ACTA issues

## Invertible Toeplitz operators and uniform algebras

 Abstract. Toeplitz operators $T_\phi ^M$ are defined on invariant subspaces $M$ of an arbitrary uniform algebra $A$. We give a necessary and sufficient condition for uniform invertibility of $M_\phi ^M$ with respect to some family $\cal F$ of invariant subspaces $M$. This condition is the same as a classical one in case $A$ is the disc algebra. In special uniform algebras we can choose a small family, in fact, if $A$ is a disc algebra then $\cal F$ can be a single set. Then this generalizes the Widom-Devinatz Theorem. As an application, we study a $\cal F$-union of spectra of $\{T_\phi ^M;M\in{\cal F}\}$. AMS Subject Classification (1991): 47B35, 47A10 Keyword(s): Toeplitz operator, invertible, uniform algebra, spectrum Received November 10, 1993 and in revised form March 21, 1994. (Registered under 5580/2009.)