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

## TL;DR

This paper characterizes a class of measures called horizontal Fock-Carleson type measures for derivatives in the Fock space, analyzes the boundedness of associated Toeplitz operators, and explores their algebraic properties and invariance under translations.

## Contribution

It introduces real coderivatives of these measures and demonstrates the commutative structure of the generated C*-algebra, extending results to translation-invariant measures.

## Key findings

- Boundedness conditions for Toeplitz operators are established.
- The C*-algebra generated by these operators is commutative and isomorphic to a subalgebra of L_infinity.
- Results extend to measures invariant under translations along Lagrangian planes.

## 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.