On the inverse closedness of the subalgebra of local Hilbert--Schmidt operators

E. Yu. Guseva

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 4, Tome 193 (2021), pp. 69-86

Résumé

A local Hilbert–Schmidt operator is an operator of the form \begin{equation*} (Tx)(t)=\int\limits_{-\infty}^{+\infty}k(t,s)x(s)ds \end{equation*} with a measurable kernel $k:\mathbb{R}^2\to\mathbb{C}$ under the condition that \begin{equation*} \int\limits_a^{b}\int\limits_a^{b}|k(t,s)|^2 ds dt\infty \end{equation*} for all $-\infty$. We prove that, under some additional conditions that provide the action of the operator $T$ in $L_2(\mathbb{R},\mathbb{C})$, the invertibility of the operator $\mathbf{1}+T$ implies that the inverse operator has the form $\mathbf{1}+T_1$, where $T_1$ is also a local Hilbert–Schmidt operator whose kernel $S$ satisfies the same conditions.

Détail

Keywords: Hilbert–Schmidt operator, full subalgebra, difference operator, convolution operator, operator majorized by a convolution.