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Operator Least Squares Problems and Moore–Penrose Inverses in Krein Spaces

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Abstract

Given a Krein space \(\mathcal {H}\) and B, C in \(L(\mathcal {H}),\) the bounded linear operators on \(\mathcal {H},\) the minimization/maximization of expressions of the form \((BX- C)^{\#}(BX - C)\) as X runs over \(L(\mathcal {H})\) is studied. Complete solutions are found for the problems posed, including solvability criteria and a characterization of the solutions when they exist. Min–max problems associated to Krein space decompositions of B are also considered, leading to a characterization of the Moore–Penrose inverse as the unique solution of a variational problem. Other generalized inverses are similarly described.

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References

  1. Ando, T.: Linear Operators on Krein Spaces. Hokkaido University, Sapporo (1979)

    MATH  Google Scholar 

  2. Ando, T.: Projections in Krein spaces. Linear Algebra Appl. 431, 2346–2358 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arias, M.L., Corach, G., Gonzalez, C.: Generalized inverses and Douglas equations. Proc. Am. Math. Soc. 136, 3177–3183 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Azizov, T.Y., Iokhvidov, I.S.: Linear Operators in Spaces with and Indefinite Metric. Wiley, New York (1989)

    MATH  Google Scholar 

  5. Bognár, J.: Indefinite Inner Product Spaces. Springer, Berlin (1974)

    Book  MATH  Google Scholar 

  6. Contino, M., Giribet, J.I., Maestripieri, A.: Weighted Procrustes problems. J. Math. Anal. Appl. 445, 443–458 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Contino, M., Giribet, J.I., Maestripieri, A.: Weighted least square solutions of the equation \(AXB-C=0\). Linear Algebra Appl. 518, 177–197 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Corach, G., Fongi, G., Maestripieri, A.: Weighted projections into closed subspaces. Stud. Math. 216, 131–148 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Douglas, R.G.: On majorization, factorization and range inclusion of operators in Hilbert space. Proc. Am. Math. Soc. 17, 413–416 (1966)

    Article  MATH  Google Scholar 

  10. Dritschel, M.A., Rovnyak, J.: Extension theorems for contraction operators on Krein spaces. Oper. Theory Adv. Appl. 47, 221–305 (1990)

    MathSciNet  MATH  Google Scholar 

  11. Dritschel, M.A., Rovnyak, J.: Operators on indefinite inner product spaces. Lect. Oper. Theory Its Appl. 3, 141–232 (1996)

    MathSciNet  MATH  Google Scholar 

  12. Giribet, J.I., Maestripieri, A., Martínez Pería, F.: Indefinite least-squares problems and pseudo-regularity. J. Math. Anal. Appl. 430, 895–908 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Giribet, J.I., Maestripieri, A., Martínez Pería, F.: A geometrical approach to indefinite least squares problems. Acta Appl. Math 111, 65–81 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gheondea, A.: On the geometry of pseudo-regular subspaces of a Krein space. In: Operator Theory: Advances and Applications, vol. 14. Birkhäuser, Basel (1984)

  15. Hassi, S., Nordström, K.: On projections in a space with an indefinite metric. Linear Algebra Appl. 208–209, 401–417 (1974)

    MathSciNet  MATH  Google Scholar 

  16. Hassibi, B., Sayed, A.H., Kailath, T.: Linear estimation in Krein spaces—part I: theory. IEEE Trans. Autom. Control 41, 18–33 (1996)

    Article  MATH  Google Scholar 

  17. Hassibi, B., Sayed, A.H., Kailath, T.: Linear estimation in Krein spaces—part II: application. IEEE Trans. Autom. Control 41, 33–49 (1996)

    MATH  Google Scholar 

  18. Hassibi, B., Sayed, A.H., Kailath, T.: Indefinite-Quadratic Estimation and Control. A Unified Approach to \({\cal{H}}^2\) and \({\cal{H}}^\infty \) Theories. Studies in Applied and Numerical Mathematics (1999)

  19. Izumino, S.: The product of operators with closed range and an extension of the reverse order law. Tohoku Math. J. 34, 43–52 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  20. Maestripieri, A., Martínez Pería, F.: Decomposition of selfadjoint projections in Krein spaces. Acta Sci. Math. (Szeged) 72, 611–638 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Maestripieri, A., Martínez Pería, F.: Normal projections in Krein spaces. Integral Equ. Oper. Theory 76, 357–380 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mary, X.: Moore–Penrose inverse in Krein spaces. Integral Equ. Oper. Theory 60, 419–433 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mitra, S.K., Rao, C.R.: Projections under seminorms and generalized Moore–Penrose inverses and operator ranges. Linear Algebra Appl. 9, 155–167 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Instituto Argentino de Matemática “Alberto P. Calderón”, Saavedra 15, Piso 3 (1083), Buenos Aires, Argentina

    Maximiliano Contino, Alejandra Maestripieri & Stefania Marcantognini

  2. Departamento de Matemática, Facultad de Ingeniería, Universidad de Buenos Aires, Paseo Colón 850 (1063), Buenos Aires, Argentina

    Maximiliano Contino & Alejandra Maestripieri

  3. Departamento de Matemática, Instituto Venezolano de Investigaciones Científicas, Km 11 Carretera Panamericana, Caracas, Venezuela

    Stefania Marcantognini

Authors
  1. Maximiliano Contino
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  2. Alejandra Maestripieri
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  3. Stefania Marcantognini
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Correspondence to Maximiliano Contino.

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Contino, M., Maestripieri, A. & Marcantognini, S. Operator Least Squares Problems and Moore–Penrose Inverses in Krein Spaces. Integr. Equ. Oper. Theory 90, 32 (2018). https://doi.org/10.1007/s00020-018-2456-4

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  • DOI: https://doi.org/10.1007/s00020-018-2456-4

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