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On the Gelfand-Hille theorems. (English) Zbl 0822.47005

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, 369-385 (1994).
Let \(T\) be a bounded linear operator on a complex Banach space \(X\), with the smallest possible spectrum, for instance \(\sigma(T)= \{1\}\). This paper deals with the additional conditions under which then \(T= I\), or \(T- I\) is a nilpotent operator. If \(X\) is a finite-dimensional space and \(\sigma(T)= 1\), then \(T- I\) is always a nilpotent operator, while in the infinite-dimensional case, it is not always true.
The first result concerning the infinite-dimensional case was published by I. M. Gelfand [Mat. Sb., N. Ser. 9(51), 41-48 (1941)], who proved that if \(\sigma(T)= 1\) and \(\sup\{\| T^ n\|\): \(n\in \mathbb{Z}\}<\infty\), then \(T= I\). Related theorems are called by the author, Gelfand-Hille theorems.
In this reporting paper, the author describes some of the most important results from this topics. The bibliography includes a large number of papers (178 units) concerning the Gelfand-Hille theorems, which are published till 1994. It also includes a lot of instructive remarks, questions and suggestions about the mentioned theorems, which can serve as items for the future work.
For the entire collection see [Zbl 0792.00007].

MSC:

47A10 Spectrum, resolvent
47A35 Ergodic theory of linear operators
47D03 Groups and semigroups of linear operators
30B30 Boundary behavior of power series in one complex variable; over-convergence
30D15 Special classes of entire functions of one complex variable and growth estimates
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