Trace estimates of Toeplitz operators on Bergman spaces and applications to composition operators

Let $\Omega$ be a subdomain of $\mathbb{C}$ and let $\mu$ positive Borel measure on $\Omega$. In this paper, we study the asymptotic behavior eigenvalues compact Toeplitz operators $T\_\mu$ acting Bergman spaces $(\lambda n(T\mu))$ decreasing sequence $T\_\mu$, $\rho$ an increasing function such that $\rho (n)/n^A$ is for some $A>0$. We give explicit necessary sufficient geometric condition in order to have $\lambda n(T\mu)\asymp 1/\rho (n)$. As applications, consider composition $C\_\varphi$, standard analytic unit disc $\mathbb{D}$. First, general criterion ensuring singular values $C\_\varphi$ satisfy $s\_n(C\_\varphi ) \asymp 1/\rho(n)$. Next, focus our attention with univalent symbols, where express terms harmonic $\varphi (\mathbb{D})$. finally case $\partial \varphi (\mathbb{D})$ meets circle one point several concrete examples. Our method based upper lower estimates trace $h(T\_\mu)$, $h$ suitable concave or convex function.