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Spectral properties of the Cauchy transform on \(L_2(\mathbb C,e^{-|z|^2}\lambda(z))\). (English) Zbl 1100.47029

Let \(d\mu\left(z\right)= e^{-\left| z\right| ^2}\,d\lambda\left(z\right)\) be the Gaussian density on \({\mathbb C}\), where \(\lambda\left(z\right)\) is the usual Lebesgue measure on \({\mathbb C}\) (like on \({\mathbb R}^2\)). The Cauchy transform of a function \(f\in L_2\left({\mathbb C}, d\mu\right)\) is defined by \[ Cf\left(z\right)={1\over\pi}\int_{{\mathbb C}} {f\left(w\right)\over{z-w}}\,d\mu\left(w\right). \] The set \[ \Lambda_0\left({\mathbb C}\right)=\left\{f\in L_2\left({\mathbb C}, d\mu\right),\, f\text{ is entire}\right\} \] is called Bargmann space.
The present article is devoted to investigation of some spectral properties of the Cauchy transform \(C\) on the orthogonal complement \(\Lambda_0^\bot\left({\mathbb C}\right)\) of the Bargmann space \(\Lambda_0\left({\mathbb C}\right)\) in \(L_2\left({\mathbb C}, d\mu\right)\). This is possible by using the following Hilbertian orthogonal decomposition of \(L_2\left({\mathbb C}, d\mu\right)\): \[ L_2\left({\mathbb C}, d\mu\right)= \bigoplus_{m\in{\mathbb Z}_+}\Lambda_m\left({\mathbb C}\right), \] where \[ \Lambda_m\left({\mathbb C}\right)= \left\{f\in L_2\left({\mathbb C}, d\mu\right),\, A^*Af=mf\right\}. {(1)} \] Here the operators \(A^*\) and \(A\) are given by \(A^*=-{\partial\over{\partial z}}+\overline{z}\) and \(A={\partial\over{\partial\overline{z}}}\), respectively.
The main result of the article is the following statement.
Theorem. Let \(m\in{\mathbb Z}_+\) and \(m\) be fixed. Let \(\lambda_{m,p}\), \(p=0,1,2,\dots\), be the sequence of nonzero eigenvalues of the positive operator \[ \left| P_m C\right| := \sqrt{\left(P_m C\right)^*\left(P_m C\right)}, \] where \(P_m\) is the orthogonal projection operator from \(L_2\left({\mathbb C}, d\mu\right)\) onto the space \(\Lambda_m\left({\mathbb C}\right)\) in \(L_2\left({\mathbb C}, d\mu\right)\) and \(C\) is the Cauchy transform in \(L_2\left({\mathbb C}, d\mu\right)\). Then we have:
(i) The eigenvalues \(\lambda_{m,p}\), \(p=0,1,2,\dots\), of \(\left| P_m C\right| \) are given explicitly via the following formula: \[ \left(\lambda_{m,p}\right)^2= \gamma_{m-1,p}\,{}_2F_1\left(-\min\left(m-1,p\right);-\min\left(m-1,p\right); \left| m-1-p\right| +1; {1\over 4}\right), \] where \[ \gamma_{m-1,p}=\left({2\over 3}\right)^{2\min\left(m-1,p\right)} 3^{-\left(\left| m-1-p\right| +1\right)} {\left(\max\left(m-1,p\right)!\right)^2\over{m!p!\left| m-1-p\right| !}}, \] and \({}_2F_1\left(\alpha;\beta;\gamma;x\right)\) stands for the usual Gauss hypergeometric function.
(ii) For \(m\in{\mathbb Z}_+\) and \(p=0,1,2,\dots\), let \(\phi_{m,p}\) be the functions of Hermite type defined on \({\mathbb C}\) by \[ \phi_{m,p}\left(z\right)=\left\{ \begin{matrix} {-1\over{p+1}}{}_1F_1\left(1;p+2;\left| z\right| ^2\right)z^{p+1}, & m=0,\\ -\left(-1\right)^{m+p} {\partial^{m-1+p}\over{\partial^{m-1}z\partial^p\overline{z}}} e^{-\left| z\right| ^2}, & m=1,2,\dots. \end{matrix} \right. \] Then for each \(p=0,1,2,\dots\), the above function \(\phi_{m,p}\left(z\right)\) is an eigenvector of the operator \(\left| P_m C\right| \) with \(\lambda_{m,p}\) as eigenvalue.
As \(p\to\infty\) for fixed \(m\in{\mathbb Z}_+\), the authors obtain the following asymptotics: \[ \lambda_{m,p}\sim {1\over{2\sqrt{m!}}}\left({4\over 3}\right)^{m/2} e^{-p\log\sqrt{3}}\sqrt{{\Gamma\left(p+1\right)\over{\Gamma\left(p-m+2\right)}}}. \]
The authors obtain explicit formulae for the Schwartz kernel functions of \(P_m C\) as well as \(\left(P_m C\right)^*\left(P_m C\right)\). This helps to prove the main result. Proceeding in the same way, the authors deal with the Green transform on \(L_2\left({\mathbb C}, d\mu\right)\) and state its main spectral properties that correspond to the analogue of what they have done for the Cauchy transform \(C\) on \(L_2\left({\mathbb C}, d\mu\right)\).
The article contains several interesting results that could be useful for specialists in complex analysis and operator theory.

MSC:

47B38 Linear operators on function spaces (general)
46E20 Hilbert spaces of continuous, differentiable or analytic functions
47A10 Spectrum, resolvent
47G10 Integral operators
47A75 Eigenvalue problems for linear operators
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
47B65 Positive linear operators and order-bounded operators

Citations:

Zbl 0976.47019
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References:

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