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The Bauer-Fike theorem of eigen-value pertubation for matrices $n \times n$ states:

Let $A$ be a $n \times n$ matrix diagonalizable satisfying $A = X D X^{-1}$, where $D$ is diagonal and $n \times n$ matrix. Let $B$ be another $n \times n$ matrix, then every eigenvalue $\phi$ of the matrix $A+B$ satisfies the inequality

$ |\phi - \lambda | \leq \|X\|\cdot \|X^{-1} \|\cdot \|B\| $ where $\lambda$ is some eigenvalue of $A$, and $\|\cdot\|$ is a matrix norm.

I wonder if this theorem is still valid for self adjoint and bounded operators.

Thanks in advance.

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    Do you want to extend to infinite dimensional Banach/Hilbert spaces? In this case when you say eigenvalue do you still want to think about eigenvalues or do you want to talk about elements of the spectrum? – 2017-01-14
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    I am interested to show if the limit of quadratic form $\lim_{n \rightarrow \infty} w_n^T(A_n+E_n)w_n$ is positive, where $A_n \in [0,1]^{n \times n}$ is a diagonal matrix and $E_n \in [0,1]^{n \times n}$ a symmetric matrix and $w_n$ a vector $n \times 1$. I proved through Bauer-Fike theorem that $A_n+B_n$ has positive eigenvalues for $n > n_0$. – 2017-01-14

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