Abstract
A remarkable theorem of Domar asserts that the lattice of the invariant subspaces of the right shift semigroup {Sτ≥ 0 in L2(ℝ+, w(t)dt) consists of just the “standard invariant subspaces” whenever w is a positive continuous function in ℝ+ such that
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(1)|log w is concave in [c,℞) for some c ≥ 0
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(2)|\({\lim _{t \to \infty }}\frac{{ - \log w\left( t \right)}}{t} = \infty \), and \({\lim _{t \to \infty }}\frac{{\log \left| {\log w\left( t \right)} \right| - \log t}}{{\sqrt {\log t} }} = \infty \)
We prove an extension of Domar’s Theorem to a strictly wider class of weights w, answering a question posed by Domar in [6].
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Communicated by L. Kérchy
Authors are partially supported by Plan Nacional I+D grant no. MTM2013-42105-P.
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Gallardo-Gutiérrez, E.A., Partington, J.R. & Rodríguez, D.J. An extension of a theorem of Domar on invariant subspaces. ActaSci.Math. 83, 271–290 (2017). https://doi.org/10.14232/actasm-015-837-7
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DOI: https://doi.org/10.14232/actasm-015-837-7