Abstract
We consider a complex symmetric sequence space E that possesses the Fatou property and is different from l2. We prove that, for every surjective linear isometry V on E, there exist λ n ∈ ℂ with |λ n | = 1 and a bijective mapping π on the set ℕ of natural numbers such that
for every {ξ n {n∈ℕ ∈ E.
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References
J. Arazy, “Isometries on complex symmetric sequence spaces,” Math. Z. 188, 427 (1985).
S. Banach, Theory of Linear Operations (North-Holland, Amsterdam–New York–Oxford–Tokyo, 1987).
F. F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras (Cambridge Univ. Press, London, 1971).
M. Sh. Braverman and E. M. Semenov, “Isometries on symmetric spaces,” Dokl. Akad. Nauk SSSR 217, 257 (1974) [Sov.Math., Dokl. 15, 1027 (1974)].
M. Sh. Braverman and E. M. Semenov, “Isometries on symmetric spaces,” Trudy NII Matem. Voronezh. Gos. Univ. 7, 17 (1975) [in Russian].
J. Dixmier, Les C*-algébres et leurs représentations (Gauthier-Villars, Paris, 1964) [C*-algebras (North-Holland, Amsterdam, 1983)].
R. J. Fleming and J. E. Jamison, Isometries on Banach Spaces: Function Spaces (Chapman & Hall/CRC, Boca Raton, FL, 2003).
G. Godefroy and N. J. Kalton, “The ball topology and its applications,” Contemp. Math. 85, 195 (1989).
R. C. James, “Orthogonality and linear functionals in normed linear spaces,” Trans. Amer. Math. Soc. 61, 285 (1947).
N. J. Kalton and B. Randrianantoanina, “Surjective isometries on rearrangement-invariant spaces,” Quart. J.Math. Oxford (2), 45, 301 (1994).
L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1984) [Functional Analysis (Pergamon Press, Oxford, 1982)].
J. L. Kelley, General Topology (Dover Publications, Mineola, NY, 2017).
G. Köthe, Topologische lineare Räume. I (Springer-Verlag, Berlin–Heidelberg–New York, 1966) [Topological Vector Spaces. I (Springer-Verlag, New York–Heidelberg–Berlin, 1969)].
S. G. Kreĭn, Yu. I. Petunin, and E.M. Semenov, Interpolation of Linear Operators (Nauka,Moscow, 1978) [Interpolation of Linear Operators (Amer.Math. Soc., Providence, RI, 1982)].
J. Lamperti, “On the isometries on certain function-spaces,” Pacific J.Math. 8, 459 (1958).
G. Lumer, “On the isometries on reflexive Orlicz spaces,” Ann. Inst. Fourier 13, 99 (1963).
W. A. J. Luxemburg and A. C. Zaanen, “Compactness of integral operators in Banach function spaces,” Math. Ann. 149, 150 (1963).
L. Molnár, “2-Local isometries on some operator algebras,” Proc. EdinburghMath. Soc. (2) 45, 349 (2002).
G. Pólya and G. Szegő, Aufgaben und Lehrsätze aus der Analysis. Bd. 1. Reihen, Integralrechnung, Funktionentheorie (Springer-Verlag, Berlin–Heidelberg–New York, 1970) [Problems and Theorems in Analysis. I. Series, Integral Calculus, Theory of Functions (Springer, Berlin, 1998)].
A. R. Sourour, “Isometries on norm ideals of compact operators,” J. Funct. Anal. 43, 69 (1981).
P. Wojtaszczyk, Banach Spaces for Analysts (Cambridge Univ. Press, Cambridge, 1996).
M. G. Zaidenberg, “On isometric classification of symmetric spaces,” Dokl. Akad. Nauk SSSR 234, 283 (1977) [Sov.Math., Dokl. 18, 636 (1977)].
M. G. Zaidenberg, “A representation of isometries on function spaces,” Mat. Fiz. Anal. Geom. 4, 339 (1997).
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Original Russian Text © B.R. Aminov and V.I. Chilin, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 1, pp. 21–42.
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Aminov, B.R., Chilin, V.I. Isometries and Hermitian operators on complex symmetric sequence spaces. Sib. Adv. Math. 27, 239–252 (2017). https://doi.org/10.3103/S1055134417040022
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DOI: https://doi.org/10.3103/S1055134417040022