Compressions, Graphs, and Hyperreflexivity

Abstract

In this note we extend the ideas in [11] to prove reflexivity and especially hyperreflexivity results that pertain to compressions rather than summands. These results in turn are relevant for some remarkable classes of operators.1997 Academic Press

General reflexivity and pattern subspaces

“…Hence by the preceding lemma either S or S has a separating vector. It follows that S has property ( A l ( r ) ) for some r > 1 and, it follows from Theorem 4.10 in [9] that S is 2-hyperreflexive.…”

“…It was proved in [3] that, in the Hilbert space setting, property (B12) implies 2-reflcxivity. Statement (2) in the following lemma was proved in [9]; the remainder of the proofs are omitted.…”

“…For example suppose e is a nonzero vector in H and q5@ = (Sue) @ e (where 6@ is the Kronecker S function). However, these ideas were successfully used in [9] to define an analogue of strict cyclicity in reflexivity triples.…”

n-Reflexivity for linear spaces of operators

“…It is known that every algebraic operator with property (A2) is reflexive [5,Corollary 6]. By [11,Theorem 3.14], every reflexive algebraic operator is hyperreflexive. Hence we have the corollary.…”

Hyper-Reflexivity and the Factorization of Linear Functionals

“…The reflexivity of L n was first proved in [14] for n=1, and in [1] for n 2. In a related recent result, it is shown in [9] that the direct sum of two operators in A + 0 is hyper-reflexive.…”