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GdTProbaTESD20191125
Double and joint coboundaries of irrational circle rotations
Salle de séminaire M.0.1.
(Ben-Gurion University, Israel)
Let T and S be contractions on a Banach space X. Elements of (I−T)X
are called coboundaries of T; the elements of (I−T)X∩(I−S)X are called
joint coboundaries of T and S. If T and S commute, then obviously
the elements of (I−T)(I−S)X, called double coboundaries, are joint coboundaries.
It is natural to ask if there exist joint coboundaries which are not double coboundaries.
We prove that if Tα and Tβ are induced on Lp(T) or C(T)
by two irrational rotations, α and β, of the unit circle T,
then there are joint coboundaries which are not double coboundaries.
Moreover, we prove that there are continuous functions ψ,f,g∈C(T) such that
(I−Tα)f=ψ=(I−Tβ)g (i.e. ψ is a joint C(T) coboundary), but
there is NO measurable h satisfying (I−Tα)(I−Tβ)h=ψ.
For non-commuting transformations, the problem of existence of joint coboundaries
was studied by T. Adams and J. Rosenblatt.
(Joint work with Guy Cohen)