Rochberg’s Abstract Coboundary Theorem Revisited

Catalin Badea () and Oscar Devys
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Catalin Badea: CNRS UMR 8524 - Laboratoire Paul Painlevé, University of Lille
Oscar Devys: CNRS UMR 8524 - Laboratoire Paul Painlevé, University of Lille

A chapter in Multivariable Operator Theory, 2023, pp 19-35 from Springer

Abstract: Abstract Rochberg’s coboundary theorem provides conditions under which the equation ( I - T ) y = x $$(I-T)y = x$$ is solvable in y. Here T is a unilateral shift on Hilbert space, I is the identity operator and x is a given vector. The conditions are expressed in terms of Wold-type decomposition determined by T and growth of iterates of T at x. We revisit Rochberg’s theorem and prove the following result. Let T be an isometry acting on a Hilbert space H $$\mathcal H$$ and let x ∈ H $$x \in \mathcal H$$ . Suppose that ∑ k = 0 ∞ k ‖ T ∗ k x ‖

Keywords: Coboundary theorems; Unilateral shifts; Wold decomposition; Functional equations; 47A05; 47A35; 39B05 (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-50535-5_2

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DOI: 10.1007/978-3-031-50535-5_2

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