Abstract
The composition operator induced by a hyperbolic Möbius transform ϕ on the classical Hardy space H2 is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function u in H2 if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of u is continuously extendable at one of the fixed points of ϕ and its value at the point is nonzero, then the cyclic subspace generated by u is minimal if and only if u is constant.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 837-841 |
| Number of pages | 5 |
| Journal | Proceedings of the American Mathematical Society |
| Volume | 119 |
| Issue number | 3 |
| DOIs | |
| State | Published - 1993 |
| Externally published | Yes |
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ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
Cite this
On the minimal invariant subspaces of the hyperbolic composition operator. / Matache, Valentin.
In: Proceedings of the American Mathematical Society, Vol. 119, No. 3, 1993, p. 837-841.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On the minimal invariant subspaces of the hyperbolic composition operator
AU - Matache, Valentin
PY - 1993
Y1 - 1993
N2 - The composition operator induced by a hyperbolic Möbius transform ϕ on the classical Hardy space H2 is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function u in H2 if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of u is continuously extendable at one of the fixed points of ϕ and its value at the point is nonzero, then the cyclic subspace generated by u is minimal if and only if u is constant.
AB - The composition operator induced by a hyperbolic Möbius transform ϕ on the classical Hardy space H2 is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function u in H2 if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of u is continuously extendable at one of the fixed points of ϕ and its value at the point is nonzero, then the cyclic subspace generated by u is minimal if and only if u is constant.
UR - http://www.scopus.com/inward/record.url?scp=84966212692&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84966212692&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-1993-1152988-8
DO - 10.1090/S0002-9939-1993-1152988-8
M3 - Article
VL - 119
SP - 837
EP - 841
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
SN - 0002-9939
IS - 3
ER -