Toeplitz operators associated to unimodular algebras
Let \(G\) a compact Hausdorff space, \(C(G)\) the space of all continuous functions on \(G\) with the supremum norm. In the present paper, the author introduces a special closed subalgebra \(A\) of \(C(G)\) that is called unimodular algebra. It is shown that this class of subalgebras is very extensive. A regular probability measure \(m\) on \(G\) is said to be a representing measure on the algebra \(A\) if \[ \int\phi\psi dm=\int\phi dm\int\psi dm\quad\text{for all }\phi,\psi\in A. \] Let \(m\) be a representing measure on the algebra \(A\) and denote by \(H^p(A,m)\) the closure of \(A\) in \(L^p(G,m)\), for \(1\leq p<\infty\). Denote by \(H^{\infty}(A,m)\) the norm-closed unital subalgebra of \(L^{\infty}(G,m)\) consisting of the functions \(\phi\) for which \(\phi H^2(A,m)\subset H^2(A,m)\). \(H^p(A,m)\) (\( 1\leq p<\infty\)) is said to be a generalized Hardy space. Under suitable hypotheses on \(m\), substantial portions of the classical Hardy space theory can be extended to the generalized Hardy spaces. Let \(P\) denote the projection from \(L^2(G,m)\) onto \(H^2(A,m)\). For \(\phi\in L^{\infty}(G,m)\), denote by \(T_{\phi}\) the bounded linear operator on \(H^2(A,m)\) defined \[ T_{\phi}f=P(\phi f),\quad\text{for all }f\in H^2(A,m). \] \(T_{\phi}\) is said to be a Toeplitz operator on \(H^2(A,m)\) with symbol \(\phi\), if \(\phi\in H^{\infty}(A,m)\), \(T_{\phi}\) is also said to be analytic. In the paper under review, the author studies Toeplitz operators on \(H^2(A,m)\). It is shown that many results of the classical Toeplitz operator theory can be extended to the context of Toeplitz operators on \(H^2(A,m)\).
scientific article; zbMATH DE number 1971843