Linear operators on Fock spaces
The aim of this paper is to extend several results about properties of some linear operators on the Fock space \(F_\alpha^2\) to linear operators on Fock spaces \(F_\alpha^p\) for \( 0<p<\infty \). The main results of the paper concern the conditions for boundedness and compactness of such operators on \(F_\alpha^p\). The obtained conditions for a linear operator \(T\) on \(F_\alpha^p\) are obtained imposing some boundedness conditions on the operator \(T_a=W_aTW_a\), where \(W_a\) is the Weil operator on \(H(\mathbb C)\) -- the linear space of all entire functions on \(\mathbb C\). Since the vector space \(\mathcal K\) of all finite linear combinations of normalized reproducing kernels is dense in the Fock space \(F_\alpha^p\) for each \(0<p<\infty\), a linear mapping \(T:{\mathcal K}\rightarrow H(\mathbb C)\) is called a bounded operator on \(F_\alpha^p\) if there exists a positive constant \(C\) such that \(\|Tf\|_{p,\alpha}\leq C\|f\|_{p,\alpha}\). Beside specific results on boundedness and compactness obtained in the paper, one should mention here Theorem 3.4 which says that, if \(T:{\mathcal K}\rightarrow H(\mathbb C)\) is linear and has the property that \(\|T_a1\|\rightarrow 0\) as \(a\rightarrow\infty\), and if there exists a positive constant \(C\) such that \(\|T^*_a1\|_{1,\alpha}\leq C\) for all \(a\in\mathbb C\), then for any \(1\leq p<\infty\) the operator \(T\) is bounded on \(F_\alpha^p\) and has the property that \(\|Tf_n\|_{p,\alpha}\rightarrow 0\) as \(a\rightarrow\infty\), whenever \(\{f_n\}\) is a bounded sequence in \(F_\alpha^p\) that converges to zero on compact sets. In particular, \(T\) is compact on \(F_\alpha^p\) for \(1<p<\infty\).
boundedness and compactness of linear operators
scientific article; zbMATH DE number 6805237