Perturbation of Toeplitz operators and reflexivity (Q482409)

For a Hilbert space \(\mathcal{H}\), let \( \mathcal{B}(\mathcal{H})\) denote the algebra of all bounded linear operators on \(\mathcal{H}\). The reflexive closure of a subspace \(\mathcal{M}\subset \mathcal{B}(\mathcal{H})\) is given by the formula \(\mathrm{ref}\mathcal{M}= \{A \in \mathcal{B}(\mathcal{H}) : Ah\in [\mathcal{M}h] \text{ for all }h \in \mathcal{H}\}\), here \([\cdot]\) denotes the norm-closure. If \(\mathcal{M}= \mathrm{ref}\mathcal{M}\) then \(\mathcal{M}\) is said to be reflexive. Let us recall the definition of \(k\)-hyperreflexivity. If \(\mathcal{M}\subset \mathcal{B}(\mathcal{H})\), \(\mathcal{M}_{\perp}\) denotes the preannihilator of \(\mathcal{M}\). Let \(F_k\) denote all operators of rank less than \(k\). For a subspace \(\mathcal{M}\subset \mathcal{B}(\mathcal{H})\) and an operator \(A \in \mathcal{B}(\mathcal{H})\), denote by \(\alpha_k(A,\mathcal{M}) = \sup\{| \mathrm{tr}(At)| : t \in \mathcal{M}_{\perp} \cap F_k, \|t\|_1\leq 1\}\). A subspace \(\mathcal{M}\) is \(k\)-hyperreflexive if there is a constant \(c > 0\) such that \(d(A,\mathcal{M}) \leq c \alpha_k(A,\mathcal{M})\) for any \(A \in \mathcal{B}(\mathcal{H})\). The constant of \(k\)-hyperreflexivity is the infimum of all constants \(c\) such that above inequality holds and is denoted by \(\mathcal{K}_k(\mathcal{M})\). In this paper, the authors show that the space of Toeplitz operators perturbated by finite rank operators is 2-hyperreflexive.