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\begin{document}
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{\Large Department of Mathematics, BGU}

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{\Huge BGU Probability and Ergodic Theory  (PET) seminar}\\[0.2\baselineskip]

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\textbf{On} \emph{Thursday, May  2, 2019}
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\textbf{At} \emph{11:10 -- 12:00}
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\textbf{In} \emph{-101}

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{\large\scshape Michael Lin  
  %
  (Ben-Gurion University)
}
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will talk about
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{\Large\bfseries Joint and double coboundaries of transformations  an application of maximal spectral type of spectral measures\par}
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\textsc{Abstract:}
Let T be a bounded linear operator on a Banach space X; the elements
of (I − T)X are called coboundaries. For two commuting operators T and
S, elements of (I − T)X ∩ (I − S)X are called joint coboundaries, and those
of (I − T)(I − S)X are double coboundaries. By commutativity, double
coboundaries are joint ones. Are there any other?
Let θ and τ be commuting invertible measure preserving transformations
of (Ω, Σ, m), with corresponding unitary operators induced on L2(m). We
prove the existence of a joint coboundary g ∈ (I − U)L2 ∩ (I − V )L2 which
is not in (I − U)(I − V )L2.
For the proof, let E be the spectral measure on T
2 obtained by Stone's
spectral theorem. Joint and double coboundaries are characterized using E,
and properties of the maximal spectral type of E, together with a result of
Foia³ on multiplicative spectral measures acting on L2, are used to show the
existence of the required function.


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