Tokyo Journal of Mathematics

Hilbert-Schmidt Hankel Operators and Berezin Iteration

Wolfram BAUER and Kenro FURUTANI

Full-text: Open access

Abstract

Let $H$ be a reproducing kernel Hilbert space contained in a wider space $L^2(X,\mu)$. We study the Hilbert-Schmidt property of Hankel operators $H_g$ on $H$ with bounded symbol $g$ by analyzing the behavior of the iterated Berezin transform. We determine symbol classes $\mathcal{S}$ such that for $g\in \mathcal{S}$ the Hilbert-Schmidt property of $H_g$ implies that $H_{\bar{g}}$ is a Hilbert-Schmidt operator as well and there is a norm estimate of the form $\|H_{\bar{g}}\|_{\text{HS}}\leq C\cdot \| H_g\|_{\text{HS}}$. Finally, applications to the case of Bergman spaces over strictly pseudo convex domains in $\mathbb{C}^n$, the Fock space, the pluri-harmonic Fock space and spaces of holomorphic functions on a quadric are given.

Article information

Source
Tokyo J. of Math., Volume 31, Number 2 (2008), 293-319.

Dates
First available in Project Euclid: 5 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1233844053

Digital Object Identifier
doi:10.3836/tjm/1233844053

Mathematical Reviews number (MathSciNet)
MR2477873

Zentralblatt MATH identifier
1181.47022

Subjects
Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]
Secondary: 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20] 32A25: Integral representations; canonical kernels (Szego, Bergman, etc.) 32Q15: Kähler manifolds 53D50: Geometric quantization

Citation

BAUER, Wolfram; FURUTANI, Kenro. Hilbert-Schmidt Hankel Operators and Berezin Iteration. Tokyo J. of Math. 31 (2008), no. 2, 293--319. doi:10.3836/tjm/1233844053. https://projecteuclid.org/euclid.tjm/1233844053


Export citation

References

  • P. Ahern, M. Flores, W. Rudin, An invariant volume-mean-value property, J. Funct. Anal., 111 (1993), 380–397.
  • J. Arazy, M. Englis, Iterates and the boundary behavior of the Berezin transform, Ann. Inst. Fourier, Grenoble, 51 (4) (2001), 1101–1133.
  • S. Axler, D. Zheng, Compact operators via the Berezin transform, Indiana University Math. Journ., 47 (2) (1998), 387–400.
  • W. Bauer, Hilbert-Schmidt Hankel operators on the Segal-Bargmann space, Proc. Amer. Math. Soc., 132 (2004), 2989–2998.
  • W. Bauer, K. Furutani, Quantization operators on quadrics, Kyushu J. Math., 62 (1) (2008), 221–258.
  • W. Bauer, K. Furutani, Quantization operator on quaternion projective space and Cayley projective plane, in preparation.,
  • D. Békollé C.A. Berger, L.A. Coburn, K.H. Zhu, BMO in the Bergman metric on bounded symmetric domains, J. Funct. Anal., 93 no. 2 (1990), 310–350.
  • F.A. Berezin, Quantization, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 1116–1175.
  • F.A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1134–1167.
  • C.A. Berger, L.A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc., 301 (1987), 813–829.
  • L.A. Coburn, Sharp Berezin Lipschitz estimates, Proc. Amer. Math. Soc., 135 (2007), 1163–1168.
  • L.A. Coburn, A Lipschitz estimate for Berezin's operator calculus, Proc. Amer. Math. Soc., 133 No. 1 (2005), 127–131.
  • L.A. Coburn, Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J., 23 (1973/74), 433–439.
  • M. Englis, Compact Toeplitz operators via the Berezin transform on bounded symmetric domains, Int. Equ. Op. Theory, 33 (1999), 426–455.
  • M. Englis, Functions invariant under the Berezin transform, J. Funct. Anal., 121 no. 1 (1994), 233–254.
  • K. Furutani, Quantization of the geodesic flow on quaternion projective spaces, Ann. Global Anal. Geom., 22 no. 1, (2002), 1–27.
  • K. Furutani, S. Yoshizawa, A Kähler structure on the punctured cotangentbundle of complex and quaternion projective spaces and its application to geometric quantization II, Japan J. Math., 21 (1995), 355–392.
  • S.G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, second edition, 1992.
  • T. Mazur, P. Pflug, M. Skwarczynski, Invariant distances related to the Bergman function, Proc. Amer. Math. Soc., 94 no. 1 (1985), 72–76.
  • J. H. Rawnsley, A nonunitary pairing of polarizations for the Kepler problem, Trans. Amer. Math. Soc., 250 (1979), 167–180.
  • K. Stroethoff, Hankel and Toeplitz operators on the Fock space, Michigan Math. J., 39 no. 1 (1992), 3–16.
  • K. Stroethoff, Compact Hankel operators on weighted harmonic Bergman spaces, Glasgow Math. J., 39 no. 1 (1997), 77–84.
  • J. Xia, D. Zheng, Standard deviation and Schatten class Hankel operators on the Segal-Bargmann space, Indiana Univ. Math. J., 53 no. 5 (2004), 1381–1399.
  • U. Venugopalkrishna, Fredholm operators associated with strongly pseudoconvex domains in $\mathbbC^n$, J. Funct. Anal., 9 (1972), 349–373.
  • K. Zhu, Hilbert-Schmidt Hankel operators on the Bergman space, Proc. Amer. Math. Soc., 109 no. 3 (1990), 721–730.