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Certain Invariant Subspaces of H2 and L2 on a Bidisc

Published online by Cambridge University Press:  20 November 2018

Takahiko Nakazi
Takahiko Nakazi*
Affiliation:
Hokkaido University, Sapporo, Japan
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We let T2 be the torus that is the cartesian product of 2 unit circles in C. The usual Lebesgue spaces, with respect to the Haar measure m of T2, are denoted by Lp = Lp(T2), and Hp = Hp(T2) is the space of all f in LP whose Fourier coefficients

are 0 as soon as at least one component of (j, ℓ) is negative.

A closed subspace M of L2 is said to be invariant if

Whenever this is the case, it follows that fM ⊂ M for every f in H∞. One can ask for a classification or an explicit description (in some sense) of all invariant subspaces of L2, but this seems out of reach.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 5 , 01 October 1988 , pp. 1272 - 1280
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Curto, R. E., Muhly, P. S., Nakazi, T. and Yamamoto, T., On superalgebras of the poly disc algebra, Acta Sci. Math. 51 (1987), 413–421.Google Scholar
2. Helson, H., Lectures on invariant subspaces (Academic Press, New York and London, 1964).Google Scholar
3. Helson, H. and Lowdenslager, D., Invariant subspaces, Proc. Int. Symp. Linear Spaces, Jerusalem, 1960 (MacMillan (Pergamon), 1961), 251–262.Google Scholar
4. Nakazi, T., Invariant subspaces of unitary operators, J. Math. Soc. Japan 34 (1982), 627–635.Google Scholar
5. Nakazi, T., Invariant subspaces of weak* Dirichlet algebras, Pacific J. Math. 69 (1977), 151–167.Google Scholar
6. Rudin, W., Function theory in polydiscs (Benjamin, New York, 1969).Google Scholar