A Quasi-nilpotent Operator with Reflexive Commutant, Ii

A new example of a non-zero quasi-nilpotent operator T with reflexive commutant is presented. Norms ‖T n‖ converge to zero arbitrarily fast. Let H be a complex separable Hilbert space and let B(H) denote the algebra of all continuous linear operator on H. If T ∈ B(H) then {T}′ = {A ∈ B(H) : AT = TA} is called the commutant of T . By a subspace we always mean a closed linear subspace. If A ⊂ B(H) then AlgA denotes the smallest weakly closed subalgebra of B(H) containing the identity I and A, and LatA denotes the set of all subspaces invariant for each A ∈ A. If L is a set of subspaces of H, then AlgL = {T ∈ B(H) : L ⊂ Lat{T}}. T is said to be hyperreflexive if {T}′ = Alg Lat{T}′, i.e., if the algebra {T}′ is reflexive. It can be shown (see [1]) that if T is a nilpotent hyperreflexive operator on a separable Hilbert space then T = 0. This is not true for quasinilpotent operators. An example of a non-zero quasinilpotent hyperreflexive operator was given in [5] using a modification of an idea of Wogen [4]. The powers of the example converged to zero slowly, more precisely the following inequality was true for all positive integers: