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**Positive Toeplitz operators on the Bergman space of a minimal bounded homogeneous domain.**
*(English)*
Zbl 1248.47032

In this paper, the author obtains necessary and sufficient conditions for a positive Toeplitz operator to be bounded or compact on the Bergman space of a minimal bounded homogeneous domain. The characterization is in terms of the Berezin transform, the averaging function and a Carleson measure property. The idea of proof is to use the strategy of proof used by K. Zhu [J. Oper. Theory 20, No. 2, 329–357 (1988; Zbl 0676.47016)] in obtaining necessary and sufficient conditions for a Toeplitz operator to be bounded or compact on a bounded symmetric domain.

Let \(D\) be a bounded homogeneous domain in \(\mathbb{C}^n\), \(dV\) the Lebesgue measure and for, \(p\geq 1\), \(L^p_a(D)\) the Bergman space. The function \(K_D\) will denote the Bergman kernel of \(D\). A minimal bounded homogeneous domain \(\mathcal{U}\) with center point \(t\) is one for which \(K_D(z,t)=K_D(t,t)\) for any \(z\in\mathcal{U}\).

For a complex Borel measure \(\mu\) on \(\mathcal{U}\), the Toeplitz operator \(T_\mu\) is defined by \[ T_\mu f(z)=\int_{\mathcal{U}} K_\mathcal{U}(z,w)f(w)\, d\mu(w) \] for \(z\in\mathcal{U}\) and \(f\in L^2_a(\mathcal{U})\). A Toeplitz operator \(T_\mu\) is called positive if its symbol (measure) is positive.

To state the characterizations of positive Toeplitz operators requires a little more notation. Let \(k_z\) denote the \(L^2_a(\mathcal{U})\) normalized Bergman kernel. Then the Berezin transform of \(T_\mu\) is given by \[ \tilde{T}_\mu(z)=\langle T_\mu k_z, k_z\rangle. \] For \(z\in\mathcal{U}\) and \(r>0\), let \(B(z,r)=\{w\in\mathcal{U}:\beta(z,w)\leq r\}\), where \(\beta\) is the Bergman metric. For fixed \(\rho>0\), define the averaging function of the measure \(\mu\) \[ \hat{\mu}(z)=\frac{\mu\left(B(z,\rho)\right)}{\left |B(z,\rho)\right|}. \]

A measure \(\mu\) is called a \(p\)-Carleson measure if there exists a constant \(M\) such that \[ \int_{\mathcal{U}}\left| f(z)\right|^p d\mu(z)\leq m\int_{\mathcal{U}}\left | f(z)\right|^p dV(z) \] for all \(f\in L^p_a(\mathcal{U})\). Note that this is equivalent to the inclusion \(L^p_a(\mathcal{U})\subset L^p(\mathcal{U};\mu)\). When \(p=2\), if \(\mu\) is a Carleson measure for \(L^2_a(\mathcal{U})\), then \(\mu\) is a vanishing Carleson measure if the inclusion \(L^2_a(\mathcal{U})\subset L^2(\mathcal{U};\mu)\) is compact.

The first main theorem of the paper is the following. Suppose that \(\mathcal{U}\subset\mathbb{C}^n\) is a minimal bounded homogeneous domain and \(\mu\) a finite positive Borel measure. Then the following conditions are equivalent:

(a) \(T_\mu\) is a bounded operator on \(L^2_a(\mathcal{U})\);

(b) the Berezin transform \(\tilde{T}_\mu\) is a bounded function on \(\mathcal{U}\);

(c) for all \(p\geq 1\), \(\mu\) is a Carleson measure for \(L^p_a(\mathcal{U})\);

(d) the averaging function \(\hat{\mu}\) is bounded on \(\mathcal{U}\).

The author also obtains an analogous compactness statement. Suppose that \(\mathcal{U}\subset\mathbb{C}^n\) is a minimal bounded homogeneous domain and \(\mu\) a finite positive Borel measure. The following conditions are equivalent:

(a) \(T_\mu\) is a bounded operator on \(L^2_a(\mathcal{U})\);

(b) the Berezin transform \(\tilde{T}_\mu\) tends to \(0\) as \(z\to\partial\mathcal{U}\);

(c) \(\mu\) is a vanishing Carleson measure for \(L^2_a(\mathcal{U})\);

(d) the averaging function \(\hat{\mu}\) tends to \(0\) as \(z\to\partial\mathcal{U}\).

