Asymptotic behavior of eigenvalues of Toeplitz operators on the weighted analytic spaces (Q279800)

The authors study Toeplitz operators defined by positive Borel measures \(\mu\) that act on a large variety of the (not necessarily standard) weighted Bergman spaces on the unit disk \(\mathbb {D}\) and the Fock spaces. The aim of the paper is to give a geometric characterization of compact Toeplitz operators \(T_{\mu}\) whose eigenvalues have a prescribed slow decay to zero. An example of the result obtained gives the following assertion. Let \(\eta: [1,+\infty) \rightarrow (0, \infty)\) be a decreasing function that satisfies the conditions that \(\eta(\infty) = 0\) and that \(\eta (x^2)\) is equivalent to \(\eta (x)\). Let then \(\mu\) be a positive Borel measure on \(\mathbb {D}\) such that the Toeplitz operator \(T_{\mu}\) is compact on a standard weighted Bergman space on the unit disk. The the decreasing sequence \(\{\lambda(T_{\mu})\}\) of the eigenvalues of the operator \(T_{\mu}\) satisfies the condition \(\lambda(T_{\mu}) = O(\eta(n))\) if and only if there is a positive constant \(c\) such that NEWLINE\[NEWLINE \sum h_{\eta} \left(c\frac{\mu(R_n)}{|R_n|}\right)< \infty,NEWLINE\]NEWLINE where \(h_{\eta}\) is defined by \(h_{\eta}(\eta(x)) = 1/x\), \(|R_n|\) is the area of \(R_n\), and \((R_n)\) is a numeration of certain dyadic disks in \(\mathbb {D}\).