On Neumann operators (Q1919582)

Let \(X\) denote a complex Banach space and \(L(X)\) the set of bounded linear operators defined on \(X\) with values in \(X\). An operator \(T\in L(X)\) is called a Neumann operator if for each \(x\in X\), the Neumann series \(\sum^\infty_{n=0} T^nx\) converges whenever \(\lim_{n\to\infty} T^nx=0\). It was shown by \textit{J. Schulz} [Beitr. Anal. 12, 177-183 (1978; Zbl 0425.47005)] that if some power of \(T\) is strictly singular or strictly cosingular, then \(T\) is a Neumann operator. More recently, \textit{D. Medkova} [Czech. Math. J. 41(116), No. 2, 312-316 (1991; Zbl 0754.47005)] proved that \(T\) is a Neumann operator if its distance from the set of all compact operators is less than 1. Other examples of such operators are projections and quasi-Riesz operators. In the paper under review the authors show that if there exists a nonnegative integer \(k\) such that the range \(R[(I-T)^k]\) is closed and the restriction of \(I-T\) to \(R[(I-T)^k]\) is injective, then \(T\) is a Neumann operator. This result is used to derive various sufficient conditions previously obtained by other authors, including the two referred to above.