Trace ideal criteria for Toeplitz operators. (English) Zbl 0618.47018

For a complex measure \(\mu\) on the open unit disk U define an operator \(T_{\mu}\) on a Hilbert space H of analytic functions with reproducing kernel k(z,w) by \(T_{\mu}f(w)=\int f(z)\overline{k(z,w)}d\mu (z)\). For a certain scale of Hilbert spaces \(H_{\alpha}\), \(\alpha <1\), which includes the Hardy space \(H^ 2\) and weighted Bergman spaces \(A^{2,\beta}\), conditions are obtained which imply \(T_{\mu}\) belongs to a Schatten ideal \({\mathcal S}_ p\). If \(\mu\) is a positive measure then these conditions are necessary and sufficient. Application to composition operators and restriction operators are indicated.


47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46E20 Hilbert spaces of continuous, differentiable or analytic functions
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