Analytic Toeplitz operators on the Hardy space \(H^p\): A survey
This is a neat overview of results on analytic Toeplitz operators -- that~is, operators \(M_\phi\) of multiplication by analytic functions~\(\phi\) -- on~the Hardy spaces~\(H^p\), \(1\leq p<\infty\), of~the unit circle. The~topics addressed include criteria for boundedness and identification of the operator norm \(\| M_\phi\| \) with \(\| \phi\| _\infty\), the characterization of \(\{M_\phi\), \(\phi\in H^\infty\}\) as the commutant of~\(M_z\), compactness and the equality of the essential norm \(\| M_\phi\| _e\) and~\(\| \phi\| _\infty\), a description of the spectrum and the essential spectrum, criteria for closedness of the range, and Fredholmness and index theory. While all these results are very classical for the Hilbert space case \(p=2\), the information on the general case \(1\leq p<\infty\) has hitherto been rather scattered, and it is therefore very useful to have all these facts gathered in one place.
scientific article; zbMATH DE number 2031057