Generalized inverses and approximation numbers

We derive estimates for approximation numbers of bounded linear operators between normed linear spaces. As special cases of our general results, approximation numbers of some weighted shift operators on p and those of isometries andprojections of norm 1 are found. In the case of finite-rank operators, we obtain estimatesfor the smallest nonzero approximation number in terms of their generalizedinverses. Also, we prove some results regarding the relation between approximationnumbers and the closedness of the range of an operator.We recall that the closednessof the range is a necessary condition for the boundedness of a generalized inverse.We give examples illustrating the results and also show that certain inequalities neednot hold.