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1999 | 80 | 1 | 1-22
Tytuł artykułu

Operators commuting with translations, and systems of difference equations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let ${\mathcal B} ={f:ℝ → ℝ: f is bounded}$, and ${\mathcal M} ={f:ℝ → ℝ: f is Lebesgue measurable}$. We show that there is a linear operator $Φ :{\mathcal B} → {\mathcal M}$ such that Φ(f)=f a.e. for every $f ∈ {\mathcal B} ∩ {\mathcal M}$, and Φ commutes with all translations. On the other hand, if $Φ : {\mathcal B} → {\mathcal M}$ is a linear operator such that Φ(f)=f for every $f ∈ {\mathcal B} ∩ {\mathcal M}$, then the group $G_Φ$ ={ a ∈ ℝ:Φ commutes with the translation by a} is of measure zero and, assuming Martin's axiom, is of cardinality less than continuum. Let Φ be a linear operator from $ℂ^ℝ$ into the space of complex-valued measurable functions. We show that if Φ(f) is non-zero for every $f(x)=e^{cx}$, then $G_Φ$ must be discrete. If Φ(f) is non-zero for a single polynomial-exponential f, then $G_Φ$ is countable, moreover, the elements of $G_Φ$ are commensurable. We construct a projection from $ℂ^ℝ$ onto the polynomials that commutes with rational translations. All these results are closely connected with the solvability of certain systems of difference equations.
Słowa kluczowe
Rocznik
Tom
80
Numer
1
Strony
1-22
Opis fizyczny
Daty
wydano
1999
otrzymano
1996-08-26
poprawiono
1997-02-15
Twórcy
  • Department of Analysis, Eötvös Loránd University, Rákóczi út 5 Budapest, Hungary 1088
Bibliografia
  • [1] S. A. Argyros, On the space of bounded measurable functions, Quart. J. Math. Oxford (2) 34 (1983), 129-132.
  • [2] K. Kunen, Set Theory. An Introduction to Independence Proofs, Stud. Logic Found. Math. 102, North-Holland, 1980.
  • [3] M. Laczkovich, Decomposition using measurable functions, C. R. Acad. Sci. Paris Sér. I 323 (1996), 583-586.
  • [4] D. S. Passman, The Algebraic Structure of Group Rings, Wiley, 1977.
  • [5] W. Sierpiński, Sur les translations des ensembles linéaires, Fund. Math. 19 (1932), 22-28.
  • [6] S. Wagon, The Banach-Tarski Paradox, Cambridge Univ. Press, 1993.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv80i1p1bwm
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