Recent Results on Reflexivity of Operators and Algebras of Operators

Abstract

Let H be a complex Hilbert space and let L(H) denote the algebra of all bounded linear operators on H. For any subset S of L (H) let, as usual, Lat (S) be the set of (closed) subspaces M of H invariant under any element of S (i.e. TM⊂M for any T e S) and A1gLat(S) the subalgebra of L(H) consisting of those operators T such that Lat(T)⊃ ⊃Lat(S). Of course Alg Lat(S) is closed in the weak operator topology of L(H) (WOT for short). A (necessarily WOT-closed) subalgebra A of L(H) is reflexive if A = AlgLat(A). An operator T is reflexive if WT (the unital WOT-closed algebra generated by T) is reflexive. Thus roughly speaking an operator is reflexive if its lattice of invariant subspaces is rich enough so as to determine the WOT-closed subalgebra it generates. This is indirectly confirmed by the fact that a unicellular operator (i.e. an operator T such that Lat(T) is linearly ordered) cannot be reflexive.