Spectral theory of Toeplitz and Hankel operators on the Bergman space A1

摘要

The Fredholm properties of Toeplitz operators on the Bergman spaceA2 have been well-known for continuous symbolssince the 1970s. We investigate the case p=1 with continuoussymbols under a mild additional condition, namely that of thelogarithmic vanishing mean oscillation in the Bergman metric. Mostdifferences are related to boundedness properties of Toeplitzoperators acting on Ap that arise when we no longer have 1p∞; inparticular bounded Toeplitz operators on A1 were characterizedcompletely very recently but only for bounded symbols. We alsoconsider compactness of Hankel operators on A1.