Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 016, 9 pages      arXiv:1011.6584      https://doi.org/10.3842/SIGMA.2011.016

On the Complex Symmetric and Skew-Symmetric Operators with a Simple Spectrum

Sergey M. Zagorodnyuk
School of Mathematics and Mechanics, Karazin Kharkiv National University, 4 Svobody Square, Kharkiv 61077, Ukraine

Received December 14, 2010, in final form February 11, 2011; Published online February 16, 2011

Abstract
In this paper we obtain necessary and sufficient conditions for a linear bounded operator in a Hilbert space H to have a three-diagonal complex symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in H. It is shown that a set of all such operators is a proper subset of a set of all complex symmetric operators with a simple spectrum. Similar necessary and sufficient conditions are obtained for a linear bounded operator in H to have a three-diagonal complex skew-symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in H.

Key words: complex symmetric operator; complex skew-symmetric operator; cyclic operator; simple spectrum.

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References

  1. Garcia S.R., Putinar M., Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), 1285-1315.
  2. Garcia S.R., Putinar M., Complex symmetric operators and applications. II, Trans. Amer. Math. Soc. 359 (2007), 3913-3931.
  3. Zagorodnyuk S.M., On a J-polar decomposition of a bounded operator and matrices of J-symmetric and J-skew-symmetric operators, Banach J. Math. Anal. 4 (2010), no. 2, 11-36.
  4. Akhiezer N.I., Classical moment problem, Fizmatlit, Moscow, 1961 (in Russian).
  5. Geronimus Ya.L., Orthogonal polynomials on the circle and on an interval. Estimates, asymptotic formulas, orthogonal series, Fizmatlit, Moscow, 1958 (in Russian).
  6. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
  7. Guseinov G.Sh., Determination of an infinite non-self-adjoint Jacobi matrix from its generalized spectral function, Mat. Zametki 23 (1978), 237-248 (English transl.: Math. Notes 23 (1978), 130-136).
  8. Zagorodnyuk S.M., Direct and inverse spectral problems for (2N+1)-diagonal, complex, symmetric, non-Hermitian matrices, Serdica Math. J. 30 (2004), 471-482.
  9. Zagorodnyuk S.M., Integral representations for spectral functions of some nonself-adjoint Jacobi matrices, Methods Funct. Anal. Topology 15 (2009), 91-100.
  10. Zagorodnyuk S.M., The direct and inverse spectral problems for (2N+1)-diagonal complex transposition-antisymmetric matrices, Methods Funct. Anal. Topology 14 (2008), 124-131.

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