## 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.

### MSC:

 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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### References:

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