Carleson's Corona theorem states that if \(F: \mathbb{D}\to\mathbb{C}^n\) is bounded and analytic on the unit disc \(\mathbb{D}\) then the Carleson condition \[ F^\ast(z) F(z)\geq \delta^2 \tag{C} \] is sufficient for the existence of a bounded analytic solution \(G\) of the Bézout equation \[ G(z)\cdot F(z)= 1\quad\text{on } \mathbb{D}. \tag{B} \] Cases in which \(F\) is infinite dimensional-valued were proved by \textit{V. A. Tolokonnikov} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 113, 178--198 (1981; Zbl 0472.46024)] and Uchiyama. The operator corona problem considers the case in which \(F:E\to E_*\) is a bounded analytic function between Hilbert spaces \(E\) and \(E^\ast\) (\(F\in H^\infty (E,E_\ast)\)). Then (C) is sufficient for (B) when \(E\) is finite dimensional but not when \({\text{ dim}}\,E=\infty\) as the author previously proved [Sov. Math., Dokl. 38, No. 2, 394--399 (1989); translation from Dokl. Akad. Nauk SSSR 302, No. 5, 1063--1068 (1988; Zbl 0687.47004)]. In the present paper the result (C) implies (B) is shown to be stable in a certain sense: if there is a constant operator \(A\) on \(E\) such that either of the trace norms \(\| A - F^\ast(z) F(z)\| _{\text{Tr}}\) or \(\| A - F(z) F^\ast(z)\| _{\text{Tr}}\) are uniformly bounded for all \(z\in\mathbb{D}\) then the operator corona theorem holds in the sense that condition (C) implies the left invertibility of \(F\) in \(H^\infty\). The author also proves a perturbation result in terms of Hilbert-Schmidt norms. The methods do not reduce to simple matrix perturbation arguments. Rather, they rely on the so-called Tolokonnikov lemma which states that \(F\in H^\infty(E,E_\ast)\) is left invertible precisely when it can be extended to an invertible mapping.