On some properties of Deddens algebras (Q1430427)

The algebra \[ B_A = \{X\in B(H) : \sup_n \| A^nXA^{-n}\| < +\infty\}, \] where \(A\) is an invertible operator acting on a Hilbert space \(H\), was introduced and studied by \textit{J. A. Deddens} [Lect. Notes Math., 693, 77--86 (1978; Zbl 0405.47029)]. It was shown in [\textit{J. A. Deddens} and \textit{T. K. Wong}, Trans. Am. Math. Soc. 184, 261--273 (1974; Zbl 0273.47017)] that if \(A\) is of the form \(A = \lambda I + N\), where \(N\) is a nilpotent operator, then \(B_A\) coincides with the commutant \(\{A\}'\) of \(A\). In the paper under review, the authors extend this result to arbitrary Banach algebras and determine the Deddens algebras of the form \(B_{e+p}\), where \(e\) is the unit of the algebra and \(p\) is an idempotent. Furthermore, if \(N\) is not necessarily a nilpotent operator, they show that the commutant of \(A\) is equal to the intersection of \(B_A\) and the Shulman subspace \(\mathcal{U}(N,M) = \{N\}' + \{N\}M\), where \(M\) is a bounded operator with the property that the commutator \([N,M]\) of \(N\) and \(M\) commutes with \(N\). Applications to the study of the Volterra integration operator are provided.