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Multipliers of invariant subspaces in the bidisc

Published online by Cambridge University Press:  20 January 2009

Takahiko Nakazi
Takahiko Nakazi
Affiliation:
Department of Mathematics Faculty of ScienceHokkaido UniversitySapporo 060, Japan
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Abstract

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For any nonzero invariant subspace M in H2(T2), set . Then Mx is also an invariant subspace of H2(T2) that contains M. If M is of finite codimension in H2(T2) then Mx = H2(T2) and if M = qH2(T2) for some inner function q then Mx = M. In this paper invariant subspaces with Mx = M are studied. If M = q1H2(T2) ∩ q2H2(T2) and q1, q2 are inner functions then Mx = M. However in general this invariant subspace may not be of the form: qH2(T2) for some inner function q. Put (M) = {ø ∈ L ∞: ø M ⊆ H2(T2)}; then (M) is described and (M) = (Mx) is shown. This is the set of all multipliers of M in the title. A necessary and sufficient condition for (M) = H∞(T2) is given. It is noted that the kernel of a Hankel operator is an invariant subspace M with Mx = M. The argument applies to the polydisc case.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 37 , Issue 2 , June 1994 , pp. 193 - 199
Copyright
Copyright © Edinburgh Mathematical Society 1994

References

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