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Abstract
We show that for an entire function φ belonging to the Fock space F^{2}(C^{n}) on the complex Euclidean space C^{n}, the integral operator S_{φ}F(z)=∫C^{n}F(w)e^{z⋅w¯}φ(z−w¯)dλ(w),z∈C^{n}, is bounded on F^{2}(C^{n}) if and only if there exists a function m∈L^{∞}(R^{n}) such that φ(z)=∫R^{n}m(x)e^{−2(x−[Formula presented]z)2 }dx,z∈C^{n}. Here dλ(w)=π^{−n}e^{−w2 }dw is the Gaussian measure on C^{n}. With this characterization we are able to obtain some fundamental results of the operator S_{φ}, including the normality, the C^{⁎} algebraic properties, the spectrum and its compactness. Moreover, we obtain the reducing subspaces of S_{φ}. In particular, in the case n=1, we give a complete solution to an open problem proposed by K. Zhu for the Fock space F^{2}(C) on the complex plane C (Zhu (2015) [30]).
Original language  English 

Article number  107001 
Pages (fromto)  133 
Number of pages  33 
Journal  Advances in Mathematics 
Volume  363 
DOIs  
Publication status  Published  25 Mar 2020 
Keywords
 Fock space
 Singular integral operator
 Bargmann transform
 Riesz transform
 Spectrum
 Reducing subspace
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 2 Active

Harmonic analysis: function spaces and partial differential equations
Duong, X., Hofmann, S., Ouhabaz, E. M. & Wick, B.
11/02/19 → 10/02/22
Project: Other

Harmonic analysis and dispersive partial differential equations
Li, J., Guo, Z., Kenig, C. & Nakanishi, K.
31/01/17 → …
Project: Research