## Doctoral Dissertations

Spring 1994

Dissertation

Mathematics

#### Degree Name

Doctor of Philosophy

Let H be a separable, complex, Hilbert space and let ${\cal B}(H$) be the algebra of all (bounded linear) operators on H. We define a function$$\kappa:{\cal B}(H) \to \lbrack 1,\infty\rbrack;\qquad \kappa(T) = K({\cal A}\sb{w}(T)),\qquad \forall T \in {\cal B}(H),$$where ${\cal A}\sb{w}(T$) is the unital weakly closed algebra generated, in ${\cal B}(H$), by T, and $K({\cal A}\sb{w}(T$)) is the constant of hyperreflexivity of ${\cal A}\sb{w}(T$). If H is finite-dimensional, we show that $\kappa$ is continuous at $T \in {\cal B}(H$) if and only if T is non-reflexive or has dimH distinct eigenvalues (Theorem 2.6). An auxiliary result (Theorem 2.1) states that the closure of the non-reflexive operators on a finite-dimensional space is the complement of the set of operators with distinct eigenvalues. A consequence of our results is that, in case dimH = 2, the function $\kappa$ is surjective.
If H is infinite-dimensional, we show that the set of points of continuity for $\kappa$ is a dense $G\sb\delta$ set in ${\cal B}(H$) (Theorem 3.15), is included in the set of all non-hyperreflexive operators (Theorem 3.1 and Corollary 3.3) and is closed under similarity (Theorem 3.7). Also, its complement, the set of points of discontinuity for $\kappa$, is also closed under similarity and contains, at least, the hyperreflexive operators (consequence of Theorem 3.1) and the non-hyperreflexive operators whose $C\sp\*$-algebra contains no non-zero compact operators (Theorem 3.5).
The paper contains results about the stability of another problem related to constants of hyperreflexivity, specifically the continuity properties of functions of the type\eqalign{f\sb{n,p}:M\sbsp{n}{p} \to \lbrack 1,\infty\rbrack\cr f\sb{n,p}(T\sb1,T\sb2,\...,T\sb{p}) = K(Sp(T\sb1, T\sb2,\...,T\sb p)),\qquad \forall T\sb1,T\sb2,\...,T\sb{p} \in M\sb n,\cr}where $Sp(T\sb1,T\sb2,\...,T\sb{p}$) is the subspace generated by $\{T\sb1,T\sb2,\...,T\sb{p}\}$ in $M\sb n$. Denote by ${\cal C}\sb{n,p}$ the set of points of continuity for $f\sb{n,p}$. For arbitrary n, the results obtained describe ${\cal C}\sb{n,p}$, for all $p\geq$ 1 (Theorem 4.8), ${\cal C}\sb{n,n\sp2-1}$ (Theorem 4.10) and ${\cal C}\sb{n,1}$ (Theorem 4.11, which gives an equivalent formulation to the conjecture that all one-dimensional subspaces have constant of hyperreflexivity equal to one). For n = 2, we give complete characterization of the continuity of all $f\sb{2,p}$, for all $p\geq$ 1.