ACTA issues

## Estimates of inverses and finite section inverses of Toeplitz operators on the Wiener algebra

Ludmila N. Nikolskaia

Acta Sci. Math. (Szeged) 67:1-2(2001), 395-409
2791/2009

 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.)