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On absolutely norm attaining operators

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Abstract

We give the necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss the structure of such operators in the case of self-adjoint and normal operators separately. Finally, we discuss several properties of absolutely norm attaining operators.

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References

  1. Bouldin R, The essential minimum modulus, Indiana Univ. Math. J. 30(4) (1981) 513–517

    Article  MathSciNet  Google Scholar 

  2. Bouldin R, Distance to invertible linear operators without separability, Proc. Amer. Math. Soc. 116(2) (1992) 489–497

    Article  MathSciNet  Google Scholar 

  3. Carvajal X and Neves W, Operators that achieve the norm, Integr. Equ. Oper. Th. 72(2) (2012) 179–195

    Article  MathSciNet  Google Scholar 

  4. Conway John B, A course in functional analysis, volume 96 of Graduate Texts in Mathematics, second edition (1990) (New York: Springer-Verlag)

  5. Enflo P, Kover J and Smithies L, Denseness for norm attaining operator-valued functions, Linear Algebra Appl. 338 (2001) 139–144

    Article  MathSciNet  Google Scholar 

  6. Ganesh J, Ramesh G and Sukumar D, A characterization of absolutely minimum attaining operators, J. Math. Anal. Appl. 468 (1) (2018) 567–583

    Article  MathSciNet  Google Scholar 

  7. Groetsch C W, Elements of applicable functional analysis, Monographs and Textbooks in Pure and Applied Mathematics, 55 (1980) (New York: Dekker)

  8. Halmos P R, A Hilbert space problem book, volume 19 of Graduate Texts in Mathematics, second edition, Encyclopedia of Mathematics and its Applications, 17 (1982) (New York: Springer-Verlag)

  9. Konrad S, Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics, 265 (2012) (Dordrecht: Springer) pp. xx+432

  10. Pandey S K and Paulsen V I, A spectral characterization of \({\cal{AN}}\) operators, J. Aust. Math. Soc. 102(3) (2017) 369–391

    Article  MathSciNet  Google Scholar 

  11. Ramesh G, Structure theorem for \({\cal{AN}}\)-operators, J. Aust. Math. Soc. 96(3) (2014) 386–395

    Article  MathSciNet  Google Scholar 

  12. Rudin W, Functional analysis (1973) (New York: McGraw-Hill)

    MATH  Google Scholar 

  13. Taylor A E and Lay D C, Introduction to functional analysis, second edition (1980) (New York: John Wiley & Sons)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the referee for useful comments which improved the clarity of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Hyderabad, Kandi (V), Sangareddy, Telangana, 502 285, India

    D Venku Naidu & G Ramesh

Authors
  1. D Venku Naidu
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  2. G Ramesh
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Correspondence to G Ramesh.

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Communicating Editor: B V Rajarama Bhat

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Venku Naidu, D., Ramesh, G. On absolutely norm attaining operators. Proc Math Sci 129, 54 (2019). https://doi.org/10.1007/s12044-019-0491-y

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  • DOI: https://doi.org/10.1007/s12044-019-0491-y

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