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The Eigenstructure of Complex Symmetric Operators

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Book cover Recent Advances in Matrix and Operator Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 179))

Abstract

We discuss several algebraic and analytic aspects of the eigenstructure (si.e., eigenvalues, eigenvectors, and generalized eigenvectors) of complex symmetric operators. In particular, we examine the relationship between the bilinear form [x,y] = <x, Cy> induced by a conjugation C on a complex Hilbert space H and the eigenstructure of a bounded linear operator T: H → H which is C-symmetric (T = CT*C).

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References

  1. Christensen, O., An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003.

    MATH  Google Scholar 

  2. Craven, B.D., Complex symmetric matrices, J. Austral. Math. Soc. 10 (1969), 341–354.

    Article  MATH  MathSciNet  Google Scholar 

  3. Danciger, J., Garcia, S.R., Putinar, M., Variational principles for symmetric bilinear forms, Math. Nachr., to appear.

    Google Scholar 

  4. Dunford, N., Schwartz, J.T., Linear operators, Part I: General Theory, Wiley, New York, 1988.

    MATH  Google Scholar 

  5. Garcia, S.R., Conjugation and Clark operators, Contemp. Math. 393 (2006), 67–112.

    Google Scholar 

  6. Garcia, S.R., Putinar, M., Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), 1285–1315.

    Article  MATH  MathSciNet  Google Scholar 

  7. Garcia, S.R., Putinar, M., Complex symmetric operators and applications II, Trans. Amer. Math. Soc. 359 (2007), 3913–3931.

    MATH  MathSciNet  Google Scholar 

  8. Halmos, P.R., A Hilbert Space Problem Book (Second Edition), Springer-Verlag, 1982.

    Google Scholar 

  9. Peller, V.V., Hankel Operators and Their Applications, Springer Monographs in Mathematics, Springer-Verlag, 2003.

    Google Scholar 

  10. Prodan, E., Garcia, S.R., Putinar, M., Norm estimates of complex symmetric operators applied to quantum systems, J. Phys. A: Math. Gen. 39 (2006), 389–400.

    Article  MATH  MathSciNet  Google Scholar 

  11. Riss, U.V., Extension of the Hilbert space by J-unitary transformations, Helv. Phys. Acta 71 (1998), 288–313.

    MATH  MathSciNet  Google Scholar 

  12. Scott, N.H., A theorem on isotropic null vectors and its application to thermoelasticity, Proc. Roy. Soc. London Ser. A 440 no. 1909 (1993), 431–442.

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Pomona College, 610 North College Avenue, Claremont, CA, 91711, USA

    Stephan Ramon Garcia

Authors
  1. Stephan Ramon Garcia
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Joseph A. Ball

  2. School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Israel

    Yuli Eidelman

  3. Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, California, 92093-0112, USA

    J. William Helton

  4. Department of Mathematics, University of Connecticut, 196 Auditorium Road, U-9, Storrs, CT, 06269, USA

    Vadim Olshevsky

  5. Department of Mathematics, University of Virginia, P. O. Box 400137, Charlottesville, VA, 22904-4137, USA

    James Rovnyak

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© 2007 Birkhäuser Verlag Basel/Switzerland

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Garcia, S.R. (2007). The Eigenstructure of Complex Symmetric Operators. In: Ball, J.A., Eidelman, Y., Helton, J.W., Olshevsky, V., Rovnyak, J. (eds) Recent Advances in Matrix and Operator Theory. Operator Theory: Advances and Applications, vol 179. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8539-2_10

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