Let \(D\) be a bounded homogeneous domain in \(\mathbb{C}^n\), \(dV\) the Lebesgue measure and for, \(p\geq 1\), \(L^p_a(D)\) the Bergman space. The function \(K_D\) will denote the Bergman kernel of \(D\). A minimal bounded homogeneous domain \(\mathcal{U}\) with center point \(t\) is one for which \(K_D(z,t)=K_D(t,t)\) for any \(z\in\mathcal{U}\).

For a complex Borel measure \(\mu\) on \(\mathcal{U}\), the Toeplitz operator \(T_\mu\) is defined by \[ T_\mu f(z)=\int_{\mathcal{U}} K_\mathcal{U}(z,w)f(w)\, d\mu(w) \] for \(z\in\mathcal{U}\) and \(f\in L^2_a(\mathcal{U})\). A Toeplitz operator \(T_\mu\) is called positive if its symbol (measure) is positive.

To state the characterizations of positive Toeplitz operators requires a little more notation. Let \(k_z\) denote the \(L^2_a(\mathcal{U})\) normalized Bergman kernel. Then the Berezin transform of \(T_\mu\) is given by \[ \tilde{T}_\mu(z)=\langle T_\mu k_z, k_z\rangle. \] For \(z\in\mathcal{U}\) and \(r>0\), let \(B(z,r)=\{w\in\mathcal{U}:\beta(z,w)\leq r\}\), where \(\beta\) is the Bergman metric. For fixed \(\rho>0\), define the averaging function of the measure \(\mu\) \[ \hat{\mu}(z)=\frac{\mu\left(B(z,\rho)\right)}{\left |B(z,\rho)\right|}. \]

A measure \(\mu\) is called a \(p\)-Carleson measure if there exists a constant \(M\) such that \[ \int_{\mathcal{U}}\left| f(z)\right|^p d\mu(z)\leq m\int_{\mathcal{U}}\left | f(z)\right|^p dV(z) \] for all \(f\in L^p_a(\mathcal{U})\). Note that this is equivalent to the inclusion \(L^p_a(\mathcal{U})\subset L^p(\mathcal{U};\mu)\). When \(p=2\), if \(\mu\) is a Carleson measure for \(L^2_a(\mathcal{U})\), then \(\mu\) is a vanishing Carleson measure if the inclusion \(L^2_a(\mathcal{U})\subset L^2(\mathcal{U};\mu)\) is compact.

The first main theorem of the paper is the following. Suppose that \(\mathcal{U}\subset\mathbb{C}^n\) is a minimal bounded homogeneous domain and \(\mu\) a finite positive Borel measure. Then the following conditions are equivalent:

(a) \(T_\mu\) is a bounded operator on \(L^2_a(\mathcal{U})\);

(b) the Berezin transform \(\tilde{T}_\mu\) is a bounded function on \(\mathcal{U}\);

(c) for all \(p\geq 1\), \(\mu\) is a Carleson measure for \(L^p_a(\mathcal{U})\);

(d) the averaging function \(\hat{\mu}\) is bounded on \(\mathcal{U}\).

The author also obtains an analogous compactness statement. Suppose that \(\mathcal{U}\subset\mathbb{C}^n\) is a minimal bounded homogeneous domain and \(\mu\) a finite positive Borel measure. The following conditions are equivalent:

(a) \(T_\mu\) is a bounded operator on \(L^2_a(\mathcal{U})\);

(b) the Berezin transform \(\tilde{T}_\mu\) tends to \(0\) as \(z\to\partial\mathcal{U}\);

(c) \(\mu\) is a vanishing Carleson measure for \(L^2_a(\mathcal{U})\);

(d) the averaging function \(\hat{\mu}\) tends to \(0\) as \(z\to\partial\mathcal{U}\).

Reviewer: Brett Wick (Atlanta)

### MSC:

47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |

32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |