Integral Equations and Operator Theory

, Volume 84, Issue 4, pp 487–500 | Cite as

Positive Isometric Averaging Operators on \({\ell^2(\mathbb{Z}, \mu)}\)



We show that positive isometric averaging operators on the sequence space \({\ell^2(\mathbb{Z}, \mu)}\) are determined by very subtle arithmetic conditions on \({\mu}\) (even for very simple examples), contrary to what happens in the continuous case \({L^2({\mathbb{R}}^+)}\), where any possible average value is realized by a suitable positive isometry.


Isometries Averaging operators 

Mathematics Subject Classification

Primary 46E30 Secondary 46B04 


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  1. 1.
    Ash P., Marshall Ash J., Ogden R.D.: A characterization of isometries. J. Math. Anal. Appl. 60, 417–428 (1977)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Boza S., Soria J.: Solution to a conjecture on the norm of the Hardy operator minus the identity. J. Funct. Anal. 260(4), 1020–1028 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Boza S., Soria J.: Isometries on \({L^2(X)}\) and monotone functions. Math. Nach. 287, 160–172 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brown A., Halmos P.R., Shields A.L.: Cesàro operators. Acta Sci. Math. (Szeged) 26, 125–137 (1965)MathSciNetMATHGoogle Scholar
  5. 5.
    Cerdà, J.: Linear functional analysis. In: Graduate Studies in Mathematics, vol. 116. American Mathematical Society, Providence, Real Sociedad Matemática Española, Madrid (2010)Google Scholar
  6. 6.
    Kaiblinger, N., Maligranda, L., Persson, L.E.: Norms in weighted \({L^2}\)-spaces and Hardy operators. In: Function Spaces, The Fifth Conference (Poznań, 1998), Lecture Notes in Pure and Appl. Math., vol. 213, pp. 205–216. Dekker, New York (2000)Google Scholar
  7. 7.
    Kalton N.J., Randrianantoanina B.: Surjective isometries on rearrangement-invariant spaces. Q. J. Math. Oxford Ser. (2) 45(179), 301–327 (1994)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Zaidenberg M.G.: A representation of isometries on function spaces. Mat. Fiz. Anal. Geom. 4, 339–347 (1997)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Applied Mathematics IV, EPSEVGPolytechnical University of CataloniaVilanova i GeltrúSpain
  2. 2.Department of Applied Mathematics and AnalysisUniversity of BarcelonaBarcelonaSpain

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