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On theorems of Halmos and Roth. (English) Zbl 1275.15008

Dym, Harry (ed.) et al., Mathematical methods in systems, optimization, and control. Festschrift in honor of J. William Helton. Basel: Birkhäuser (ISBN 978-3-0348-0410-3/hbk; 978-3-0348-0411-0/ebook). Operator Theory: Advances and Applications 222, 173-187 (2012).
Summary: This paper was motivated by a result by P. R. Halmos [Linear Algebra Appl. 4, 11–15 (1971; Zbl 0264.15001)] 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 by W. E. Roth [Proc. Am. Math. Soc. 3, 392–396 (1952; Zbl 0047.01901)]. The method presented here has the advantage of generalizing to a class of infinite-dimensional shift operators.
For the entire collection see [Zbl 1250.00008].

MSC:

15A24 Matrix equations and identities
15A54 Matrices over function rings in one or more variables
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References:

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