A Volterra-type integration operator on Fock spaces (Q2846851)

For \(0 < p < \infty\) the Fock space \(\mathcal{F}^p\) is the space of entire functions \(f\) for which NEWLINE\[NEWLINE \| f\|^p_{p}=\left(\frac{p}{2\pi}\right)\int_{\mathbb{C}}|f(z)|^p e^{-\frac{p}{2}|z|^2}dA(z)<\infty,NEWLINE\]NEWLINE where \(dA(z)\) denotes the Lebesgue area measure on \(\mathbb{C}\).NEWLINENEWLINEThe author studies boundedness and compactness of the Volterra-type integral operators \(T_g\) acting on the Fock spaces, where \(T_g\) is defined in the following way: For \(f, g\) analytic, NEWLINE\[NEWLINET_gf(z) = \int_0^z f(\zeta)g'(\zeta) d\zeta.NEWLINE\]NEWLINE The author shows that if \(p \leq q\), then \(T_g :\mathcal{F}^p \rightarrow \mathcal{F}^q\) is bounded if and only if the symbol \(g\) is a polynomial of degree \(\leq 2\), while it is compact if and only if \(g\) is a polynomial of degree \(\leq 1\). For \(p > q\), the operator \(T_g\) is bounded if and only if it is compact. The uthor also shows that if the operator \(T_g\) acting on \(\mathcal{F}^2\) is compact, then it belongs to the Schatten class \(\mathcal{S}^p\) for all \(p > 2\), but it fails to be Hilbert-Schmidt unless \(g\) is constant. Finally the author characterizes the invariant subspaces of the Volterra operator on \(\mathcal{F}^p\) using an old result of N. K. Nikolski.