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A trace formula for Hankel operators


Authors: Aurelian Gheondea and Raimund J. Ober
Journal: Proc. Amer. Math. Soc. 127 (1999), 2007-2012
MSC (1991): Primary 47B35; Secondary 47A56, 93B28
DOI: https://doi.org/10.1090/S0002-9939-99-04669-9
Published electronically: February 26, 1999
MathSciNet review: 1476131
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that if $G$ is an operator valued analytic function in the open right half plane such that the Hankel operator $H_G$ with symbol $G$ is of trace-class, then $G$ has continuous extension to the imaginary axis,

\begin{displaymath}G(\infty):=\lim\limits _{r \rightarrow \infty \atop r \in {\Bbb R}} G(r)\end{displaymath}

exists in the trace-class norm, and $\operatorname{tr}(H_G)={1\over 2}\, \operatorname{tr}(G(0)-G(\infty))$.


References [Enhancements On Off] (What's this?)

  • [1] K.V. Fernando, H. Nicholson, On the structure of balanced and other principal representa-
    tions of SISO systems, IEEE Transactions on Automatic Control, 28(1983), 228-231.
    MR 84i:93028
  • [2] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, Vol. 18, Amer. Math. Soc., Providence RI 1969. MR 39:7447
  • [3] J.S. Howland, Trace class Hankel operators, Quart. J. Math. Oxford, 22(1971), 147-159. MR 44:5826
  • [4] S.S. Mahil, F.W. Fairman, B.S. Lee, Some integral properties for balanced realizations of scalar systems, IEEE Transactions on Automatic Control, 29(1984), 181-183. MR 85b:93016
  • [5] R.J. Ober, Balanced parametrization of classes of linear systems, SIAM Journal on Control and Optimization, 29(1991), 1251-1287. MR 92j:93028
  • [6] R.J. Ober, S. Montgomery-Smith, Bilinear transformation of infinite-dimensional state-space systems and balanced realizations of nonrational transfer functions, SIAM Journal on Control and Optimization, 28(1990), 438-465. MR 91d:93019
  • [7] R.J. Ober, On Stieltjes functions and Hankel operators, Systems and Control Letters, 27(1996), 275-277. MR 97a:93034
  • [8] J.R. Partington, An introduction to Hankel operators, Cambridge University Press, 1988. MR 90c:47047

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Additional Information

Aurelian Gheondea
Affiliation: Center for Engineering Mathematics EC35, University of Texas at Dallas, Richardson, Texas 75083-0688
Email: gheondea@imar.ro

Raimund J. Ober
Affiliation: Center for Engineering Mathematics EC35, University of Texas at Dallas, Richardson, Texas 75083-0688
Email: ober@utdallas.edu

DOI: https://doi.org/10.1090/S0002-9939-99-04669-9
Received by editor(s): May 29, 1997
Received by editor(s) in revised form: September 10, 1997
Published electronically: February 26, 1999
Additional Notes: This research was supported in part by NSF grant DMS-9501223.
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1999 American Mathematical Society

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