Discrete convolution operators on the quarter plane and their indices

R. V. Duduchava

Izvestiya. Mathematics , Tome 11 (1977) no. 5, pp. 1072-1084

Résumé

Let $\Gamma^2=\Gamma\times\Gamma$, where $\Gamma$ is the unit circle, and let $L_2^m(\Gamma^2)$ be the Hilbert space of vector-valued functions $\varphi=(\varphi_1,\dots,\varphi_m)$ whose components $\varphi_k(\zeta_1,\zeta_2)$ are complex-valued square integrable functions on $\Gamma^2$. The author considers the subspace $H_2^m(\Gamma^2)$ of functions in $L_2^m(\Gamma^2)$ having analytic continuations into the torus $\{(z_1,z_2):|z_k|1\}$; let $P$ be the projection of $L_2^m(\Gamma^2)$ onto $H_2^m(\Gamma^2)$. For a bounded measurable matrix-valued function $a(\zeta_1,\zeta_2)$ of order $m$ on $\Gamma^2$ having limits $a(\zeta\pm0,t)$ and $a(t,\zeta\pm0)$ ($\zeta\in\Gamma)$ uniform in $t\in\Gamma$, the bounded operator $T_a^2=PaP$ is defined in $H_2^m(\Gamma^2)$. In this paper a homotopy method is described for computing the index of Noetherian operators in the $C^*$-algebra generated by the operators $T_a^2$. In the case where $a(\zeta_1,\zeta_2)$ is continuous a simple formula for computing the index of $T_a^2$ is indicated. Bibliography: 24 titles.