Rank-one perturbation of Toeplitz operators and reflexivity (Q2901967)

Let \(\mathcal{H}\) be a Hilbert space. By \(\mathcal{B}(\mathcal{H})\) we 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 NEWLINE\[NEWLINE\mathrm{ref}\mathcal{M}=\{T\in \mathcal{B}(\mathcal{H}):Tx\in [\mathcal{M}x]\text{ for all }x \in \mathcal{H}\},NEWLINE\]NEWLINE where \([.]\) denotes the norm-closure. A subspace \(\mathcal {M}\) is called reflexive if \(\mathcal{M}=\mathrm{ref} \mathcal{M}.\) A subspace \(\mathcal {M}\subset \mathcal{B}(\mathcal{H})\) is called \(k\)-reflexive if \(\mathcal{M}^{(k)}=\{M^{(k)}:M\in \mathcal{M}\} \) is reflexive in \(\mathcal{B}(\mathcal{H}^{(k)})\).NEWLINENEWLINEIn this paper, it is shown that a rank-one perturbation of the space of Toeplitz operators on the Hardy space preserves 2-hyperreflexivity.