Generalized Inverses and Approximation Numbers

K. P. Deepesh (), S. H. Kulkarni () and M. T. Nair ()
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K. P. Deepesh: Indian Institute of Technology Madras, Department of Mathematics
S. H. Kulkarni: Indian Institute of Technology Madras, Department of Mathematics
M. T. Nair: Indian Institute of Technology Madras, Department of Mathematics

A chapter in Combinatorial Matrix Theory and Generalized Inverses of Matrices, 2013, pp 143-158 from Springer

Abstract: Abstract 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 ℓ p and those of isometries and projections of norm 1 are found. In the case of finite-rank operators, we obtain estimates for the smallest nonzero approximation number in terms of their generalized inverses. Also, we prove some results regarding the relation between approximation numbers and the closedness of the range of an operator. We recall that the closedness of 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 need not hold.

Keywords: Generalized inverse; Generalized inverse of operator; Normed linear space; Approximation numbers; 15A09; 47B06 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-81-322-1053-5_12

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