It is well-known that for two (positive) compact self-adjoint operators $S$ and $T$ on a Hilbert space $H$, we have \begin{equation*} |\lambda_i -\mu_i|\leq C\|S-T\|, \end{equation*} where $\lambda_i$ and $\mu_i$ are the $i$-th eigenvalues of $S$ and $T$ in a decreasing order, respectively.
In my research, I encounter the following (matrix-valued) eigenvalues:
Definition: Let $H$ be a Hilbert space, $n$ be a natural number, $H^n$ be the product space of $H$ with the natural inner product structure, i.e. \begin{equation*} \langle u,v\rangle : = \langle u_1,v_1\rangle_H + \cdots + \langle u_n,v_n\rangle_H \end{equation*} for $u = (u_1,\cdots,u_n)$ and $v = (v_1,\cdots,v_n)$. Assume that $L:H^n\rightarrow H^n$ be a compact self-adjoint operator, we say a $n\times n$ matrix $\Lambda$ is a (matrix-valued) eigenvalue of $L$, if there exists $u\in H^n$ such that $Lu=\Lambda u$.
I am wondering that if there exists some reference that introducing such kind of eigenvalues. More precisely, is the stability theorem still hold:
Let $\Lambda$ be a eigenvalue of $L$ with $m$ multiplicity, there exists $\Lambda_1,\cdots,\Lambda_m$ of eigenvalues of $T$, such that for any $i = 1,\cdots,m$, \begin{equation*} \| \Lambda - \Lambda_i \| \leq C\|S-T\|. \end{equation*} Any help will be appreciated. Thank you.