Deddens algebras and shift (Q537669)

Let \(\mathcal{H}\) be a Hilbert space, \(S\) be the unilateral shift on \(\mathcal H\), and \(\lambda\) be a complex number. The author defines \(\mathcal{D}_{S-\lambda I}\) to be the Deddens algebra associated to \(S-\lambda I\), where \(I\) is the identity operator on \(\mathcal{H}\). That is to say, \(\mathcal{D}_{S-\lambda I}\) is the algebra of all \(T\in\mathcal{L}\left( \mathcal{H}\right) \) such that there is a constant \(M>0\) for which \[ \left\| \left( S-\lambda I\right) ^{n}Tx\right\| \leq M\left\| \left( S-\lambda I\right) ^{nn}x\right\| \text{ for all }x\in \mathcal{H},\, n\in\mathbb{N}. \] It is proved in this interesting note that \(\mathcal{D}_{S-\lambda I}\) strictly contains the commutant of \(S-\lambda I\). Besides, it is shown that \(D_{S-\lambda I}\) is weakly dense in \(\mathcal{L}\left( \mathcal{H}\right) \). At the end of the paper, the author compares two Deddens algebras \(\mathcal{D}_{S-\lambda_{1}I}\) and \(\mathcal{D}_{S-\lambda_{2}I}\), is able to get some results in this regard, and leaves an open problem to the interested reader.