# Operators commuting with translations, and systems of difference equations

Colloquium Mathematicae (1999)

- Volume: 80, Issue: 1, page 1-22
- ISSN: 0010-1354

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topLaczkovich, Miklós. "Operators commuting with translations, and systems of difference equations." Colloquium Mathematicae 80.1 (1999): 1-22. <http://eudml.org/doc/210701>.

@article{Laczkovich1999,

abstract = {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.},

author = {Laczkovich, Miklós},

journal = {Colloquium Mathematicae},

keywords = {operators commuting with translation; systems of difference equations},

language = {eng},

number = {1},

pages = {1-22},

title = {Operators commuting with translations, and systems of difference equations},

url = {http://eudml.org/doc/210701},

volume = {80},

year = {1999},

}

TY - JOUR

AU - Laczkovich, Miklós

TI - Operators commuting with translations, and systems of difference equations

JO - Colloquium Mathematicae

PY - 1999

VL - 80

IS - 1

SP - 1

EP - 22

AB - 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.

LA - eng

KW - operators commuting with translation; systems of difference equations

UR - http://eudml.org/doc/210701

ER -

## References

top- [1] S. A. Argyros, On the space of bounded measurable functions, Quart. J. Math. Oxford (2) 34 (1983), 129-132. Zbl0525.46017
- [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. Zbl0859.28001
- [4] D. S. Passman, The Algebraic Structure of Group Rings, Wiley, 1977. Zbl0368.16003
- [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. Zbl0569.43001

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