H., Kurc W., Wisła M., Strongly extreme points in Orlicz function spaces, J. Math. Anal. and Appl. 189 (1995), 651-670. (1995) MR1312545
  6. Jamison J., Kamińska A., Lin P.K., Isometries in Musielak-Orlicz spaces II, Studia Math. 104 (1993), 75-89. (1993) MR1208040
  7. Kalton N.J., Randrianantoanina B., Surjective isometries of rearrangement-invariant spaces, Quart. J. Math. Oxford 45 (1994), 301-327. (1994) MR1295579
  8. Krasnosel'skii M.A., Rutickii Ya.B., Convex Functions and Orlicz Spaces, P. Noordhoff LTD., Groningen, The Netherlands, 1961. MR0126722
  9. Lacey H.E., Wojtaszczyk P., Banach lattice structures on separable L p -spaces, Proc. Amer. Math. Soc. 54 (1976), 83-89. (1976) Zbl0317.46027MR0390743
  10. Lamperti J., On the isometries of some function spaces, Pacific J. Math. 8 (1958), 459-466. (1958) MR0105017
  11. Lin P.K., Elementary isometries of rearrangement invariant spaces, preprint. 
  12. Lindenstrauss J., Tzafriri L., Classical Banach Spaces, Vol. 2, Function Spaces, SpringerVerlag, Berlin-Heidelberg-New York, 1979. MR0540367
  13. Montgomery-Smith S.J., Comparison of Orlicz-Lorentz spaces, Studia Math. 11 (1993), 679-698. (1993) MR1199324
  14. Zaidenberg M.G., A representation of isometries on function spaces, I Prépublication de l'Institut Fourier des Mathématiques, 305, Grenoble, 1995. Zbl0905.47023
  15. Zaidenberg M.G., Groups of isometries of Orlicz spaces, Soviet Math. Dokl. 17 (1976), 432-436. (1976) Zbl0345.46028
  16. Zaidenberg M.G., On the isometric classification of symmetric spaces, Soviet Math. Dokl. 18 (1977), 636-640. (1977) MR0442667
  17. Zaidenberg M.G., Special representations of isometries of function spaces (in Russian), Investigations on the theory of functions of several real variables, Yaroslavl, 1980. 

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.