Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains. (English) Zbl 0676.47016

Zhu, Kehe

J. Oper. Theory 20, No. 2, 329-357 (1988).

Let \(\Omega\) be a bounded symmetric domain in \({\mathbb{C}}^ n\) with Bergman kernel K(z,w) normalized by the condition \(K(z,0)=K(0,w)=1\). Let also \(T_{\mu}\) be a Toeplitz operator with measure \(\mu\) as symbol, defined by \[ (T_{\mu}f)(z)=\int_{\Omega}K_{\lambda}(z,w)f(w)d\mu (w). \] Here \(K_{\lambda}(z,w)=[K(z,w)]^{1-\lambda}\), and \(\mu\) be a finite complex Borel measure on \(\Omega\). In the case \(\mu\) positive the author obtains the necessary and sufficient conditions for boundedness, compactness and membership in Schatten classes \({\mathcal S}_ p\) of the operator \(T_{\mu}\).

Reviewer: N.K.Karapetianc

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.)

Keywords:

bounded symmetric domain; Bergman kernel; Toeplitz operator; boundedness; compactness; Schatten classes