A dense set of operators with tiny commutants

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 $\mathcal {A}’(T)$ and $\mathcal {A}''(T)$ denote the commutant and, respectively, the double commutant of $T$. We say that $T$ has a tiny commutant if $\mathcal {A}’(T)= {\mathcal {A}^a}(T)$. 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 ${\{ T \in \mathcal {L}(\mathcal {H}):\mathcal {A}(T)= {\mathcal {A}^a}(T)\} ^ - } = {\{ T \in \mathcal {L}(\mathcal {H}):\operatorname {Lat}\;\mathcal {A}(T)= \operatorname {Lat}\;{\mathcal {A}^a}(T)\} ^ - }= \{ A \in \mathcal {L}(\mathcal {H})$ 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 $\hat \gamma \cap {\rho _{s - F}}(A)\} \subset {\{ T \in \mathcal {L}(\mathcal {H}):\mathcal {A}(T)= \mathcal {A}’(T)\} ^ - } \subset \{ T \in \mathcal {L}(\mathcal {H}):\operatorname {Lat}\;\mathcal {A}(T)= \operatorname {Lat}\;\mathcal {A}’(T)\} \subset \{ A \in \mathcal {L}(\mathcal {H})$: 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 $\operatorname {Lat}\;\mathcal {A}''(T)$.