R\'esultats sur quelques op\'erateurs dans l'espace de Hilbert \`a poids $L^2(\mathbb{C}, \mathrm{e}^{-|z|^2})$

TL;DR

This paper investigates certain operators in a weighted Hilbert space of complex functions, establishing the existence and boundedness of their inverses using Hörmander's $L^2$-method.

Contribution

It demonstrates the invertibility and boundedness of specific operators in a weighted $L^2$ space on the complex plane, extending previous results.

Findings

Existence of bounded inverses for studied operators

Application of Hörmander's $L^2$-method to operator analysis

Operators are invertible with bounded inverses in the weighted space

Abstract

By H\"ormander's $L^2$-m\'ethode, we study some operators in the Hilbert space of weight $L^2(\mathbb{C}, \mathrm{e}^{-|z|^2})$. We prove in each case of operator the existence of its inverse which is also a bounded operator.