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Generalized Inverses and Approximation Numbers

In: Combinatorial Matrix Theory and Generalized Inverses of Matrices

Author

Listed:
  • 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)

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.

Suggested Citation

  • K. P. Deepesh & S. H. Kulkarni & M. T. Nair, 2013. "Generalized Inverses and Approximation Numbers," Springer Books, in: Ravindra B. Bapat & Steve J. Kirkland & K. Manjunatha Prasad & Simo Puntanen (ed.), Combinatorial Matrix Theory and Generalized Inverses of Matrices, edition 127, pages 143-158, Springer.
    • Handle: RePEc:spr:sprchp:978-81-322-1053-5_12
    DOI: 10.1007/978-81-322-1053-5_12
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