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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.

MSC:

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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References:

[1] J.A. Deddens, Another description of nest algebras , Lecture Notes in Math., vol. 693, Springer, New York, 1978, pp. 77-86. · Zbl 0405.47029
[2] J.A. Deddens and T.K. Wong, The commutant of analytic Toeplitz operators , Trans. Amer. Math. Soc. 184 (1973), 261-273. · Zbl 0273.47017
[3] P.R. Halmos, A Hilbert space problem book , 2nd. ed., 1982. · Zbl 0496.47001
[4] D. Sarason, A remark on the Volterra operator , J. Math. Anal. Appl. 12 (1965), 244-246. · Zbl 0138.38801
[5] V.S. Shulman, On transitivity of some space of operators , Functional Anal. Appl. 16 (1982), 91-92.
[6] ——–, Invariant subspace and spectral mapping theorem , Banach Center Publ. vol. 30, PWN, Warsaw, 1994, pp. 313-325. · Zbl 0822.47007
[7] I. Todorov, Bimodules over nest algebras and Deddens theorem , Proc. Amer. Math. Soc. 127 (1999), 1771-1780. JSTOR: · Zbl 0915.47033
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