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Uniqueness, strong forms of uniqueness and negative powers of contractions. (English) Zbl 0893.46043
Zemánek, Jaroslav (ed.), Functional analysis and operator theory. Proceedings of the 39th semester at the Stefan Banach International Mathematical Center in Warsaw, Poland, held March 2-May 30, 1992. Warsaw: Polish Academy of Sciences, Banach Cent. Publ. 30, 127-145 (1994).
The author gives a review about a set of uniqueness on the unit circle $$\Gamma$$. A subset $$E$$ of $$\Gamma$$ is said to be a set of uniqueness if the zero sequence is the only sequence $$(c_n)_{n\in\mathbb{Z}}$$ of complex numbers such that $$\sum_{| n|\leq m} c_ne^{int}\to 0$$ $$(n\to\infty)$$ for every $$e^{it}\not\in E$$. A set of multiplicity is a set which is not a set of uniqueness. The author assumes that $$E$$ is closed.
There are seven sections in this paper as the following: §1. Introduction, §2. Classical uniqueness theory, §3. Strong uniqueness properties of countable sets, §4. Strong uniqueness properties of the Cantor set, §5. Distributions on Dirichlet sets, §6. Negative powers of contractions, §7. Closed ideals of $$A^+$$.
The proofs are given for several theorems. Some new results are given.
For the entire collection see [Zbl 0792.00007].
Reviewer: T.Nakazi (Sapporo)

##### MSC:
 46J20 Ideals, maximal ideals, boundaries 42A20 Convergence and absolute convergence of Fourier and trigonometric series 47D03 Groups and semigroups of linear operators