Finite Blaschke product and the multiplication operators on Sobolev disk algebra. (English) Zbl 1176.47030
Summary: Let \(R(D)\) be the algebra generated in Sobolev space \(W^{22}(D)\) by the rational functions with poles outside the unit disk \(\overline D\). In this paper, the multiplication operator \(M_{g}\) on \(R(D)\) is studied and it is proved that \(M_{g} \sim M_{z^n}\) if and only if \(g\) is an \(n\)-Blaschke product. Furthermore, if \(g\) is an \(n\)-Blaschke product, then \(M_{g}\) has uncountably many Banach reducing subspaces if and only if \(n > 1\).
MSC:
| 47B38 | Linear operators on function spaces (general) |
| 30J10 | Blaschke products |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 47A15 | Invariant subspaces of linear operators |
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