Let \(H(\Omega)\) be a reproducing kernel Hilbert space over a nonempty set \(\Omega\) and let \(\widetilde{A}\) denote the Berezin symbol of a bounded linear operator \(A\) on \(H(\Omega)\). Assume that \(| \widetilde{A}(z)| \geq\delta\) for all \(z\in\Omega\) and some \(\delta>0\). Two problems are considered. Problem 1. Find a number \(\delta_0>0\) such that \(\delta>\delta_0\) implies the invertibility of \(A\). Problem 2. Let \(E\) be a closed subspace of \(H(\Omega)\) and let \(P_E\) be the orthogonal projection from \(H(\Omega)\) onto \(E\). Find a number \(\delta_0>0\) such that \(\delta>\delta_0\) implies the invertibility of \(P_EA| _E\). The author solves these problems in some special cases. The cases of the Bergman space \(L_a^2\) over the unit disk \({\mathbb D}\) and the Hardy space \(H^2\) over the unit circle are considered in detail. In particular, suppose that \(A=T_\varphi\) is the Toeplitz operator on \(H^2\) with a symbol \(\varphi\). Then its Berezin symbol coincides with the harmonic extension \(\widetilde{\varphi}\) of \(\varphi\). The author's result in this case reads as follows. Suppose that \(\delta:=\inf\limits_{z\in{\mathbb D}}| \widetilde{\varphi}(z)| >0\) and there that exists a sequence \(\{\lambda_n\}\subset{\mathbb D}\) such that \[ \sum_{n=0}^\infty\Big(\widehat{| \widehat{\varphi}| ^2}(0)- 2\text{ Re} \overline{\widetilde{\varphi}(\lambda_n)}\widehat{\varphi}(0) +| \widetilde{\varphi}(\lambda_n)| ^2\Big)=:\delta_0\delta_0\), then \(T_\varphi\) is invertible on \(H^2\) and \(\| T_\varphi^{-1}\| \leq(\delta-\delta_0)^{-1}\).