$\mathcal{L}$-invariant Fock-Carleson type measures for derivatives of order $k$ and the corresponding Toeplitz operators

Abstract: Our purpose is to characterize the so-called horizontal Fock-Carleson type measures for derivatives of order $k$ (we write it $k$-hFC for short) for the Fock space as well as the Toeplitz operators generated by sesquilinear forms given by them. The boundedness conditions for such operators are found. We introduce real coderivatives of $k$-hFC type measures and show that the C*-algebra generated by Toeplitz operators with the corresponding class of symbols is commutative and isometrically isomorphic to certain $C^*$-subalgebra of $L_{\infty}(\mathbb{R}^{n})$. The above results are extended to measures that are invariant under translations along Lagrangian planes.