On some properties of Deddens algebras. (English) Zbl 1079.46032

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 J. A. Deddens [Lect. Notes Math., 693, 77–86 (1978; Zbl 0405.47029)]. It was shown in [J. A. Deddens and 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.


46H10 Ideals and subalgebras
47L10 Algebras of operators on Banach spaces and other topological linear spaces
46H35 Topological algebras of operators
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