Algebras of Toeplitz operators in spaces of smooth functions (Q1814558)

Let \({\mathcal A}=H(D)\) be the space of functions analytic on the open unit disc \(D\) with the topology of uniform convergence on compact sets. Let \({\mathcal A}^*=H(\overline{D})\) be the space of functions analytic on a neighborhood of the closed disc \(\overline{D}\). The duality \[ \langle f,g\rangle=\sum \hat f(n)\overline{\hat g(n)} \] holds between these spaces. Let \({\mathcal H}={\mathcal A}+\overline{\mathcal A}\) and \({\mathcal H}^*={\mathcal A}^*+\overline{\mathcal A}^*\). These spaces consist of functions harmonic on \(D\). The Banach space \(X\) is called an \({\mathcal A}\)-space provided \(X\) is contained in \({\mathcal A}\), contains \({\mathcal A}^*\) and the inclusions are continuous. The closure of \({\mathcal A}^*\) in \(X\) will be denoted by \(X_ c\). Analogously we can define an \({\mathcal H}\)-space. If the product is defined by convolution then \({\mathcal H}^*\cdot{\mathcal H}\subset{\mathcal H}\). For an arbitrary \({\mathcal H}\)-space \(X\) a function \(f\in{\mathcal H}\) is said to belong to the space \(M(X)\) of all multipliers, if the operator of multiplication by \(f:{\mathcal H}^*\to{\mathcal H}\) has a continuous extension, \(X_ c\to X\). For an \({\mathcal H}\)-space \(X\) the function \(f\) is said to belong to the space \(ST(X)\), the symbols of all Toeplitz operators, provided the Toeplitz operator \(T_ f(h)=P(fh)\), \(h\in{\mathcal A}^*\), has a continuous extension \(X_ c\to X\), where \(P\) is the orthogonal projection of \(L^ 2\) onto \(H^ 2\) restricted to \({\mathcal H}\). The space \(ST(X)\cap{\mathcal A}\) is denoted by \(M(X)\). The norm of \(L(X)\) makes \(M(X)\) and \(ST(X)\) into Banach spaces. The \({\mathcal A}\)-space \(X\) is called a \({\mathcal A}t\)-space provided \(z,\bar z\in ST(X)\) and \(\sup\{\| z^ n\|_{ST(X)}+\|\bar z^ n\|_{ST(X)}\}<\infty\). The Toeplitz operators on spaces \(\ell_ A^ p,H^ p,\dots\) are studied in [\textit{A. Böttcher} and \textit{B. Silbermann}, Invertibility and asymptotics of Toeplitz matrices, Berlin (1983; Zbl 0578.47015)]. The present author shows that if \(f,g,fg\in ST(X)\) then \(T_ f T_ g- T_{fg}\in K_ 0(X)\) where \(K_ 0(X)\) is a certain closed left ideal of \(L(X)\). For the special spaces of Besov, \(B_{pq}^ s\), \(s<0\), \(1<p,q<\infty\) and Sobolev, \(W_ p^ s\), \(s<0\), \(1<p<\infty\), the author determines \(ST(X)\) and conditions are given for \(T_ f\) to be Fredholm.