부산대학교 도서관

The author considers pseudo-differential operators on the unit circle $\Bbb S^1\simeq \Bbb T^1$ given by the toroidal quantization $$ (T_\sigma f)(\theta)=\sum_{n\in\Bbb Z} e^{in\theta}\sigma(\theta,n)\widehat{f}(n), $$ with symbols $\sigma(\theta,n)$ satisfying $$ |\partial_\theta^\alpha\Delta_n^\beta \sigma(\theta,n)|\leq C_{\alpha\beta} (1+|n|)^{m-|\beta|},\tag1 $$ where $\Delta_n$ is a first-order difference operator on $\Bbb Z$. The author shows that the minimal and maximal closed extensions of an elliptic $T_\sigma$ with $m>0$ on the space $L^p(\Bbb T^1)$ coincide, so that the closed extension is unique and its domain is the Sobolev space $H^{m,p}$. The author then confirms the Fredholmness of the minimal operator and investigates the essential spectra of $T_\sigma$.