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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A dense set of operators with tiny commutants


Author: Domingo A. Herrero
Journal: Trans. Amer. Math. Soc. 327 (1991), 159-183
MSC: Primary 47A99; Secondary 47A15, 47C05, 47D99
DOI: https://doi.org/10.1090/S0002-9947-1991-1022867-8
MathSciNet review: 1022867
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Abstract: For a (bounded linear) operator $ T$ on a complex, separable, infinite-dimensional Hilbert space $ \mathcal{H}$, let $ \mathcal{A}\,(T)$ and $ {\mathcal{A}^a}(T)$ denote the weak closure of the polynomials in $ T$ and, respectively, the weak closure of the rational functions with poles outside the spectrum of $ T$. Let $ \mathcal{A}^{\prime}(T)$ and $ \mathcal{A}''(T)$ denote the commutant and, respectively, the double commutant of $ T$. We say that $ T$ has a tiny commutant if $ \mathcal{A}^{\prime}(T)= {\mathcal{A}^a}(T)$. By constructing a large family of "models" and by using standard techniques of approximation, it is shown that $ T \in \mathcal{L}\,(\mathcal{H}):T$ has a tiny commutant is norm-dense in the algebra $ \mathcal{L}\,(\mathcal{H})$ of all operators acting on $ \mathcal{H}$. Other related results: Let $ \operatorname{Lat}\;\mathcal{B}$ denote the invariant subspace lattice of a subalgebra $ \mathcal{B}$ of $ \mathcal{L}(\mathcal{H})$. For a Jordan curve $ \gamma \subset {\mathbf{C}}$, let $ \hat \gamma $ denote the union of $ \gamma $ and its interior; for $ T \in \mathcal{L}\;(\mathcal{H})$, let $ {\rho _{s - F}}\,(T)= \{ \lambda \in {\mathbf{C}}:\lambda - T$ is a semi-Fredholm operator, and let $ \rho _{s - F}^ + (T)(\rho _{s - F}^ - (T))= \{ \lambda \in {\rho _{s - F}}(T):{\text{ind}}(\lambda - T) > 0\;(< 0,{\text{resp.)\} }}$. With this notation in mind, it is shown that $ {\{ T \in \mathcal{L}(\mathcal{H}):\mathcal{A}(T)= {\mathcal{A}^a}(T)\} ^ - } ... ...peratorname{Lat}\;{\mathcal{A}^a}(T)\} ^ - }= \{ A \in \mathcal{L}(\mathcal{H})$ if $ \gamma $ (Jordan curve) $ \subset\rho _{s - F}^ \pm (A)$, then $ \hat \gamma \subset \sigma (A)\} $; moreover, $ \{ A \in \mathcal{L}(\mathcal{H})$: if $ \gamma $ (Jordan curve) $ \subset\rho _{s - F}^ \pm (A)$, then $ {\text{ind}}(\lambda - A)$ is constant on $ \hat \gamma \cap {\rho _{s - F}}(A)\} \subset {\{ T \in \mathcal{L}(\mathcal{H... ...orname{Lat}\;\mathcal{A}^{\prime}(T)\} \subset\{ A \in \mathcal{L}(\mathcal{H})$: if $ \gamma $ (Jordan curve) $ \subset\rho _{s - F}^ \pm (A)$, then $ \hat \gamma \cap {\rho _{s - F}}(A) \subset\rho _{s - F}^ \pm (A)\} \subset \{ T \in \mathcal{L}(\mathcal{H}):\mathcal{A}(T)= {\mathcal{A}^a}(T)\} $. (The first and the last inclusions are proper.) The results also include a partial analysis of $ \operatorname{Lat}\;\mathcal{A}''(T)$.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1022867-8
Keywords: Hubert space operator, commutant, double commutant, weak closure of the polynomials, weak closure of the rational functions with poles outside the spectrum, invariant subspace lattices, tiny commutants, semi-Fredholm domain, approximation of operators
Article copyright: © Copyright 1991 American Mathematical Society