2019/07/01 by Cao, Guangfu, Li, Ji, Shen, Minxing +2 · 1 citation
#30H20 #42A38 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1907.00574
We show that for an entire function φ belonging to the Fock space \mathscr F2(ℂn) on the complex Euclidean space ℂn, the integral operator SφF(z)=∫ℂn F(w) e^z ⋅w φ(z- w) dλ(w), z∈ ℂn, is bounded on \mathscr F2(ℂn) if and only if there exists a function m∈ L∞(ℝn) such that φ(z)=∫ℝn m(x)e-2(x-(i)/(2) z )⋅ (x-(i)/(2) z ) dx, z∈ ℂn. Here dλ(w)= π-ne-\vert w\vert2dw is the Gaussian measure on \mathbb Cn. With this characterization we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator Sφ. Moreover, we obtain the reducing subspaces of Sφ. In particular, in the case n=1, we give a complete solution to an open problem proposed by K. Zhu for the Fock space \mathscr F2(ℂ) on the complex plane \mathbb C (Integr. Equ. Oper. Theory \bf 81 (2015), 451--454).