On the continuity of spectra of Toeplitz operators

Abstract.

This paper concerns the continuity of spectra of Toeplitz operators $T_{\varphi}$, on the Hardy space $H^2({\Bbb T})$. The main results are as follows. The restriction of the spectrum $\sigma$, a set valued function, to $\frak {T}_{GD}$ is continuous at each Toeplitz operator with Douglas symbol, where $\frak {T}_{GD}$ denotes the set of Toeplitz operators with generalized Douglas symbols. Also the restriction of $\sigma$ to the set of all Toeplitz operators having symbols in $AP_{\Bbb T}+C({\Bbb T}),H^ {\infty} +C({\Bbb T})$, PQC, and $LS({\Bbb T})$, respectively, is continuous.