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isometries; rearrangement-invariant function spaces; Orlicz spaces; Lorentz spaces
Suppose that a real nonatomic function space on $[0,1]$ is equipped with two re\-arran\-ge\-ment-invariant norms $\|\cdot\|$ and $|\kern -0.5pt |\kern -0.5pt |\cdot|\kern -0.5pt |\kern -0.5pt |$. We study the question whether or not the fact that $(X,\|\cdot\|)$ is isometric to $(X,|\kern -0.5pt |\kern -0.5pt |\cdot|\kern -0.5pt |\kern -0.5pt |)$ implies that $\|f\|= |\kern -0.5pt |\kern -0.5pt |f|\kern -0.5pt |\kern -0.5pt |$ 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, resp\. Lorentz 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 $\varphi \neq\psi$ with the property that $L_\varphi$ and $L_\psi$ are isometric.
[1] Abramovich Y.A., Zaidenberg M.: Rearrangement invariant spaces. Notices Amer. Math. Soc. 26(2):A (1979), 207.
[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. MR 1358141
[3] Baron K., Hudzik H.: Orlicz spaces which are $L_p$-spaces. Aequationes Math. 48 (1994), 254-261. MR 1295094
[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. MR 1244671
[5] Hudzik H., Kurc W., Wisła M.: Strongly extreme points in Orlicz function spaces. J. Math. Anal. and Appl. 189 (1995), 651-670. MR 1312545
[6] Jamison J., Kamińska A., Lin P.K.: Isometries in Musielak-Orlicz spaces II. Studia Math. 104 (1993), 75-89. MR 1208040
[7] Kalton N.J., Randrianantoanina B.: Surjective isometries of rearrangement-invariant spaces. Quart. J. Math. Oxford 45 (1994), 301-327. MR 1295579
[8] Krasnosel'skii M.A., Rutickii Ya.B.: Convex Functions and Orlicz Spaces. P. Noordhoff LTD., Groningen, The Netherlands, 1961. MR 0126722
[9] Lacey H.E., Wojtaszczyk P.: Banach lattice structures on separable $L_p$-spaces. Proc. Amer. Math. Soc. 54 (1976), 83-89. MR 0390743 | Zbl 0317.46027
[10] Lamperti J.: On the isometries of some function spaces. Pacific J. Math. 8 (1958), 459-466. MR 0105017
[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. MR 0540367
[13] Montgomery-Smith S.J.: Comparison of Orlicz-Lorentz spaces. Studia Math. 11 (1993), 679-698. MR 1199324
[14] Zaidenberg M.G.: A representation of isometries on function spaces. I Prépublication de l'Institut Fourier des Mathématiques, 305, Grenoble, 1995. Zbl 0905.47023
[15] Zaidenberg M.G.: Groups of isometries of Orlicz spaces. Soviet Math. Dokl. 17 (1976), 432-436. Zbl 0345.46028
[16] Zaidenberg M.G.: On the isometric classification of symmetric spaces. Soviet Math. Dokl. 18 (1977), 636-640. MR 0442667
[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.
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