Abstract
In this paper we consider a class of weighted integral operators onL 2 (0, ∞) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.
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References
[AS] Abramowitz, M. and Stegun, I. A., “Handbook of Mathematical Functions,” Dover, New York, 1970.
[B] Bonsall, F. F.,Hankel operators on the Bergman space for the disc, J. London Math. Soc. (2)33 (1986), 355–364.
[BW] Bonsall, F. F., and Walsh, D.,Symbols for trace class Hankel operators with good estimates for norms, Glasgow Math. J.28 (1986), 47–54.
[GCP] Glover, K., Curtain, R. F., and Partington, J. R.,Realization and approximation of linear infinite dimensional systems with error bounds, SIAM Journal of control and Optimization26 (1988), 863–898.
[GLP1] Glover, K., Lam, J., and Partington, J. R.,Rational approximation of a class of infinite dimensional systems 1: Singular values of Hankel operators, Mathematics of Control, Signals and Systems3 (1990), 325–344.
[GLP2] Glover, K., Lam, J., and Partington, J. R.,Rational approximation of a class of Infinite dimensional systems: The L 2 case, “Progress in approximation theory,” Academic Press, Inc., 1991, pp. 405–440.
[HS] Halmos, P.R., and Sunder, V. S., “Bounded Integral operators onL 2 spaces,” Springer-Verlag, Berlin-Heidelberg-New York, 1978.
[J] Janson, S.,Hankel operators between weighted Bergman spaces, Ark. Mat.26 (1988), 205–219.
[JPR] Janson, S., Peetre, J., and Rochberg, R.,Hankel forms and the Fock space, Revista Mat. Ibero-Amer.3 (1987), 61–138.
[MM] Margenau, H. and Murphy, G. M., “The Mathematics of physics and chemistry,” Van Nostrand, Princeton, NJ, 1964.
[O] Olver, F. W. J., “Asymptotics and special functions,” Academic Press, 1974.
[P] Partington, J. R., “An Introduction to Hankel operators,” London Math. Soc. Student Texts 13, Cambridge Univ. Press, Cambridge, 1988.
[R] Rochberg, R., “Operators and Function Theory,” (S. C. Power, editor) D.Reidel Publishing Company, 1985, pp. 225–277.
[T] Triebel, H., “Theory of Function Spaces II,” Birkhäuser, 1992.