Zemánek, Jaroslav 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]. Reviewer: A.Torgašev (Beograd) Cited in 2 ReviewsCited in 14 Documents 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 Keywords:Gelfand-Hille theorems PDFBibTeX XMLCite \textit{J. Zemánek}, Banach Cent. Publ. 30, 369--385 (1994; Zbl 0822.47005)