Non-synthetic diagonal operators on the space of functions analytic on the unit disk. (English) Zbl 1325.30057
Summary: Examples are given of continuous operators of functions analytic on the unit disk having the monomials as eigenvectors which fail spectral synthesis (that is, which have closed invariant subspaces which are not the closed linear span of collections of eigenvectors). Examples include the diagonal operator having as eigenvalues an enumeration of \(\mathbb Z\times i \mathbb Z\equiv \{m+in: m,n\in\mathbb{Z}\}\) and diagonal operators having as eigenvalues enumerations of \(\{n^{1/p}e^{2\pi ij/3p}: 0\leq j <p\}\) where \(p\) is an integer at least 2.
MSC:
| 30H99 | Spaces and algebras of analytic functions of one complex variable |
| 47A16 | Cyclic vectors, hypercyclic and chaotic operators |
| 47B38 | Linear operators on function spaces (general) |
| 47B40 | Spectral operators, decomposable operators, well-bounded operators, etc. |
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