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Invariant Subspaces on ${{\mathbb{T}}^{N}}$ and ${{\mathbb{R}}^{N}}$

Published online by Cambridge University Press:  20 November 2018

Michio Seto
Michio Seto*
Affiliation:
Mathematical Institute Tohoku University Sendai 980-8578 Japan, email: [email protected]
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Abstract

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Let $N$ be an integer which is larger than one. In this paper we study invariant subspaces of ${{L}^{2}}({{\mathbb{T}}^{N}})$ under the double commuting condition. A main result is an $N$-dimensional version of the theorem proved by Mandrekar and Nakazi. As an application of this result, we have an $N$-dimensional version of Lax's theorem.

Keywords

47A1547B47invariant subspaces
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 47 , Issue 1 , 01 March 2004 , pp. 100 - 107
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Helson, H., Lectures on Invariant Subspaces. Academic Press Inc., New York. London, 1964.Google Scholar
[2] Hoffman, K., Banach spaces of analytic functions. Prentice-Hall, Englewood Cliffs, N.J., 1962.Google Scholar
[3] Izuchi, K., Invariant subspaces on a torus. Lecture notes (1998), unpublished.Google Scholar
[4] Lax, P. D., Translation invariant subspace. Acta Math. 101 (1959), 163–178.Google Scholar
[5] Mandrekar, V., The validity of Beurling theorems in polydiscs. Proc. Amer. Math. Soc. 103 (1988), 145–148.Google Scholar
[6] Merrill, S. III, and Lal, N., Characterization of certain invariant subspace of Hp and Lp spaces derived from logmodular algebras. Pacific J. Math. 30 (1969), 463–474.Google Scholar
[7] Nakazi, T., Invariant subspaces of weak-*Dirichlet algebras. Pacific J. Math. 69 (1977), 151–167.Google Scholar
[8] Nakazi, T., Certain invariant subspaces of H2 and L2 on a bidisc. Canad. J. Math. 40 (1988), 1272–1280.Google Scholar
[9] Nakazi, T., Invariant subspaces in the bidisc and commutators. J. Austral. Math. Soc. Ser. A 56 (1994), 232–242.Google Scholar
[10] Ohno, Y., Some Invariant Subspaces in LH 2 . Interdiscip. Inform. Sci. 2 (1996), 131–137.Google Scholar
[11] Rudin, W., Function theory in polydiscs. Benjamin Inc., New York—Amsterdam, 1969.Google Scholar
[12] Yoshino, T., Introduction to operator theory. Pitman Res. Notes Math. Ser. 300, Longman Scientific and Technical, Harlow, and John Wiley & Sons, New York, 1993.Google Scholar