## 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

### Keywords:

Deddens algebra; Banach algebra; Shulman subspace; commutant

### Citations:

Zbl 0405.47029; Zbl 0273.47017
Full Text:

### 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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