##
**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.

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.

Reviewer: Ivan G. Todorov (Belfast)

### 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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\textit{M. T. Karaev} and \textit{H. S. Mustafayev}, Rocky Mt. J. Math. 33, No. 3, 915--926 (2003; Zbl 1079.46032)

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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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