Let us suppose that $A,A_1,A_2,\ldots$ are non-negative definite self-adjoint bounded linear operators in $L(\mathbb H)$, where $\mathbb H$ is a separable Hilbert space. $(v_j)_{j\ge1}$ and $(\lambda_j)_{j\ge1}$ are the eigenvectors and the eigenvalues of $A$ such that $\lambda_1>\lambda_2>\ldots$ and $\sum\lambda_j<\infty$. For $n\ge1$, $(v_{nj})_{j\ge1}$ and $(\lambda_{nj})_{j\ge1}$ are the eigenvectors and the eigenvalues of $A_n$.

Suppose that $\|A-A_n\|_{op}=o(a_n)$ as $n\to\infty$, where $\|\cdot\|_{op}$ is the operator norm and $(a_n)_{n\ge1}$ is a sequence such that $a_n\to0$ as $n\to\infty$. Can we say anything about the convergence rate of $\|v_k-v_{nk}\|$ as $n\to\infty$ with $k\ge1$? Is it true that $\|v_k-v_{nk}\|\le C\|A-A_n\|_{op}$ for $k\ge1$, where $C$ is some positive constant? Do we need stronger assumptions to say anything about the convergence rate of the eigenvectors?

It seems that if $\|A-A_n\|_{op}\to0$ as $n\to\infty$, then $\lambda_{nk}\to\lambda_k$ and $\|v_{nk}-v_k\|\to0$ as $n\to\infty$ for $k\ge1$ (see this paper by Joachim Weidmann). I am interested in the convergence rate of the eigenvectors given the convergence rate of the operators. Maybe some additional assumptions are needed to establish the convergence rate of eigenvectors (for example, a stronger convergence of the operators).

References are very welcome. Any help is much appreciated!