Abstract. Let $ W$ be the Wiener algebra of absolutely convergent Fourier series on the unit circle $ {\msbm T}$, and let $ T_{\varphi }\colon W_{+}\longrightarrow W_{+}$, $ T_{\varphi }f=: P_{+}(\varphi f)$ ($ \varphi\in W$) be an invertible Toeplitz operator, that is, $ \delta(T_{\varphi })= \inf_{\msbm T}| \varphi | =\inf\{| \lambda | :\lambda\in \sigma(T_{\varphi })\} >0$ and $ \mathop{\rm Ind}(\varphi )=0$. The problems of estimates of inverses $ \|T_{\varphi }^{-1}\|$ and of finite section inverses $ \|(P_{n}T_{\varphi }| {\cal P}_{n})^{-1}\|$, in terms of the lower spectral bound $ \delta(T_{\varphi })$, are considered. It is shown that for $ \delta(T_{\varphi })\geq\delta \|T_{\varphi }\|$ and $ \delta > 1/{\sqrt{2}} $ there exists a constant $ c(\delta )$ such that $ \|T_{\varphi }^{-1}\|$, $ \|(P_{n}T_{\varphi }| {\cal P}_{n})^{-1}\|\leq\|T_{\varphi }\|^{-1}c(\delta )$ for every $ n\geq0$, and that for $ 0< \delta\leq 1/2$ such an estimate is impossible: $ \sup\{\|T_{\varphi }^{-1}\|: \delta(T_{\varphi })\geq\delta, \mathop{\rm Ind}(\varphi )=0, \|T_{\varphi }\|\leq1\} =\infty $ and $ \sup\{\overline{\lim }_{n}\|(P_{n}T_{\varphi }| {\cal P}_{n})^{-1}\|: \delta(T_{\varphi })\geq\delta, \mathop{\rm Ind}(\varphi )=0, \|T_{\varphi }\|\leq1\} =\infty $. For analytic symbols $ \varphi\in W_{+}$, the similar constant $ c_{+}(\delta )$ is finite iff $ 1/2< \delta\leq 1$. Similar results are obtained for Fredholm regularizers of operators $ T$ from the Toeplitz algebra $ \mathop{\rm Alg}(T_{W})$.

AMS Subject Classification (1991): 47B35, 46J15, 65J10

Received April 19, 1999, and in revised form March 14, 2000. (Registered under 2791/2009.)