Journal of Operator Theory

Volume 32, Issue 2, Fall 1994 pp. 353-379.

Continuity of the constant of hyperreflexivity

Authors: Ileana Ionaşcu
Author institution:Pure Mathematics Department, University of Waterloo, Waterloo, Ontario, N2L 3G1, CANADA and The Fields Institute, 180 Columbia St. West, Waterloo, Ontario N2L 5Z5, CANADA. On leave from: The Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700 Bucharest, ROMANIA

Summary: Starting from the question of what are the possible values of the constant of hyperreflexivity for subspaces of $\mathcal B(H)$, where H is a separable complex Hilbert space, the paper considers the continuity of the function $\kappa {\rm{ : }} \mathcal B(H) \to \bar R$, defined by $\kappa(T) = K(\mathcal A_w (T)), \mathcal A_w (T)) denoting the unital weakly closed algebra generated by T. As a consequence, it is shown that any number bigger than or equal to one is a constant of hyperreflexivity of a subspace. Besides several results concerning the continuity of the function $\kappa$, the paper contains also more general results, like those determining the closures (in the norm topology) or the set of reflexive, respectively non-reflexive, operators.

Keywords: Hyperreflexivity, invariant subspace, Hilbert space.

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