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Spectral picture for rationally multicyclic subnormal operators. (English) Zbl 07002036

Summary: For a pure bounded rationally cyclic subnormal operator \(S\) on a separable complex Hilbert space \(\mathcal{H}\), Conway and Elias showed that \(\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))=\operatorname{clos}(\operatorname{Int}(\sigma(S)))\). This article examines the property for rationally multicyclic (\(N\)-cyclic) subnormal operators. We show that there exists a \(2\)-cyclic irreducible subnormal operator \(S\) with \(\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))\neq\operatorname{clos}(\operatorname{Int}(\sigma(S)))\). We also show the following. For a pure rationally \(N\)-cyclic subnormal operator \(S\) on \(\mathcal{H}\) with the minimal normal extension \(M\) on \(\mathcal{K}\supset\mathcal{H}\), let \(\mathcal{K}_{m}=\operatorname{clos}(\operatorname{span}\{(M^{*})^{k}x:x\in\mathcal{H},0\leq k\leq m\}\). Suppose that \(M|_{\mathcal{K}_{N-1}}\) is pure. Then \(\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))=\operatorname{clos}(\operatorname{Int}(\sigma(S)))\).

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47A16 Cyclic vectors, hypercyclic and chaotic operators
30H99 Spaces and algebras of analytic functions of one complex variable
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References:

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