Hyperreflexivity and operator ideals

Abstract

Suppose ( B , β ) is an operator ideal, and A is a linear space of operators between Banach spaces X and Y. Modifying the classical notion of hyperreflexivity, we say that A is called B -hyperreflexive if there exists a constant C such that, for any T ∈ B ( X , Y ) with α = sup β ( q T i ) < ∞ (the supremum runs over all isometric embeddings i into X, and all quotient maps of Y, satisfying q A i = 0 ), there exists a ∈ A , for which β ( T − a ) ⩽ C α . In this paper, we give examples of B -hyperreflexive spaces, as well as of spaces failing this property. In the last section, we apply S E -hyperreflexivity of operator algebras ( S E is a regular symmetrically normed operator ideal) to constructing operator spaces with prescribed families of completely bounded maps.