Abstract
This paper was motivated by a result of Halmos [1971] on the characterization of invariant subspaces of finite-dimensional, complex linear operators. It presents a purely algebraic approach, using polynomial and rational models over an arbitrary field, that yields a functional proof of an extension of the result by Halmos. This led to a parallel effort to give a simplified, matrix-oriented, proof. In turn, we explore the connection of Halmos’ result with a celebrated Theorem of Roth [1952]. The method presented here has the advantage of generalizing to a class of infinite-dimensional shift operators.
Mathematics Subject Classification. 15A24, 15A54.
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Dedicated to Bill Helton on the occasion of his 65th birthday
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Fuhrmann, P.A., Helmke, U. (2012). On Theorems of Halmos and Roth. In: Dym, H., de Oliveira, M., Putinar, M. (eds) Mathematical Methods in Systems, Optimization, and Control. Operator Theory: Advances and Applications, vol 222. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0411-0_13
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DOI: https://doi.org/10.1007/978-3-0348-0411-0_13
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