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Invariant subspaces in the bidisc and commutators

Part of: Holomorphic functions of several complex variables Special classes of linear operators Commutative Banach algebras and commutative topological algebras General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Takahiko Nakazi
Takahiko Nakazi
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060, Japan
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Abstract

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Let M be an invariant subspace of L2 (T2) on the bidisc. V1 and V2 denote the multiplication operators on M by coordinate functions z and ω, respectively. In this paper we study the relation between M and the commutator of V1 and , For example, M is studied when the commutator is self-adjoint or of finite rank.

MSC classification

Secondary: 47A15: Invariant subspaces 47B47: Commutators, derivations, elementary operators, etc. 47B20: Subnormal operators, hyponormal operators, etc. 46J15: Banach algebras of differentiable or analytic functions, $H^p$-spaces 32A35: $H^p$-spaces, Nevanlinna spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 56 , Issue 2 , April 1994 , pp. 232 - 242
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Agrawal, O. P., Clark, D. N. and Douglas, R. G., ‘Invariant subspaces in the polydisk’, Pacific J. Math. 121 (1986), 1–11.CrossRefGoogle Scholar
[2]Ahern, P. R. and Clark, D. N., ‘Invariant subspaces and analytic continuation in several variables’, J. Math. Mech. 19 (1970), 963–969.Google Scholar
[3]Curto, R. E., Muhly, P. S., Nakazi, T. and Yamamoto, T., ‘On superalgebras of the polydisc algebra’, Acta Sci. Math. (Szeged) 51 (1987), 413–421.Google Scholar
[4]Helson, H., ‘Analyticity on compact abelian groups’, in: Algebras in Analysis-Proceedings of the instructional conference and NATO advanced study institute, Birmingham, 1973 (Academic Press, London, 1975) pp. 1–62.Google Scholar
[5]Izuchi, K., ‘Unitarily equivalence of invariant subspaces in the polydisk’, Pacific J. Math. 130 (1987), 351–358.CrossRefGoogle Scholar
[6]Mandrekar, V., ‘The validity of Beurling theorems in polydiscs’, Proc. Amer. Math. Soc. 103 (1988), 145–148.CrossRefGoogle Scholar
[7]Nakazi, T., ‘Invariant subspaces of weak* Dirichlet algebras’, Pacific J. Math. 69 (1977), 151–167.CrossRefGoogle Scholar
[8]Nakazi, T., ‘Certain invariant subspaces of H2 and L2 on a bidisc’, Canad. J. Math. XL (1988), 1272–1280.CrossRefGoogle Scholar
[9]Rudin, W., ‘Invariant subspaces of H2 on a torus’, J. Funct. Anal. 61 (1985), 378–384.CrossRefGoogle Scholar