Tokyo Journal of Mathematics

Hilbert-Schmidt Hankel Operators and Berezin Iteration

Wolfram BAUER and Kenro FURUTANI

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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

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

First available in Project Euclid: 5 February 2009

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Zentralblatt MATH identifier

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


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.

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