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Generalized Inverses and Approximation Numbers

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Book cover Combinatorial Matrix Theory and Generalized Inverses of Matrices

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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.

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Acknowledgements

K.P. Deepesh acknowledges the financial support received from the Council of Scientific and Industrial Research(CSIR), Government of India (File No: 09/084(0391)/2006-EMR-I).

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India

    K. P. Deepesh, S. H. Kulkarni & M. T. Nair

Authors
  1. K. P. Deepesh
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  2. S. H. Kulkarni
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  3. M. T. Nair
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Corresponding author

Correspondence to S. H. Kulkarni .

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Editors and Affiliations

  1. Indian Statistical Institute, S.J.S. Sansanwal Marg 7, New Delhi, 110016, India

    Ravindra B. Bapat

  2. Hamilton Institute, National University of Ireland, Kilkenny, Maynooth, Co. Kildar, Kilkenny, Ireland

    Steve J. Kirkland

  3. Department of Statistics, Manipal University, manipal.edu, Manipal, 576104, Karnataka, India

    K. Manjunatha Prasad

  4. , School of Information Sciences, University of Tampere, Kalevantie 4,, Tampere, 33014, Finland

    Simo Puntanen

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Deepesh, K.P., Kulkarni, S.H., Nair, M.T. (2013). Generalized Inverses and Approximation Numbers. In: Bapat, R., Kirkland, S., Prasad, K., Puntanen, S. (eds) Combinatorial Matrix Theory and Generalized Inverses of Matrices. Springer, India. https://doi.org/10.1007/978-81-322-1053-5_12

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