Let $H(\lambda):D\to\mathcal H$ be an analytic family of [self-adjoint operators](/page/Self-Adjoint%20Operators) of type A near $\lambda=0$, with common dense domain $D\subset\mathcal H$. Suppose $E_0$ is an isolated eigenvalue of $H(0)$ with finite multiplicity $m$. Then, for $|\lambda|$ sufficiently small, the part of the spectrum of $H(\lambda)$ near $E_0$ consists of $m$ eigenvalues counted with multiplicity, and these eigenvalues can be parameterised by real analytic functions of $\lambda$. The associated total spectral projection is also analytic in $\lambda$.